A New Robust Iterative Scheme Applied in Solving a Fractional Diffusion Model for Oxygen Delivery via a Capillary of Tissues

Abstract

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example.

1. Introduction

Let $\mathcal{B}$ be a nonempty closed convex subset of a Banach space $\mathbb{E}$. A self-mapping $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ is called a contraction if for any $\delta \in [0,1)$, $x,y\in \mathbb{E}$,

$$\parallel \mathcal{T}x-\mathcal{T}y\parallel \le \delta \parallel x-y\parallel ,$$

(1)

holds. An element ${\tau }^{*}\in \mathcal{B}$ is called a fixed point of $\mathcal{T}$ if it satisfies the operator equation

$$\mathcal{T}{\tau }^{*}={\tau }^{*}.$$

(2)

The set of all fixed point of $\mathcal{T}$ is denoted by $\mathcal{F}\left(\mathcal{T}\right)$.

A Banach space $\mathbb{E}$ is said to be uniformly convex (as introduced by Clarkson [1] in 1936) if for any $ϵ$, $0<ϵ\le 2$, the inequalities $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge ϵ$ imply that there exists a $\delta =\delta \left(ϵ\right)>0$ such that $\parallel \frac{(x+y)}{2}\parallel \le 1-\delta$.

Quite a number of physical problems are modeled as partial differential equations and ordinary differential equations in the form of initial value problems (IVPs) or boundary value problems (BVPs). Most times, these equations are in the form of nonlinear problems that are usually difficult (if not impossible) to solve through the use of analytical methods. In a way to circumvent this difficulty of obtaining the solutions to nonlinear problems through the use of the analytical method, the fixed point theory approach becomes useful by way of proving the existence and uniqueness of solutions to the problems of concern. Due to this advantage, since inception, fixed point theory has made a remarkable impact in mathematics and other areas of applied science, including its application to BVPs. For instance, the papers [2,3,4] and the references therein have been dedicated specifically to the use of fixed point iterative processes in approximating the solutions to BVPs via the use of Green’s functions. The idea involved in fixed point theory is to transform any problem of focus into a fixed point equation as in (2) which thereafter solved for ${\tau }^{*}$ being the fixed point. In practice, the fixed point represents the approximate solution which is obtained more suitably through the use of a robust fixed point iterative scheme. The fixed point theory method has been applied in solving diverse problems in science and engineering (see, e.g., [5]) and the references therein.

While an ordinary differential equation is an equation involving differential coefficients to the integer order, a fractional differential equation exists as an equation involving differential coefficients to the fractional order.

The remarkable thing about a fractional differential equation is its wide range of applications. For example, it can be applied in modeling some physical phenomena in science and engineering, such as in optics [6]; in electrochemistry; in viscoelasticity [7]; in control theory; in biology; in fluid flow; and in other fields (see, for example, [8,9] and other references therein).

In the sequel, $0<{\alpha }_n, {\beta }_n, {\gamma }_n<1$ are parametric sequences of real numbers.

The aim of this paper is to answer the following question.

Question 

Is there a fixed point iterative scheme that can converge faster than other existing schemes in the literature and solve some problems in its application?

To answer the question above, we construct the following fixed point iterative scheme and call it the UO iterative scheme:

$$\left\{\begin{array}{c}{v}_o\in \mathcal{B}\hfill \\ r_n=\mathcal{T}{v}_n\hfill \\ s_n=(1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n\hfill \\ t_n=\mathcal{T}{s}_n\hfill \\ u_n=(1-{\beta }_n)t_n+{\beta }_n\mathcal{T}{t}_n\hfill \\ v_{n+1}=(1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(3)

which, as will be shown in the subsequent sections, converges faster than some existing iterative schemes in the literature as outlined in the next section.

Our iterative scheme generalizes and extends other existing iterative schemes in the literature.

Remark 1. 

Observe that the UO iterative scheme (3) is a five-step iterative iterative scheme, which is not as simple as one-step or two-step iterative schemes such as the Mann and Ishikawa iterative schemes.

The remaining part of this paper is arranged as follows: Section 2 is dedicated to preliminary definitions and lemmas. Section 3 is for the main results which comprise the convergence results, rate of convergence, stability and data dependence. In Section 4, the application to an oxygen diffusion model is covered. Meanwhile, Section 5 is assigned to the application of our new scheme to BVPs, where the construction of the Green’s function, the UO–Green iterative scheme, convergence analysis of the UO–Green iterative and numerical example are considered. Section 6 contains the conclusion.

2. Preliminary

Research in the area of fixed point theory via use of iterative scheme has experienced a surge due to construction of varying forms of iterative scheme that have been useful in application.

In 2012, Chugh et al. [10] introduced the CR iterative as follows:

$$\left\{\begin{array}{c}{p}_0\in \mathcal{B}\hfill \\ r_n=(1-{\gamma }_n)p_n+{\gamma }_n\mathcal{T}{p}_n\hfill \\ q_n=(1-{\beta }_n)\mathcal{T}{p}_n+{\beta }_n\mathcal{T}{r}_n\hfill \\ p_{n+1}=(1-{\alpha }_n)q_n+{\alpha }_n\mathcal{T}{q}_n,n\in \mathbb{N}.\hfill \end{array}\right.$$

(4)

Another iterative scheme is Picard-S, which was introduced by Gürsoy et al. [11] in 2014 and defined thus:

$$\left\{\begin{array}{c}{z}_n=(1-{\beta }_n)x_n+{\beta }_n\mathcal{T}{x}_n\hfill \\ y_n=(1-{\alpha }_n)\mathcal{T}{x}_n+{\alpha }_n\mathcal{T}{z}_n\hfill \\ x_{n+1}=\mathcal{T}{y}_n,n\in \mathbb{N}.\hfill \end{array}\right.$$

(5)

Abbas et al. (2022) [12] constructed the following AA iteration scheme:

$$\left\{\begin{array}{c}{z}_{n+1}=\mathcal{T}{y}_n\hfill \\ y_n=\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\hfill \\ w_n=\mathcal{T}[(1-{\beta }_n)d_n+{\beta }_n\mathcal{T}{d}_n]\hfill \\ d_n=(1-{\alpha }_n)z_n+{\alpha }_n\mathcal{T}{z}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(6)

which was used to approximate the solution to a delay fractional differential equation.

Uddin et al. [13], in 2022, introduced the following iterative scheme:

$$\left\{\begin{array}{c}{g}_0\in \mathcal{B}\hfill \\ e_n=\mathcal{T}[(1-{\alpha }_n)g_n+{\alpha }_n\mathcal{T}{g}_n]\hfill \\ f_n=\mathcal{T}{e}_n\hfill \\ g_{n+1}=\mathcal{T}{f}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(7)

and they were able to show that it converges faster than Thakur New, Vatan, M and M* iterations.

The following are, respectively, the $F^{*}$, Modified-SP [14] and Picard–Ishikawa [15] iterative schemes:

$$\left\{\begin{array}{c}{p}_0\in \mathcal{B}\hfill \\ p_{n+1}=\mathcal{T}{q}_n\hfill \\ q_n=\mathcal{T}[(1-{\alpha }_n)p_n+{\alpha }_n\mathcal{T}{p}_n],n\in \mathbb{N},\hfill \end{array}\right.$$

(8)

$$\left\{\begin{array}{c}{x}_0\in \mathcal{B}\hfill \\ x_{n+1}=\mathcal{T}{y}_n\hfill \\ y_n=(1-{\alpha }_n)z_n+{\alpha }_n\mathcal{T}{z}_n\hfill \\ z_n=(1-{\beta }_n)x_n+{\beta }_n\mathcal{T}{x}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(9)

$$\left\{\begin{array}{c}{w}_n=(1-{\xi }_n)u_n+{\xi }_n\mathcal{T}{u}_n,\hfill \\ v_n=(1-{\varpi }_n)u_n+{\varpi }_n\mathcal{T}{w}_n,\hfill \\ u_{n+1}=\mathcal{T}{v}_n,n\in \mathbb{N}.\hfill \end{array}\right.$$

(10)

Recently in 2023, Okeke et al. [16] introduced the AG iterative scheme, defined thus:

$$\left\{\begin{array}{c}{u}_0=u\in \mathcal{B}\hfill \\ u_{n+1}=\mathcal{T}{v}_n\hfill \\ v_n=\mathcal{T}[(1-{\alpha }_n)w_n+{\alpha }_n\mathcal{T}{w}_n]\hfill \\ w_n=(1-{\beta }_n)\mathcal{T}{u}_n+{\beta }_n\mathcal{T}{x}_n\hfill \\ x_n=(1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(11)

which was used to approximate the fixed point of contraction mapping in a uniformly convex Banach space with applications.

Definition 1. 

Two sequences, $\left\{u_n\right\}$ and $\left\{v_n\right\}$, are said to guarantee equivalence if $\underset{n\to \infty }{lim}\parallel u_n-v_n\parallel =0$.

Definition 2 

([17]). Let $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ be an operator on a real Banach space E. Assume that $v_n\in \mathcal{B}$ and $v_{n+1}=f(\mathcal{T},v_n)$ defines an iterative scheme which generates a sequence ${\left\{v_n\right\}}_{n=0}^{\infty }$ in $\mathcal{B}$. Assume, furthermore, that ${\left\{v_n\right\}}_{n=0}^{\infty }$ converges strongly to ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)\ne \varnothing$, where $\mathcal{F}\left(\mathcal{T}\right)$ is the set of all fixed points of $\mathcal{T}$. Assume that ${\left\{p_n\right\}}_{n=0}^{\infty }$ is an arbitrary bounded sequence in E and set ${ϵ}_n=\parallel p_{n+1}-f(\mathcal{T},p_n)\parallel$. Then,

1.

The iterative scheme ${\left\{v_n\right\}}_{n=0}^{\infty }$ in a real Banach space E defined by $v_{n+1}=f(\mathcal{T},v_n)$ is said to be $\mathcal{T}$-stable if $\underset{n\to \infty }{lim}{ϵ}_n=0$ implies $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$.

2.

The iterative scheme ${\left\{v_n\right\}}_{n=0}^{\infty }$ defined by $v_{n+1}=f(\mathcal{T},v_n)$ is said to be almost $\mathcal{T}$-stable if ${\sum }_{n=0}^{\infty }{ϵ}_n<\infty$ implies that $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$.

Definition 3 

([18]). Let ${\left\{a_n\right\}}_{n=0}^{\infty }$ and ${\left\{b_n\right\}}_{n=0}^{\infty }$ be two sequences of real numbers converging to a and b, respectively. If

$$\underset{n\to \infty }{lim}\frac{\parallel a_n-a\parallel }{\parallel b_n-b\parallel }=0,$$

(12)

then $\left\{a_n\right\}$ is said to converge to a faster than $\left\{b_n\right\}$ to b.

Definition 4 

([18]). Suppose that for two fixed-point iterative processes ${\left\{u_n\right\}}_{n=0}^{\infty }$ and ${\left\{v_n\right\}}_{n=0}^{\infty }$, both converging to the same fixed point p, the error estimates

$$\parallel u_n-p\parallel \le a_n,\mathit{for}\mathit{all}n\in \mathbb{N},$$

$$\parallel v_n-p\parallel \le b_n,\mathit{for}\mathit{all}n\in \mathbb{N},$$

exist, where ${\left\{a_n\right\}}_{n=0}^{\infty }$ and ${\left\{b_n\right\}}_{n=0}^{\infty }$ are two sequences of positive numbers converging to zero. If ${\left\{a_n\right\}}_{n=0}^{\infty }$ converges faster than ${\left\{b_n\right\}}_{n=0}^{\infty }$ then ${\left\{u_n\right\}}_{n=0}^{\infty }$ converges faster than ${\left\{v_n\right\}}_{n=0}^{\infty }$ to p.

Lemma 1 

([19]). If $\rho \in [0,1)$ is a real number and ${\left\{{ϵ}_n\right\}}_{n=0}^{\infty }$ is a sequence of positive numbers such that $\underset{n\to \infty }{lim}{ϵ}_n=0$, then for any sequence of positive numbers, ${\left\{p_n\right\}}_{n=0}^{\infty }$ satisfying $p_{n+1}\le \rho p_n+{ϵ}_n,(n=0,1,2,...)$ such that $\underset{n\to \infty }{lim}{p}_n=0$.

Lemma 2 

([20]). Let ${\left\{p_n\right\}}_{n=0}^{\infty }$ and ${\left\{{ϵ}_n\right\}}_{n=0}^{\infty }$ be sequences of nonnegative numbers and $\delta \in [0,1)$ such that

$${\nu }_{n+1}=\delta {\nu }_n+{ϵ}_{n}n\ge 0.$$

If ${\sum }_{n=0}^{\infty }{ϵ}_n<\infty$, then ${\sum }_{n=0}^{\infty }{\nu }_n<\infty$.

Lemma 3 

([21]). Let $\left\{{\xi }_n\right\}$ be a nonnegative sequence for which one assumes there exists $n_0\in \mathbb{N}$ such that all $n\ge n_0$, and suppose the following inequality is satisfied:

$${\xi }_{n+1}\le (1-{\phi }_n){\xi }_n+{\phi }_n{\varrho }_n$$

where ${\phi }_n\in (0,1)$, $\forall n\in \mathbb{N}$, ${\sum }_{n=0}^{\infty }{\phi }_n=\infty$ and ${\varrho }_n\ge 0$ $\forall n\in \mathbb{N}$. Then,

$$0\le \underset{n\to \infty }{lim\; sup}{\xi }_n\le \underset{n\to \infty }{lim\; sup}{\varrho }_n.$$

Lemma 4 

([22]). Let ${\sigma }_n$ be a nonnegative sequence satisfying the inequality

$${\sigma }_{n+1}\le (1-{\eta }_n){\sigma }_n+{\lambda }_n$$

with ${\eta }_n\in [0,1]$, ${\sum }_{j=0}^{\infty }{\eta }_j=\infty$ and ${\lambda }_n=o\left({\eta }_n\right)$. Then, $\underset{n\to \infty }{lim}{\sigma }_n=0$.

3. Main Results

We begin this section by establishing some useful convergence results for our newly developed iteration process in Banach spaces.

3.1. Convergence Analysis of the UO Iterative Process

Theorem 1. 

Assume $\mathcal{B}$ is a nonempty closed convex subset of a Banach space $\mathbb{E}$ and $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ is a contraction mapping satisfying condition (1). Assume ${\left\{v_n\right\}}_{n=0}^{\infty }$ is an iterative sequence generated by the UO iterative scheme (3) with real sequences $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$, satisfying ${\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n=\infty$. Then, ${\left\{v_n\right\}}_{n=0}^{\infty }$ converges to a unique fixed point ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)$.

Proof. 

It can easily be verified that the Banach contraction principle guarantees the existence and uniqueness of ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)$. What is left is to show is that $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$.

Using the contraction condition (1) and the iterative scheme (3), we have the following estimates:

$$\begin{array}{cc}\hfill \parallel r_n-{\tau }^{*}\parallel & = \parallel \mathcal{T}{v}_n-{\tau }^{*}\parallel \hfill \\ \hfill & = \parallel \mathcal{T}{v}_n-\mathcal{T}{\tau }^{*}\parallel \hfill \\ \hfill & \le \delta \parallel v_n-{\tau }^{*}\parallel .\hfill \end{array}$$

(13)

Using (3) and (13),

$$\begin{array}{cc}\hfill \parallel s_n-{\tau }^{*}\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\tau }^{*}\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\tau }^{*}\parallel + {\alpha }_n\delta \parallel r_n-{\tau }^{*}\parallel \hfill \\ & =[(1-{\alpha }_n)+{\alpha }_n\delta ]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le \delta [1-(1-\delta ){\alpha }_n]\parallel v_n-{\tau }^{*}\parallel ,\hfill \end{array}$$

(14)

again, using (3) and (14),

$$\begin{array}{cc}\hfill \parallel t_n-{\tau }^{*}\parallel & = \parallel \mathcal{T}{s}_n-{\tau }^{*}\parallel \hfill \\ & \le \delta \parallel s_n-{\tau }^{*}\parallel \hfill \\ & \le \delta [1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-{\tau }^{*}\parallel ,\hfill \end{array}$$

(15)

furthermore, using (3) and (15),

$$\begin{array}{cc}\hfill \parallel u_n-{\tau }^{*}\parallel & = \parallel (1-{\beta }_n)u_n+{\beta }_n\mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\parallel \mathcal{T}{t}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\delta \parallel t_n-{\tau }^{*}\parallel \hfill \\ & =[(1-{\beta }_n)+{\beta }_n\delta ]\parallel t_n-{\tau }^{*}\parallel \hfill \\ & \le [1-(1-\delta ){\beta }_n]\parallel t_n-{\tau }^{*}\parallel \hfill \\ & \le {\delta }^2[1-(1-\delta ){\alpha }_n][1-(1-\delta ){\beta }_n]\parallel v_n-{\tau }^{*}\parallel .\hfill \end{array}$$

(16)

Finally,

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & = \parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\parallel \mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\delta \parallel u_n-{\tau }^{*}\parallel \hfill \\ & =[(1-{\gamma }_n)+{\gamma }_n\delta ]\parallel u_n-{\tau }^{*}\parallel \hfill \\ & \le [1-(1-\delta ){\gamma }_n]\parallel u_n-{\tau }^{*}\parallel \hfill \\ & \le {\delta }^2[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-{\tau }^{*}\parallel .\hfill \end{array}$$

(17)

Since $\delta \in [0,1)$, $[1-(1-\delta ){\beta }_n]<1$ and $[1-(1-\delta ){\gamma }_n]<1$, we have that

$$\parallel v_{n+1}-{\tau }^{*}\parallel \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-{\tau }^{*}\parallel .$$

Via induction, we have the following inequalities:

$$\begin{array}{cc}\hfill \parallel v_n-{\tau }^{*}\parallel & \le [1-{\alpha }_{n-1}(1-\delta )]\parallel v_{n-1}-{\tau }^{*}\parallel \hfill \\ \hfill \parallel v_{n-1}-{\tau }^{*}\parallel & \le {\delta }^2[1-{\alpha }_{n-2}(1-\delta )]\parallel v_{n-2}-{\tau }^{*}\parallel \hfill \\ & ⋮\hfill \\ \hfill \parallel v_1-{\tau }^{*}\parallel & \le {\delta }^2[1-{\alpha }_0(1-\delta )]\parallel v_0-{\tau }^{*}\parallel .\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & \le {\delta }^{2(n+1)}\prod _{k=0}^n[1-{\alpha }_k(1-\delta )]\parallel v_0-{\tau }^{*}\parallel \hfill \\ & ={\delta }^{2(n+1)}\parallel v_0-{\tau }^{*}\parallel {[1-(1-\delta )\alpha ]}^{n+1}.\hfill \end{array}$$

(18)

From elementary analysis, it is clear that $1-q\le e^{-q}$ for $q\in (0,1)$. Consequent upon that fact and inequality (18), we have

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & \le \prod _{k=0}^n{e}^{-{\alpha }_k(1-\delta )}\parallel v_0-{\tau }^{*}\parallel \hfill \\ & = \parallel v_0-{\tau }^{*}\parallel e^{-(1-\delta ){\sum }_{k=0}^n{\alpha }_k}.\hfill \end{array}$$

(19)

Taking the limit as $n\to \infty$ of both sides of (19), then, $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$.    □

Theorem 2. 

Suppose $\mathcal{B}$ is a nonempty closed convex subset of a Banach space $\mathbb{E}$ and $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ is a mapping satisfying condition (1) with a unique fixed point ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)$. Suppose that $\left\{v_n\right\}$ and $\left\{z_n\right\}$ are two iterative sequences generated by the UO iterative scheme and the AA iterative scheme, respectively, with real sequences $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$ satisfying ${\sum }_{k=0}^{\infty }{\alpha }_k=\infty$. Then, the following are equivalent:

1.

$\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$

2.

$\underset{n\to \infty }{lim}\parallel z_n-{\tau }^{*}\parallel =0$.

Proof. 

We start by showing that $\left(1\right)\Rightarrow \left(2\right)$; that is, if the UO iterative scheme converges to the fixed point ${\tau }^{*}$, then the $AA$ iterative also converge to the same fixed point ${\tau }^{*}$.

$$\begin{array}{cc}\hfill \parallel v_{n+1}-z_{n+1}\parallel & = \parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-\mathcal{T}{y}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{y}_n\parallel + {\gamma }_n\parallel \mathcal{T}{u}_n-\mathcal{T}{y}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{y}_n\parallel + {\gamma }_n\delta \parallel u_n-y_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta \parallel u_n-y_n\parallel .\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \parallel u_n-y_n\parallel & = \parallel (1-{\beta }_n)t_n+{\beta }_n\mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n+\mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + (1-{\beta }_n)\parallel \mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \parallel \mathcal{T}{t}_n-\mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta \parallel t_n-[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta (1-{\gamma }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta {\gamma }_n\parallel t_n-\mathcal{T}{w}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta (1-{\gamma }_n)\parallel t_n-\mathcal{T}{t}_n+\mathcal{T}{t}_n-\mathcal{T}{d}_n\parallel \hfill \\ & + \delta {\gamma }_n\parallel t_n-\mathcal{T}{t}_n+\mathcal{T}{t}_n-\mathcal{T}{w}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta (1-{\gamma }_n)\parallel t_n-\mathcal{T}{t}_n\parallel + \delta (1-{\gamma }_n)\parallel \mathcal{T}{t}_n-\mathcal{T}{d}_n\parallel \hfill \\ & + \delta {\gamma }_n\parallel t_n-\mathcal{T}{t}_n\parallel + \delta {\gamma }_n\parallel \mathcal{T}{t}_n-\mathcal{T}{w}_n\parallel \hfill \\ & \le \{(1-{\beta }_n) + \delta (1-{\gamma }_n) + \delta {\gamma }_n\}\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^2(1-{\gamma }_n)\parallel t_n-d_n\parallel \hfill \\ & + {\delta }^2{\gamma }_n\parallel t_n-w_n\parallel \hfill \\ & \le [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^2(1-{\gamma }_n)\parallel t_n-d_n\parallel + {\delta }^2{\gamma }_n\parallel t_n-w_n\parallel .\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \parallel t_n-w_n\parallel & = \parallel \mathcal{T}{s}_n-\mathcal{T}[(1-{\beta }_n)d_n+{\beta }_n\mathcal{T}{d}_n]\parallel \hfill \\ & \le \delta \parallel s_n-[(1-{\beta }_n)d_n+{\beta }_n\mathcal{T}{d}_n]\parallel \hfill \\ & \le \delta (1-{\beta }_n)\parallel s_n-d_n\parallel + \delta {\beta }_n\parallel s_n-\mathcal{T}{d}_n\parallel \hfill \\ & \le \delta (1-{\beta }_n)\parallel s_n-d_n\parallel + \delta {\beta }_n\parallel s_n-\mathcal{T}{s}_n+\mathcal{T}{s}_n-\mathcal{T}{d}_n\parallel \hfill \\ & \le \delta (1-{\beta }_n)\parallel s_n-d_n\parallel + \delta {\beta }_n\parallel s_n-\mathcal{T}{s}_n\parallel + {\delta }^2{\beta }_n\parallel s_n-d_n\parallel \hfill \\ & \le \delta {\beta }_n\parallel s_n-\mathcal{T}{s}_n\parallel + [\delta (1-{\beta }_n)+{\delta }^2{\beta }_n]\parallel s_n-d_n\parallel .\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \parallel s_n-d_n\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-(1-{\alpha }_n)z_n-{\alpha }_n\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-z_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-z_n\parallel + \delta {\alpha }_n\parallel r_n-z_n\parallel \hfill \\ & =[(1-{\alpha }_n) + \delta {\alpha }_n]\parallel r_n-z_n\parallel .\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \parallel r_n-z_n\parallel & = \parallel \mathcal{T}{v}_n-z_n\parallel \hfill \\ & \le \parallel \mathcal{T}{v}_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-z_n\parallel \hfill \\ & \le \delta \parallel v_n-z_n\parallel + \parallel z_n-\mathcal{T}{z}_n\parallel .\hfill \end{array}$$

(24)

Put (24) in (23):

$$\begin{array}{c}\hfill \parallel s_n-d_n\parallel \le \delta [(1-{\alpha }_n) + \delta {\alpha }_n]\parallel v_n-z_n\parallel + [(1-{\alpha }_n) + \delta {\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel .\end{array}$$

(25)

Put (25) in (22):

$$\begin{array}{cc}\hfill \parallel t_n-w_n\parallel & \le \delta {\beta }_n\parallel s_n-\mathcal{T}{s}_n\parallel +[\delta (1-{\beta }_n) + \delta {\beta }_n]\times \hfill \\ & \{\delta [(1-{\alpha }_n) + \delta {\alpha }_n]\parallel v_n-z_n\parallel +[(1-{\alpha }_n) + \delta {\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \parallel t_n-d_n\parallel & = \parallel \mathcal{T}{s}_n-(1-{\alpha }_n)z_n-{\alpha }_n\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel \mathcal{T}{s}_n-z_n\parallel + {\alpha }_n\parallel \mathcal{T}{s}_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel \mathcal{T}{s}_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-z_n\parallel + {\alpha }_n\delta \parallel s_n-z_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel s_n-z_n\parallel + (1-{\alpha }_n)\parallel \mathcal{T}{z}_n-z_n\parallel + {\alpha }_n\delta \parallel s_n-z_n\parallel \hfill \\ & \le \delta \parallel s_n-z_n\parallel + (1-{\alpha }_n)\parallel \mathcal{T}{z}_n-z_n\parallel \hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \parallel s_n-z_n\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-z_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-z_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-z_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-z_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-z_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-z_n\parallel + {\alpha }_n\delta \parallel r_n-z_n\parallel + {\alpha }_n\parallel \mathcal{T}{z}_n-z_n\parallel \hfill \\ & \le [(1-{\alpha }_n)+{\alpha }_n\delta ]\parallel r_n-z_n\parallel + {\alpha }_n\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le [(1-{\alpha }_n)+{\alpha }_n\delta ]\{\delta \parallel v_n-z_n\parallel + \parallel z_n-\mathcal{T}{z}_n\parallel \}+{\alpha }_n\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le \delta [(1-{\alpha }_n)+{\alpha }_n\delta ]\parallel v_n-z_n\parallel + [1+{\alpha }_n\delta ]\parallel z_n-\mathcal{T}{z}_n\parallel .\hfill \end{array}$$

(28)

Putting (28) in (27),

$$\begin{array}{cc}\hfill \parallel t_n-d_n\parallel & \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel + [\delta +{\delta }^2{\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + (1-{\alpha }_n)\parallel \mathcal{T}{z}_n-z_n\parallel ,\hfill \end{array}$$

(29)

and, putting (26) and (29) in (21),

$$\begin{array}{cc}\hfill \parallel u_n-y_n\parallel & \le [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^4(1-{\gamma }_n)[1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel \hfill \\ & + {\delta }^2(1-{\gamma }_n)[\delta +{\delta }^2{\gamma }_n]\parallel z_n-\mathcal{T}{z}_n\parallel + {\delta }^2(1-{\gamma }_n)(1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\delta }^3{\beta }_n{\gamma }_n\parallel s_n-\mathcal{T}{s}_n\parallel + {\delta }^4{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel \hfill \\ & + {\delta }^3{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel .\hfill \end{array}$$

(30)

Putting (30) in (20),

$$\begin{array}{cc}\hfill \parallel v_{n+1}-z_{n+1}\parallel & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel \hfill \\ & + {\delta }^5(1-{\gamma }_n)[1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel + {\delta }^3(1-{\gamma }_n)[\delta +{\delta }^2{\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\delta }^3(1-{\gamma }_n)(1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel + {\delta }^4{\beta }_n{\gamma }_n\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ & + {\delta }^5{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel \hfill \\ & + {\delta }^4{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^4{\beta }_n{\gamma }_n\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ & + \{{\delta }^5(1-{\gamma }_n)[1-(1-\delta ){\alpha }_n]\hfill \\ & + {\delta }^5{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel v_n-z_n\parallel \hfill \\ & + \{{\delta }^3(1-{\gamma }_n)[\delta +{\delta }^2{\alpha }_n]\hfill \\ & + {\delta }^4{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel .\hfill \end{array}$$

(31)

Since $\delta \in [0,1)$ and ${\delta }^5(1-{\gamma }_n)+{\delta }^5{\gamma }_n[1-(1-\delta ){\beta }_n]<1$, we have

$$\begin{array}{cc}\hfill \parallel v_{n+1}-z_{n+1}\parallel & \le (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^4{\beta }_n{\gamma }_n\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ & + \{{\delta }^3(1-{\gamma }_n)[\delta +{\delta }^2{\alpha }_n]+{\delta }^4{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + [1-(1-\delta ){\alpha }_n]\parallel v_n-z_n\parallel .\hfill \end{array}$$

(32)

Let

$$\begin{array}{cc}\hfill {\eta }_n& =(1-\delta ){\alpha }_n\in (0,1)\hfill \\ \hfill {\pi }_n& = \parallel v_n-z_n\parallel \hfill \\ \hfill c_n& =(1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta [1-{\beta }_n + \delta ]\parallel t_n-\mathcal{T}{t}_n\parallel + {\delta }^4{\beta }_n{\gamma }_n\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ & +\{{\delta }^3(1-{\gamma }_n)[\delta +{\delta }^2{\alpha }_n]+{\delta }^4{\gamma }_n[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel .\hfill \end{array}$$

Moreover, using $\mathcal{T}{\tau }^{*}={\tau }^{*}$ and $\parallel z_n-{\tau }^{*}\parallel \to 0$,

$$\underset{n\to \infty }{lim}\parallel z_n-\mathcal{T}{z}_n\parallel = \underset{n\to \infty }{lim}\parallel s_n-\mathcal{T}{s}_n\parallel = \underset{n\to \infty }{lim}\parallel t_n-\mathcal{T}{t}_n\parallel = \underset{n\to \infty }{lim}\parallel u_n-\mathcal{T}{u}_n\parallel =0,$$

and it follows that $\frac{c_n}{{\eta }_n}\to 0$ as $n\to \infty$.

Clearly, (32) satisfies the conditions of Lemma 4 and, hence, $\underset{n\to \infty }{lim}\parallel v_n-z_n\parallel =0$. Since

$$\parallel z_n-{\tau }^{*}\parallel = \parallel v_n-z_n\parallel +\parallel z_n-{\tau }^{*}\parallel$$

we have

$$\underset{n\to \infty }{lim}\parallel z_n-{\tau }^{*}\parallel =0.$$

Next, we show that $\left(2\right)\Rightarrow \left(1\right)$:

$$\begin{array}{cc}\hfill \parallel z_{n+1}-v_{n+1}\parallel & = \parallel \mathcal{T}{y}_n-(1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel \mathcal{T}{y}_n-u_n\parallel + {\gamma }_n\parallel \mathcal{T}{y}_n-\mathcal{T}{u}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel \mathcal{T}{y}_n-u_n\parallel + \delta {\gamma }_n\parallel y_n-u_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel \mathcal{T}{y}_n-\mathcal{T}{u}_n+\mathcal{T}{u}_n-u_n\parallel + \delta {\gamma }_n\parallel y_n-u_n\parallel \hfill \\ & \le (1-{\gamma }_n)\delta \parallel y_n-u_n\parallel + (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel + \delta {\gamma }_n\parallel y_n-u_n\parallel \hfill \\ & =\delta \parallel y_n-u_n\parallel + (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel \hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \parallel y_n-u_n\parallel & = \parallel \mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]-(1-{\beta }_n)t_n-{\beta }_n\mathcal{T}{t}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel \mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+\gamma \mathcal{T}{w}_n]-t_n\parallel \hfill \\ & + {\beta }_n\parallel \mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]-\mathcal{T}{t}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel \mathcal{T}[(1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n]-\mathcal{T}{t}_n+\mathcal{T}{t}_n-t_n\parallel \hfill \\ & + {\beta }_n\delta \parallel (1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n-t_n\parallel \hfill \\ & \le (1-{\beta }_n)\delta \parallel (1-{\gamma }_n)\mathcal{T}{d}_n+{\gamma }_n\mathcal{T}{w}_n-t_n\parallel + (1-{\beta }_n)\parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & + {\beta }_n\delta (1-{\gamma }_n)\parallel \mathcal{T}{d}_n-t_n\parallel + {\beta }_n{\gamma }_n\delta \parallel \mathcal{T}{w}_n-t_n\parallel \hfill \\ & \le (1-{\beta }_n)(1-{\gamma }_n)\delta \parallel \mathcal{T}{d}_n-t_n\parallel + (1-{\beta }_n){\gamma }_n\delta \parallel \mathcal{T}{w}_n-t_n\parallel \hfill \\ & + (1-{\beta }_n)\parallel \mathcal{T}{t}_n-t_n\parallel + {\beta }_n{\delta }^2(1-{\gamma }_n)\parallel d_n-t_n\parallel + {\beta }_n\delta (1-{\gamma }_n)\parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & + {\beta }_n{\gamma }_n{\delta }^2\parallel w_n-t_n\parallel + {\beta }_n{\gamma }_n\delta \parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & \le (1-{\beta }_n)(1-{\gamma }_n){\delta }^2\parallel d_n-t_n\parallel + (1-{\beta }_n)(1-{\gamma }_n)\delta \parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & + (1-{\beta }_n){\gamma }_n{\delta }^2\parallel w_n-t_n\parallel + (1-{\beta }_n){\gamma }_n\delta \parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & + (1-{\beta }_n)\parallel \mathcal{T}{t}_n-t_n\parallel + {\beta }_n{\delta }^2(1-{\gamma }_n)\parallel d_n-t_n\parallel + {\beta }_n\delta (1-{\gamma }_n)\parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & + {\beta }_n{\gamma }_n{\delta }^2\parallel w_n-t_n\parallel + {\beta }_n{\gamma }_n\delta \parallel \mathcal{T}{t}_n-t_n\parallel \hfill \\ & =(1-{\gamma }_n){\delta }^2\parallel d_n-t_n\parallel + {\gamma }_n{\delta }^2\parallel w_n-t_n\parallel + [\delta +(1-{\beta }_n)]\parallel \mathcal{T}{t}_n-t_n\parallel \hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \parallel w_n-t_n\parallel & = \parallel \mathcal{T}[(1-{\beta }_n)d_n+{\beta }_n\mathcal{T}{d}_n]-\mathcal{T}{s}_n\parallel \hfill \\ & \le \delta \parallel (1-{\beta }_n)d_n+{\beta }_n\mathcal{T}{d}_n-s_n\parallel \hfill \\ & \le \delta (1-{\beta }_n)\parallel d_n-s_n\parallel + {\beta }_n\delta \parallel \mathcal{T}{d}_n-s_n\parallel \hfill \\ & \le \delta (1-{\beta }_n)\parallel d_n-s_n\parallel + {\beta }_n{\delta }^2\parallel d_n-s_n\parallel + {\beta }_n\delta \parallel \mathcal{T}{s}_n-s_n\parallel \hfill \\ & =\delta [(1-{\beta }_n)+{\beta }_n\delta ]\parallel d_n-s_n\parallel + {\beta }_n\delta \parallel \mathcal{T}{s}_n-s_n\parallel \hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \parallel d_n-s_n\parallel & = \parallel (1-{\alpha }_n)z_n+{\alpha }_n\mathcal{T}{z}_n-(1-{\alpha }_n)r_n-{\alpha }_n\mathcal{T}{r}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel z_n-r_n\parallel + {\alpha }_n\parallel \mathcal{T}{z}_n-\mathcal{T}{r}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel z_n-r_n\parallel + {\alpha }_n\delta \parallel z_n-r_n\parallel \hfill \\ & =[1-(1-\delta ){\alpha }_n]\parallel z_n-r_n\parallel \hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{v}_n\parallel \hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-\mathcal{T}{v}_n\parallel \hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel + \delta [1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \hfill \end{array}$$

(36)

Putting (36) in (35),

$$\begin{array}{cc}\hfill \parallel w_n-t_n\parallel & \le \delta [1-(1-\delta ){\beta }_n]\left\{[1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel +\delta [1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \right\}\hfill \\ & + {\beta }_n\delta \parallel \mathcal{T}{s}_n-s_n\parallel \hfill \\ & \le \delta [1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\delta }^2[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel + {\beta }_n\delta \parallel \mathcal{T}{s}_n-s_n\parallel \hfill \end{array}$$

(37)

Next,

$$\begin{array}{cc}\hfill \parallel d_n-t_n\parallel & = \parallel (1-{\alpha }_n)z_n+{\alpha }_n\mathcal{T}{z}_n-\mathcal{T}{s}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel z_n-\mathcal{T}{s}_n\parallel + {\alpha }_n\parallel \mathcal{T}{z}_n-\mathcal{T}{s}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel z_n\mathcal{T}{z}_n+\mathcal{T}{z}_n-\mathcal{T}{s}_n\parallel + {\alpha }_n\delta \parallel z_n-s_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel + (1-{\alpha }_n)\delta \parallel z_n-s_n\parallel + {\alpha }_n\delta \parallel z_n-s_n\parallel \hfill \\ & =\delta \parallel z_n-s_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le \delta \parallel z_n-(1-{\alpha }_n)r_n-{\alpha }_n\mathcal{T}{r}_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-r_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{r}_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-\mathcal{T}{v}_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-\mathcal{T}{r}_n\parallel \hfill \\ & + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-\mathcal{T}{v}_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{z}_n\parallel + {\alpha }_n{\delta }^2\parallel z_n-r_n\parallel \hfill \\ & + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-\mathcal{T}{z}_n\parallel + (1-{\alpha }_n){\delta }^2\parallel z_n-v_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\alpha }_n{\delta }^2\parallel z_n-\mathcal{T}{v}_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-\mathcal{T}{z}_n\parallel + (1-{\alpha }_n){\delta }^2\parallel z_n-v_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\alpha }_n{\delta }^2\parallel z_n-\mathcal{T}{z}_n+\mathcal{T}{z}_n-\mathcal{T}{v}_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel z_n-\mathcal{T}{z}_n\parallel + (1-{\alpha }_n){\delta }^2\parallel z_n-v_n\parallel + {\alpha }_n\delta \parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\alpha }_n{\delta }^2\parallel z_n-\mathcal{T}{z}_n\parallel + {\alpha }_n{\delta }^3\parallel z_n-v_n\parallel + (1-{\alpha }_n)\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & =[(1-{\alpha }_n)\delta +{\alpha }_n\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + [(1-{\alpha }_n){\delta }^2+{\alpha }_n{\delta }^3]\parallel z_n-v_n\parallel \hfill \\ & \le [\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\parallel z_n-\mathcal{T}{z}_n\parallel + {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \hfill \end{array}$$

(38)

Put (37) and (38) in (34):

$$\begin{array}{cc}\hfill \parallel y_n-u_n\parallel & \le (1-{\gamma }_n){\delta }^2\left\{[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\parallel z_n-\mathcal{T}{z}_n\parallel +{\delta }^2[1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \right\}\hfill \\ \hfill & + {\gamma }_n{\delta }^2\{\delta [1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ \hfill & + {\delta }^2[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel + {\beta }_n\delta \parallel s_n-\mathcal{T}{s}_n\parallel \}\hfill \\ \hfill & + [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel \hfill \\ \hfill & \le (1-\gamma ){\delta }^2[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\parallel z_n-\mathcal{T}{z}_n\parallel + (1-{\gamma }_n){\delta }^4[1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \hfill \\ \hfill & + {\gamma }_n{\delta }^3[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ \hfill & + {\gamma }_n{\delta }^4[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel + {\beta }_n{\gamma }_n{\delta }^3\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ \hfill & + [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel \hfill \\ \hfill & =\{(1-{\gamma }_n){\delta }^2[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]+{\gamma }_n{\delta }^3[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ \hfill & + \{(1-{\gamma }_n){\delta }^4[1-(1-\delta ){\alpha }_n]+{\gamma }_n{\delta }^4[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-v_n\parallel \hfill \\ \hfill & + {\beta }_n{\gamma }_n{\delta }^3\parallel s_n-\mathcal{T}{s}_n\parallel + [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel \hfill \end{array}$$

(39)

and, putting (39) in (33), we have

$$\begin{array}{cc}\hfill \parallel z_{n+1}-v_{n+1}\parallel & \le \{(1-{\gamma }_n){\delta }^3[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\hfill \\ & + {\gamma }_n{\delta }^4[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + \left\{(1-{\gamma }_n){\delta }^5[1-(1-\delta ){\alpha }_n]+{\gamma }_n{\delta }^5[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\right\}\parallel z_n-v_n\parallel \hfill \\ & + {\beta }_n{\gamma }_n{\delta }^4\parallel s_n-\mathcal{T}{s}_n\parallel + \delta [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel + (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel \hfill \\ & \le \{(1-{\gamma }_n){\delta }^3[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]\hfill \\ & + {\gamma }_n{\delta }^4[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel \hfill \\ & + {\beta }_n{\gamma }_n{\delta }^4\parallel s_n-\mathcal{T}{s}_n\parallel + \delta [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel + (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel \hfill \\ & + \{(1-{\gamma }_n){\delta }^6+{\gamma }_n{\delta }^5[1-(1-\delta ){\beta }_n]\}[1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \hfill \end{array}$$

(40)

Let

$$\begin{array}{cc}\hfill {\eta }_n& =(1-\delta ){\alpha }_n\in (0,1)\hfill \\ \hfill {\pi }_n& = \parallel z_n-v_n\parallel \hfill \\ \hfill c_n& =\{(1-{\gamma }_n){\delta }^3[\delta +{\alpha }_n{\delta }^2+(1-{\alpha }_n)]+{\gamma }_n{\delta }^4[1-(1-\delta ){\beta }_n]\times \hfill \\ & [1-(1-\delta ){\alpha }_n]\}\parallel z_n-\mathcal{T}{z}_n\parallel + {\beta }_n{\gamma }_n{\delta }^4\parallel s_n-\mathcal{T}{s}_n\parallel \hfill \\ & + \delta [\delta +(1-{\beta }_n)]\parallel t_n-\mathcal{T}{t}_n\parallel + (1-{\gamma }_n)\parallel u_n-\mathcal{T}{u}_n\parallel \hfill \\ & + \{(1-{\gamma }_n){\delta }^5+{\gamma }_n{\delta }^5[1-(1-\delta ){\beta }_n]\}[1-(1-\delta ){\alpha }_n]\parallel z_n-v_n\parallel \hfill \end{array}$$

Using $\mathcal{T}{\tau }^{*}={\tau }^{*}$ and $\parallel v_n-{\tau }^{*}\parallel \to 0$,

$$\underset{n\to \infty }{lim}\parallel z_n-\mathcal{T}{z}_n\parallel =\underset{n\to \infty }{lim}\parallel s_n-\mathcal{T}{s}_n\parallel =\underset{n\to \infty }{lim}\parallel t_n-\mathcal{T}{t}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-\mathcal{T}{u}_n\parallel =0$$

and it follows that $\frac{c_n}{{\eta }_n}\to 0$ as $n\to \infty$.

Hence, (40) satisfies the assumption of Lemma 4 and, as such, we have $\underset{n\to \infty }{lim}\parallel z_n-v_n\parallel = 0$ since $\parallel v_n-{\tau }^{*}\parallel = \parallel z_n-v_n\parallel + \parallel z_n-{\tau }^{*}\parallel \to 0$ as $n\to \infty$.

Therefore, $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$, thereby completing the proof.    □

3.2. Rate of Convergence of Some Iteration Processes

Theorem 3. 

Let $\mathcal{B}$ be a nonempty closed convex subset of a Banach space $\mathbb{E}$ and $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ be a contraction mapping satisfying condition (1) and having the fixed point ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)\ne \varnothing$. Assume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$ are real sequences for $n\in \mathbb{N}$. Given that $v_0=z_0\in \mathcal{B}$, consider the iterative sequences ${\left\{v_n\right\}}_{n=0}^{\infty }$, ${\left\{p_n\right\}}_{n=0}^{\infty }$ and ${\left\{x_n\right\}}_{n=0}^{\infty }$ defined by the UO (3), CR (4) and Picard-S (5) iterative schemes, respectively. Then, the iterative sequence $\left\{v_n\right\}$ converges faster to the fixed point ${\tau }^{*}$ than $\left\{p_n\right\}$.

Proof. 

From Theorem 1, we have that

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & \le {\delta }^{2(n+1)}\prod _{k=0}^n[1-(1-\delta ){\alpha }_k]\parallel v_0-{\tau }^{*}\parallel \hfill \\ & ={\delta }^{2(n+1)}\parallel v_0-{\tau }^{*}\parallel {[1-(1-\delta )\alpha ]}^{n+1}\hfill \end{array}$$

From the CR iterative scheme (4),

$$\begin{array}{cc}\hfill \parallel r_n-{\tau }^{*}\parallel & = \parallel (1-{\gamma }_n)p_n+{\gamma }_n\mathcal{T}{p}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel p_n-{\tau }^{*}\parallel + {\gamma }_n\parallel \mathcal{T}{p}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel p_n-{\tau }^{*}\parallel + {\gamma }_n\delta \parallel p_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\gamma }_n]\parallel p_n-{\tau }^{*}\parallel \hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \parallel q_n-{\tau }^{*}\parallel & = \parallel (1-{\beta }_n)\mathcal{T}{p}_n+{\beta }_n\mathcal{T}{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel \mathcal{T}{p}_n-{\tau }^{*}\parallel + {\beta }_n\parallel \mathcal{T}{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n\delta )\parallel p_n-{\tau }^{*}\parallel + {\beta }_n\delta \parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\delta \parallel p_n-{\tau }^{*}\parallel + {\beta }_n\delta [1-(1-\delta ){\gamma }_n]\parallel p_n-{\tau }^{*}\parallel \hfill \\ & =\{(1-{\beta }_n)\delta +{\beta }_n\delta [1-(1-\delta ){\gamma }_n]\}\parallel p_n-{\tau }^{*}\parallel \hfill \\ & \le \delta [1-(1-\delta ){\beta }_n{\gamma }_n]\parallel p_n-{\tau }^{*}\parallel \hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \parallel p_{n+1}-{\tau }^{*}\parallel & = \parallel (1-{\alpha }_n)q_n+{\alpha }_n\mathcal{T}{q}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel q_n-{\tau }^{*}\parallel + {\alpha }_n\parallel \mathcal{T}{q}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-\alpha )\parallel q_n-{\tau }^{*}\parallel + {\alpha }_n\delta \parallel q_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\alpha }_n]\parallel q_n-{\tau }^{*}\parallel \hfill \end{array}$$

(43)

Putting (42) in (43), we have

$$\parallel p_{n+1}-{\tau }^{*}\parallel \le \delta [1-(1-\delta ){\alpha }_n][1-(1-\delta ){\beta }_n{\gamma }_n]\parallel p_n-{\tau }^{*}\parallel$$

Since $\delta \in (0,1)$ and ${\alpha }_n, {\beta }_n, {\gamma }_n\in (0,1)$,

$$\parallel p_{n+1}-{\tau }^{*}\parallel \le \delta [1-(1-\delta ){\alpha }_n]\parallel p_n-{\tau }^{*}\parallel .$$

Via induction, we have

$$\begin{array}{cc}\hfill \parallel p_{n+1}-{\tau }^{*}\parallel & \le {\delta }^{(n+1)}\prod _{k=0}^n[1-{\alpha }_k(1-\delta )]\parallel p_0-{\tau }^{*}\parallel \hfill \\ & ={\delta }^{(n+1)}\parallel p_0-{\tau }^{*}\parallel {[1-\alpha (1-\delta )]}^{(n+1)}\hfill \end{array}$$

From the Picard-S iteration method (5) and the contraction condition (1), we have

$$\begin{array}{cc}\hfill \parallel z_n-{\tau }^{*}\parallel & = \parallel (1-{\beta }_n)x_n+{\beta }_n\mathcal{T}{x}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel x_n-{\tau }^{*}\parallel + {\beta }_n\parallel \mathcal{T}{x}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel x_n-{\tau }^{*}\parallel + {\beta }_n\delta \parallel x_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\beta }_n]\parallel x_n-{\tau }^{*}\parallel \hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \parallel y_n-{\tau }^{*}\parallel & = \parallel (1-{\alpha }_n)\mathcal{T}{x}_n+{\alpha }_n\mathcal{T}{z}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel \mathcal{T}{x}_n-{\tau }^{*}\parallel + {\alpha }_n\parallel \mathcal{T}{z}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-\alpha )\delta \parallel x_n-{\tau }^{*}\parallel + {\alpha }_n\delta \parallel z_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\delta \parallel x_n-{\tau }^{*}\parallel + {\alpha }_n\delta \{[1-(1-\delta ){\beta }_n]\parallel x_n-{\tau }^{*}\parallel \}\hfill \\ & =[(1-\alpha )\delta +{\alpha }_n\delta [1-(1-\delta ){\beta }_n]]\parallel x_n-{\tau }^{*}\parallel \hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \parallel x_{n+1}-{\tau }^{*}\parallel & = \parallel \mathcal{T}{y}_n-{\tau }^{*}\parallel \hfill \\ & \le \delta \parallel y_n-{\tau }^{*}\parallel \hfill \end{array}$$

(46)

Putting (45) in (46), we have

$$\parallel x_{n+1}-{\tau }^{*}\parallel \le \delta [(1-{\alpha }_n)\delta +{\alpha }_n\delta [1-(1-\delta ){\beta }_n]]\parallel x_n-{\tau }^{*}\parallel .$$

Since $\delta \in [0,1)$ and ${\alpha }_n, {\beta }_n\in (0,1)$, we have

$$[(1-{\alpha }_n)\delta +{\alpha }_n\delta [1-(1-\delta ){\beta }_n]]<1$$

and it follows that

$$\begin{array}{cc}\hfill \parallel x_{n+1}-{\tau }^{*}\parallel & \le \delta \parallel x_n-{\tau }^{*}\parallel \hfill \\ & ⋮\hfill \\ & \le {\delta }^{n+1}\parallel x_n-{\tau }^{*}\parallel \hfill \\ & \le {\delta }^{n+1}\parallel x_o-{\tau }^{*}\parallel .\hfill \end{array}$$

Let

$$a_n={\delta }^{2(n+1)}{[1-(1-\delta )\alpha ]}^{n+1}\parallel v_0-{\tau }^{*}\parallel$$

$$b_n={\delta }^{(n+1)}\parallel p_0-{\tau }^{*}\parallel {[1-\alpha (1-\delta )]}^{(n+1)}$$

and

$$c_n={\delta }^{n+1}\parallel x_o-{\tau }^{*}\parallel$$

Set

$$\frac{a_n}{b_n}=\frac{{\delta }^{2(n+1)}{[1-(1-\delta )\alpha ]}^{n+1}\parallel v_0-{\tau }^{*}\parallel }{{\delta }^{(n+1)}\parallel p_0-{\tau }^{*}\parallel {[1-\alpha (1-\delta )]}^{(n+1)}}\to 0\mathrm{as}n\to \infty$$

and

$$\frac{a_n}{c_n}=\frac{{\delta }^{2(n+1)}{[1-(1-\delta )\alpha ]}^{n+1}\parallel v_0-{\tau }^{*}\parallel }{{\delta }^{n+1}\parallel x_o-{\tau }^{*}\parallel }\to 0\mathrm{as}n\to \infty .$$

Hence, the UO iterative scheme (3) converges to ${\tau }^{*}$ faster than the CR and Picard-S iterative schemes. Therefore, the proof is complete.    □

Example 1. 

Let $\mathcal{B}=\mathbb{R}$. We define a mapping $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ by $\mathcal{T}x=\frac{3x+2}{4}$ which is a contraction mapping with the contraction constant $\delta =\frac{3}{4}$ and $\mathcal{F}\left(\mathcal{T}\right)=\left\{2\right\}$. If we choose ${\alpha }_n={\beta }_n={\gamma }_n=\frac{3}{4}$, then it is clear from

Table 1. Comparison of speed of convergence of some iterative schemes for Example 1.

Step	UO	Picard-S	F*	Picard–Ishikawa
1	2.5000000000	2.5000000000	2.5000000000	2.5000000000
2	2.1508560181	2.2416992188	2.2285156250	2.2651367188
3	2.0455150764	2.1168370247	2.1044387817	2.1405949593
4	2.0137324464	2.0564788352	2.0477317870	2.0745537723
5	2.0041432444	2.0273017807	2.0218149183	2.0395338851
6	2.0012500667	2.0131976381	2.0099700994	2.0209637692
7	2.0003771602	2.0063797176	2.0045566470	2.0111165299
8	2.0001137938	2.0030839455	2.0020825301	2.0058948005
9	2.0000343329	2.0014907745	2.0009517813	2.0031258561
10	2.0000103587	2.0007206380	2.0004349938	2.0016575585
11	2.0000031253	2.0003483553	2.0001988058	2.0008789592
12	2.0000009430	2.0001683944	2.0000908604	2.0004660887
13	2.0000002845	2.0000814016	2.0000415261	2.0002471545
14	2.0000000858	2.0000393494	2.0000189787	2.0001310595
15	2.0000000259	2.0000190214	2.0000086739	2.0000694973
16	2.0000000078	2.0000091949	2.0000039642	2.0000368526
17	2.0000000024	2.0000044448	2.0000018118	2.0000195420
18	2.0000000007	2.0000021486	2.0000008280	2.0000103626
19	2.0000000002	2.0000010386	2.0000003784	2.0000054950
20	2.0000000001	2.0000005021	2.0000001730	2.0000029139
21	2.0000000000	2.0000002427	2.0000000790	2.0000015451
22	2.0000000000	2.0000001173	2.0000000361	2.0000008193
23	2.0000000000	2.0000000567	2.0000000165	2.0000004345
24	2.0000000000	2.0000000274	2.0000000075	2.0000002304
25	2.0000000000	2.0000000133	2.0000000034	2.0000001222

and

Table 2. Comparison of speed of convergence of some iterative schemes for Example 1.

Step	UO	CR	Uddin et al.	Modified SP
1	2.5000000000	2.5000000000	2.5000000000	2.5000000000
2	2.1508560181	2.2618408203	2.1713867188	2.2475585938
3	2.0455150764	2.1371212304	2.0587468147	2.1225705147
4	2.0137324464	2.0718078709	2.0201368476	2.0606867685
5	2.0041432444	2.0376044636	2.0069023765	2.0300470621
6	2.0012500667	2.0196927672	2.0023659513	2.0148768169
7	2.0003771602	2.0103127406	2.0008109853	2.0073657677
8	2.0001137938	2.0054005929	2.0002779842	2.0036469182
9	2.0000343329	2.0028281914	2.0000952856	2.0018056519
10	2.0000103587	2.0014810719	2.0000326614	2.0008940093
11	2.0000031253	2.0007756102	2.0000111955	2.0004426394
12	2.0000009430	2.0004061728	2.0000038375	2.0002191584
13	2.0000002845	2.0002127052	2.0000013154	2.0001085091
14	2.0000000858	2.0001113898	2.0000004509	2.0000537247
15	2.0000000259	2.0000583328	2.0000001546	2.0000266000
16	2.0000000078	2.0000305478	2.0000000530	2.0000131701
17	2.0000000024	2.0000159973	2.0000000182	2.0000065208
18	2.0000000007	2.0000083775	2.0000000062	2.0000032285
19	2.0000000002	2.0000043871	2.0000000021	2.0000015985
20	2.0000000001	2.0000022975	2.0000000007	2.0000007914
21	2.0000000000	2.0000012031	2.0000000003	2.0000003919
22	2.0000000000	2.0000006301	2.0000000001	2.0000001940
23	2.0000000000	2.0000003300	2.0000000000	2.0000000961
24	2.0000000000	2.0000001728	2.0000000000	2.0000000476
25	2.0000000000	2.0000000905	2.0000000000	2.0000000235

and Figure 1 and Figure 2 that our iterative scheme converges to the fixed point, 2, faster than all of the CR [10] as in (4), F* [14] as in (8), Picard-S [11] as in (5), Modified-SP [14] as in (9), Uddin et al. [13] as in (7) and Picard–Ishikawa [15] as in (10) methods.

3.3. Stability and Data Dependence Results

Theorem 4. 

Let $\mathbb{E}$ be a Banach space. Assume that $\mathcal{T}:\mathcal{B}\to \mathcal{B}$ is a contraction mapping with $\delta \in [0,1)$ with a fixed point ${\tau }^{*}\in \mathcal{F}\left(\mathcal{T}\right)\ne \varnothing$. Assume further that ${\left\{v_n\right\}}_{n=0}^{\infty }$ is a sequence generated by the UO iterative scheme (3) and that it converges to ${\tau }^{*}$. Then, (3) is $\mathcal{T}$-stable.

Proof. 

Assume that ${\left\{p_n\right\}}_{n=0}^{\infty }$ is an arbitrary sequence in $\mathcal{B}$ and let the sequence generated by the UO iterative scheme be $v_{n+1}=f(\mathcal{T},v_n)$, which converges to a unique fixed point ${\tau }^{*}$.

Let ${ϵ}_n=\parallel p_{n+1}-f(\mathcal{T},p_n)\parallel$. Our aim is to show that $\underset{n\to \infty }{lim}{ϵ}_n=0$ if and only if $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$. Set $r_n=\mathcal{T}{p}_n$.

Suppose $\underset{n\to \infty }{lim}{ϵ}_n=0$:

$$\begin{array}{cc}\hfill \parallel p_{n+1}-{\tau }^{*}\parallel & = \parallel p_{n+1}-f(\mathcal{T},p_n)+f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-f(\mathcal{T},p_n)\parallel + \parallel f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\parallel f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+(1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\delta \parallel u_n-{\tau }^{*}\parallel \hfill \\ & ={ϵ}_n+[1-(1-\delta ){\gamma }_n]\parallel u_n-{\tau }^{*}\parallel \hfill \end{array}$$

(47)

Next,

$$\begin{array}{cc}\hfill \parallel u_n-{\tau }^{*}\parallel & = \parallel (1-{\beta }_n)y_n+{\beta }_n\mathcal{T}{t}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\parallel \mathcal{T}{t}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\delta \parallel t_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\beta }_n]\parallel t_n={\tau }^{*}\parallel \hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \parallel t_n-{\tau }^{*}\parallel & = \parallel \mathcal{T}{s}_n-{\tau }^{*}\parallel \hfill \\ & \le \delta \parallel s_n-{\tau }^{*}\parallel \hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \parallel s_n-{\tau }^{*}\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\tau }^{*}\parallel + {\alpha }_n\delta \parallel r_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \end{array}$$

(50)

and, combining (49) and (50), we have

$$\begin{array}{cc}\hfill \parallel t_n-{\tau }^{*}\parallel & \le \delta [1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le \delta [1-(1-\delta ){\alpha }_n]\parallel \mathcal{T}{p}_n-{\tau }^{*}\parallel \hfill \\ & \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel p_n-{\tau }^{*}\parallel \hfill \end{array}$$

(51)

Putting (51) in (48), we have

$$\parallel u_n-{\tau }^{*}\parallel \le {\delta }^2[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel p_n-{\tau }^{*}\parallel$$

(52)

Again, putting (52) in (47), we have

$$\parallel p_{n+1}-{\tau }^{*}\parallel \le {ϵ}_n+{\delta }^2[1-(1-\delta ){\alpha }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\gamma }_n]\parallel p_n-{\tau }^{*}\parallel$$

Since $\delta \in [0,1)$, $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$, from Lemma 1, we have that $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$.

Conversely, suppose $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$; then,

$$\begin{array}{cc}\hfill {ϵ}_n& = \parallel p_{n+1}-f(\mathcal{T},p_n)\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}+{\tau }^{*}-f(\mathcal{T},p_n)\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + \parallel {\tau }^{*}-f(\mathcal{T},p_n)\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + \parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + (1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\delta \parallel u_n-{\tau }^{*}\parallel \hfill \\ & = \parallel p_{n+1}-{\tau }^{*}\parallel + [1-(1-\delta ){\gamma }_n]\parallel u_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]\parallel t_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + \delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]\parallel s_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + \delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel p_{n+1}-{\tau }^{*}\parallel + {\delta }^2[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel p_n-{\tau }^{*}\parallel .\hfill \end{array}$$

Take the limit as $n\to \infty$ on both sides and note that $\underset{n\to \infty }{lim}{ϵ}_n=0$.

Hence, our new fixed point iterative scheme is $\mathcal{T}$-stable.    □

Next, we show the near-$\mathcal{T}$-stability of our new iterative scheme.

Theorem 5. 

Let $\mathbb{E}$, $\mathcal{B}$ and $\mathcal{T}$ remain the same as in Theorem 4, with $\mathcal{T}$ satisfying (1) for $\mathcal{F}\left(\mathcal{T}\right)\ne \varnothing$. The iterative scheme is almost $\mathcal{T}$-stable.

Proof. 

Let $\left\{p_n\right\}$ be an approximate sequence of $\left\{v_n\right\}$ in $\mathcal{B}$. Suppose that our new iterative scheme (3) is represented as $v_{n+1}=f(\mathcal{T},v_n)$, which converges to a fixed point ${\tau }^{*}$, and ${ϵ}_n=\parallel p_{n+1}-f(\mathcal{T},p_n)\parallel$, $n\in \mathbb{N}$. It is our aim to show that ${\sum }_{n=0}^{\infty }{ϵ}_n<\infty$ implies that $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$.

Let ${\sum }_{n=0}^{\infty }{ϵ}_n<\infty$; then, via (3), we have

$$\begin{array}{cc}\hfill \parallel p_{n+1}-{\tau }^{*}\parallel & = \parallel p_{n+1}-f(\mathcal{T},p_n)+f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \parallel p_{n+1}-f(\mathcal{T},p_n)\parallel + \parallel f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\parallel f(\mathcal{T},p_n)-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+(1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\delta \parallel u_n-{\tau }^{*}\parallel \hfill \\ & ={ϵ}_n+[1-(1-\delta ){\gamma }_n]\parallel u_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]\parallel t_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]\parallel s_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+\delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel r_n-{\tau }^{*}\parallel \hfill \\ & \le {ϵ}_n+{\delta }^2[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel p_n-{\tau }^{*}\parallel \hfill \end{array}$$

Set ${\nu }_n=\parallel p_n-{\tau }^{*}\parallel$ and, since $\delta \in [0,1)$, $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$, $[1-(1-\delta ){\alpha }_n]$$[1-(1-\delta ){\beta }_n][1-(1-\delta ){\gamma }_n]<1$,

$${\nu }_{n+1}\le {\delta }^2{\nu }_n+{ϵ}_n$$

Again, since ${\sum }_{n=0}^{\infty }{ϵ}_n<\infty$, then, via Lemma 2, we have ${\sum }_{n=0}^{\infty }{\nu }_n<\infty$. It implies that $\underset{n\to \infty }{lim}{\nu }_n=0$ (that is, $\underset{n\to \infty }{lim}\parallel p_n-{\tau }^{*}\parallel =0$).    □

Remark 2 

([23]). An iterative scheme $\left\{v_n\right\}$ which is $\mathcal{T}$-stable is also almost $\mathcal{T}$. However, the converse is not true.

Theorem 6. 

Suppose $\mathcal{G}$ is an approximate operator of a contraction operator $\mathcal{T}$. Let ${\left\{v_n\right\}}_{n=0}^{\infty }$ be a sequence generated by the UO iterative scheme (3) for $\mathcal{T}$. Let the approximate scheme of the sequence ${\left\{{\overline{v}}_n\right\}}_{n=0}^{\infty }$ be

$$\left\{\begin{array}{c}{\overline{v}}_0=v\in \mathcal{B}\hfill \\ {\overline{r}}_n=\mathcal{G}{\overline{v}}_n\hfill \\ {\overline{s}}_n=(1-{\alpha }_n){\overline{r}}_n+{\alpha }_n\mathcal{G}{\overline{r}}_n\hfill \\ {\overline{t}}_n=\mathcal{G}{\overline{s}}_n\hfill \\ {\overline{u}}_n=(1-{\beta }_n){\overline{t}}_n+{\beta }_n\mathcal{G}{\overline{t}}_n\hfill \\ {\overline{v}}_{n+1}=(1-{\gamma }_n){\overline{u}}_n+{\gamma }_n\mathcal{G}{\overline{u}}_n,n\in \mathbb{N}\hfill \end{array}\right.$$

(53)

where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}$ are real parametric sequences in $(0,1)$ satisfying the condition; $\frac{1}{2}\le {\alpha }_n$, $\forall n\in \mathbb{N}$. If $\mathcal{T}{\tau }^{*}={\tau }^{*}$ and $\mathcal{G}{s}^{*}=s^{*}$ such that $\underset{n\to \infty }{lim}\parallel {\overline{v}}_n-s^{*}\parallel =0$, then, for $0\le \delta <1$, $\parallel {\tau }^{*}-s^{*}\parallel \le \frac{9ϵ}{1-\delta }$, where $ϵ>0$ is a fixed constant.

Proof. 

Using (3) and (53),

$$\begin{array}{cc}\hfill \parallel r_n-{\overline{r}}_n\parallel & = \parallel \mathcal{T}{v}_n-\mathcal{G}{\overline{v}}_n\parallel \hfill \\ & \le \parallel \mathcal{T}{v}_n-\mathcal{T}{\overline{v}}_n+\mathcal{T}{\overline{v}}_n-\mathcal{G}{\overline{v}}_n\parallel \hfill \\ & \le \parallel \mathcal{T}{v}_n-\mathcal{T}{\overline{v}}_n\parallel + \parallel \mathcal{T}{\overline{v}}_n-\mathcal{G}{\overline{v}}_n\parallel \hfill \\ & \le \parallel \mathcal{T}{v}_n-\mathcal{T}{\overline{v}}_n\parallel +ϵ\hfill \\ & \le \delta \parallel v_n-{\overline{v}}_n\parallel +ϵ\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \parallel s_n-{\overline{s}}_n\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-(1-{\alpha }_n){\overline{r}}_n-{\alpha }_n\mathcal{G}{\overline{r}}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\overline{r}}_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{G}{\overline{r}}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\overline{r}}_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{T}{\overline{r}}_n+\mathcal{T}{\overline{r}}_n-\mathcal{G}{\overline{r}}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\overline{r}}_n\parallel + {\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{T}{\overline{r}}_n\parallel + {\alpha }_n\parallel \mathcal{T}{\overline{r}}_n-\mathcal{G}{\overline{r}}_n\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\overline{r}}_n\parallel + {\alpha }_n\delta \parallel r_n-{\overline{r}}_n\parallel +{\alpha }_nϵ\hfill \\ & =[1-(1-\delta ){\alpha }_n]\parallel r_n-{\overline{r}}_n\parallel +{\alpha }_nϵ\hfill \end{array}$$

(55)

Putting (54) in (55),

$$\parallel s_n-{\overline{s}}_n\parallel \le \delta [1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +[1-(1-\delta ){\alpha }_n]ϵ+{\alpha }_nϵ$$

Again, from (3) and (53),

$$\begin{array}{cc}\hfill \parallel t_n-{\overline{t}}_n\parallel & = \parallel \mathcal{T}{s}_n-\mathcal{G}{\overline{s}}_n\parallel \hfill \\ & \le \parallel \mathcal{T}{s}_n-\mathcal{T}{\overline{s}}_n+\mathcal{T}{\overline{s}}_n-\mathcal{G}{\overline{s}}_n\parallel \hfill \\ & \le \parallel \mathcal{T}{s}_n-\mathcal{T}{\overline{s}}_n\parallel + \parallel \mathcal{T}{\overline{s}}_n-\mathcal{G}{\overline{s}}_n\parallel \hfill \\ & \le \delta \parallel s_n-{\overline{s}}_n\parallel +ϵ\hfill \\ & \le \delta \{\delta [1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +[1-(1-\delta ){\alpha }_n]ϵ+{\alpha }_nϵ\}+ϵ\hfill \\ & \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +\delta [1-(1-\delta ){\alpha }_n]ϵ+\delta {\alpha }_nϵ+ϵ\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill \parallel u_n-{\overline{u}}_n\parallel & = \parallel (1-{\beta }_n)t_n+{\beta }_n\mathcal{T}{t}_n-(1-{\beta }_n){\overline{t}}_n-{\beta }_n\mathcal{G}{\overline{t}}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\overline{t}}_n\parallel + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{G}{\overline{t}}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\overline{t}}_n\parallel + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}{\overline{t}}_n+\mathcal{T}{\overline{t}}_n-\mathcal{G}{\overline{t}}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\overline{t}}_n\parallel + {\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}{\overline{t}}_n\parallel + {\beta }_n\parallel \mathcal{T}{\overline{t}}_n-\mathcal{G}{\overline{t}}_n\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\overline{t}}_n\parallel + {\beta }_n\delta \parallel t_n-{\overline{t}}_n\parallel +\beta ϵ\hfill \\ & =[1-(1-\delta ){\beta }_n]\parallel t_n-{\overline{t}}_n\parallel +{\beta }_nϵ\hfill \end{array}$$

(57)

Putting (56) in (57),

$$\begin{array}{cc}\hfill \parallel u_n-{\overline{u}}_n\parallel & \le [1-(1-\delta ){\beta }_n]\{{\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +\delta [1-(1-\delta ){\alpha }_n]ϵ\hfill \\ & + \delta {\alpha }_nϵ+ϵ\} + {\beta }_nϵ\hfill \\ & \le {\delta }^2[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +\delta [1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]ϵ\hfill \\ & + \delta [1-(1-\delta ){\beta }_n]{\alpha }_nϵ+[1-(1-\delta ){\beta }_n]ϵ+{\beta }_nϵ\hfill \end{array}$$

(58)

Next,

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\overline{v}}_{n+1}\parallel & = \parallel (1-{\gamma }_n)u_n+{\gamma }_n\mathcal{T}{u}_n-(1-{\gamma }_n){\overline{u}}_n-{\gamma }_n\mathcal{G}{\overline{u}}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\overline{u}}_n\parallel + {\gamma }_n\parallel \mathcal{T}{u}_n-\mathcal{G}{\overline{u}}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\overline{u}}_n\parallel + {\gamma }_n\parallel \mathcal{T}{u}_n-\mathcal{T}{\overline{u}}_n+\mathcal{T}{\overline{u}}_n-\mathcal{G}{\overline{u}}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\overline{u}}_n\parallel + {\gamma }_n\parallel \mathcal{T}{u}_n-\mathcal{T}{\overline{u}}_n\parallel + {\gamma }_n\parallel \mathcal{T}{\overline{u}}_n-\mathcal{G}{\overline{u}}_n\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\overline{u}}_n\parallel + {\gamma }_n\delta \parallel u_n-{\overline{u}}_n\parallel +{\gamma }_nϵ\hfill \\ & \le [1-(1-\delta ){\gamma }_n]\parallel u_n-{\overline{u}}_n\parallel +{\gamma }_nϵ\hfill \end{array}$$

(59)

Putting (58) in (59),

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\overline{v}}_{n+1}\parallel & \le [1-(1-\delta ){\gamma }_n]\{{\delta }^2[1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel \hfill \\ & + \delta [1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]ϵ+\delta [1-(1-\delta ){\beta }_n]{\alpha }_nϵ\hfill \\ & + [1-(1-\delta ){\beta }_n]ϵ+{\beta }_nϵ\}+{\gamma }_nϵ\hfill \\ & \le {\delta }^2[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel \hfill \\ & + \delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]ϵ\hfill \\ & + \delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]{\alpha }_nϵ+[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]ϵ\hfill \\ & + [1-(1-\delta ){\gamma }_n]{\beta }_nϵ+{\gamma }_nϵ\hfill \end{array}$$

Since $\delta \in [0,1)$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$, $n\in \mathbb{N}$, ${\delta }^2[1-(1-\delta ){\gamma }_n]$$[1-(1-\delta ){\beta }_n]<1$, $\delta [1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\alpha }_n]<1$, $\delta [1-(1-\delta ){\gamma }_n]$$[1-(1-\delta ){\beta }_n]<1$, $[1-(1-\delta ){\gamma }_n][1-(1-\delta ){\beta }_n]<1$, $[1-(1-\delta ){\gamma }_n]{\beta }_n<1$, ${\gamma }_n<1$ and $1-{\alpha }_n\le {\alpha }_n$, so that

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\overline{v}}_{n+1}\parallel & \le [1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +{\alpha }_nϵ+4ϵ\hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +{\alpha }_nϵ+4(1-{\alpha }_n+{\alpha }_n)ϵ\hfill \\ & \le [1-(1-\delta ){\alpha }_n]\parallel v_n-{\overline{v}}_n\parallel +{\alpha }_n(1-\delta )\frac{9ϵ}{(1-\delta )}.\hfill \end{array}$$

Let ${\xi }_n:=\parallel v_{-}{\overline{v}}_n\parallel$, ${\phi }_n:={\alpha }_n(1-\delta )\in (0,1)$ and ${\varrho }_n:=\frac{9ϵ}{(1-\delta )}$. From Lemma 3, it is clear that $0\le \underset{n\to \infty }{lim\; sup}\parallel v_n-{\overline{v}}_n\parallel \le \underset{n\to \infty }{lim\; sup}\frac{9ϵ}{1-\delta }$. Again, from Theorem 1, it is easy to confirm that $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$. Consequently, given that $\underset{n\to \infty }{lim}\parallel {\overline{v}}_n-s^{*}\parallel =0$, we have $\parallel {\tau }^{*}-s^{*}\parallel \le \frac{9ϵ}{1-\delta }$.

Therefore, the proof is complete.    □

4. Application to Oxygen Diffusion Model

Oxygen diffusion transport is a critical chain reaction happening seamlessly within the human body to make oxygen available to every cell by basically moving oxygen down a concentration gradient across tissue barriers, including the alveolar–capillary membrane, and across the extracellular matrix between the tissue capillaries and diffusion distance, which is related to the tissue capillary density. Oxygen diffuses from the air into the blood in the lungs and it does not have the same rate of consumption (see, e.g., [24]) as in a real situation, which tends to be influenced by the thermal energy of particles induced by its kinetic energy. Oxygen most particularly binds to hemoglobin (in a large volume) and dissolves in the blood plasma (in a minute volume), and it is altogether transported through the arteries to capillaries.

Our aim here is to study the model as developed by Srivastava and Rai [25]. The model shown below

$$\frac{{\partial }^{\xi }\mathcal{C}}{\partial t^{\xi }}-\lambda \frac{{\partial }^{\mu }\mathcal{C}}{\partial t^{\mu }}=\nabla (d·\nabla \mathcal{C})-K,\xi ,\mu \in (0,1]$$

(60)

is based on a fractional diffusion equation where $\frac{{\partial }^{\xi }\mathcal{C}}{\partial t^{\xi }}$ is a fractional order derivative for $0<\xi <1$ representing the subdiffusion process, $\mathcal{C}(r,z,t)$ is the concentration of oxygen, $k(r,z,t)$ is the rate of consumption per volume of tissue and d is the diffusion coefficient of oxygen. The net diffusion of oxygen to tissue is $\frac{{\partial }^{\xi }\mathcal{C}}{\partial t^{\xi }}-\lambda \frac{{\partial }^{\mu }\mathcal{C}}{\partial t^{\mu }}$, with $\lambda$ being the time lag in the concentration of oxygen $\mathcal{C}$ along the z-axis.

The equation (60) can be reduced to an integral equivalent:

$$\mathcal{C}(r,z,t)=\mathcal{C}(r,z,0)\left(1-\lambda \frac{t^{\xi -\mu }}{\mathsf{\Gamma }(\xi -\mu +1)}\right)+\lambda {\mathcal{D}}_t^{-(\xi -\mu )}\mathcal{C}+{\mathcal{D}}_t^{-\xi }(\nabla (d·\nabla \mathcal{C}))-{\mathcal{D}}_t^{-\xi }K$$

(61)

Equation (61) can alternatively be written as

$$\mathsf{\Psi }(r,z,t)=\mathsf{\Psi }(r,z,0)\left(1-\lambda \frac{t^{\xi -\mu }}{\mathsf{\Gamma }(\xi -\mu +1)}\right)+\lambda {\mathcal{D}}_t^{-(\xi -\mu )}\mathsf{\Psi }+{\mathcal{D}}_t^{-\xi }(\nabla (d·\nabla \mathsf{\Psi })-K)$$

(62)

or

$$\mathsf{\Psi }(r,z,t)=\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,\mathsf{\Psi }\left(s\right),K)ds$$

(63)

where $\mathcal{K}\left({\mathsf{\Psi }}_0\right)=\mathsf{\Psi }(r,z,0)\left[1-\lambda \frac{t^{\xi -\mu }}{\mathsf{\Gamma }(\xi -\mu +1)}\right]$ and $\mathcal{H}(s,\mathsf{\Psi },K)=\lambda \frac{{\partial }^{\mu }\mathsf{\Psi }}{\partial t^{\mu }}+[\nabla (d·\nabla \mathsf{\Psi })-K]$.

Let

$$\mathcal{T}\mathsf{\Psi }(r,z,t)=\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,\mathsf{\Psi }\left(s\right),K)ds$$

(64)

be an integral operator.

We define, for $t\in [0,T]$ and the space $\mathcal{S}=([0,T],\mathbb{R})$, the supremum norm:

$$\parallel \mathsf{\Psi }\parallel =\underset{t\in [0,T]}{sup}\left\{\right|\mathsf{\Psi }\left(t\right)|:\mathsf{\Psi }\in \mathcal{S}\}.$$

The following result will be useful in proving our main result in this section.

Theorem 7. 

Suppose that the following conditions are satisfied:

$\left(E_1\right)$

There exists a constant $L_{\mathcal{H}}>0$ such that

$$|\mathcal{H}(t,{\mathsf{\Psi }}_1\left(t\right),K)-\mathcal{H}(t,{\mathsf{\Psi }}_2\left(t\right),K)| \le L_{\mathcal{H}}|{\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2|$$

for each $\mathsf{\Psi }\in \mathcal{S}$ and $t\in [0,T]$.

$\left(E_2\right)$

$\frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}<1$.

Then, (60) has a unique solution.

Here, we are set to present our main result of this section.

Theorem 8. 

Assume that the condition of theorem 7 holds. Let $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\in (0,1)$ be real sequences of the iterative scheme (3) such that ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$. Then, the diffusion model Equation (60) has a solution c and the iterative scheme (3) converges to c.

Proof. 

Consider the space $\mathcal{S}=([0,T],\mathbb{R})$ with a supremum norm defined as

$$\parallel \mathsf{\Psi }\parallel =\underset{t\in [0,T]}{sup}\left\{\right|\mathsf{\Psi }\left(t\right)|:\mathsf{\Psi }\in \mathcal{S}\}.$$

Let $\left\{v_n\right\}$ be a sequence generated by the iterative scheme (3) for the operator $\mathcal{T}:\mathcal{S}\to \mathcal{S}$ defined by

$$\mathcal{T}\mathsf{\Psi }\left(t\right)=\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,\mathsf{\Psi }\left(s\right),K)ds.$$

We want to show that $\left\{v_n\right\}$ converges to c as n converges to ∞.

From (3), (64) and the conditions of Theorem 7, we have

$$\begin{array}{cc}\hfill \parallel r_n-c\parallel & = \parallel \mathcal{T}{v}_n-c\parallel \hfill \\ & \le \parallel \mathcal{T}{v}_n-\mathcal{T}c\parallel \hfill \\ & \le \underset{t\in [0,T]}{max}|\mathcal{T}{v}_n\left(t\right)-\mathcal{T}c\left(t\right)|\hfill \\ & =\underset{t\in [0,T]}{max}|\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,v_n\left(s\right),K)ds-\mathcal{K}\left({\mathsf{\Psi }}_0\right)-\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ & =\frac{1}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|\mathcal{H}(s,v_n\left(s\right),K)-\mathcal{H}(s,c\left(s\right),K)|ds\hfill \\ & \le \frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|v_n\left(s\right)-c\left(s\right)|ds\hfill \\ & \le \frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}t\parallel v_n-c\parallel \hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill \parallel s_n-c\parallel & = \parallel (1-{\alpha }_n)r_n+{\alpha }_n\mathcal{T}{r}_n-c\parallel \hfill \\ \hfill & \le (1-{\alpha }_n)\parallel r_n-c\parallel +{\alpha }_n\parallel \mathcal{T}{r}_n-\mathcal{T}c\parallel \hfill \\ \hfill & \le (1-{\alpha }_n)\parallel r_n-c\parallel +{\alpha }_n\underset{t\in [0,T]}{max}|\mathcal{T}{r}_n-\mathcal{T}c|\hfill \\ \hfill & \le (1-{\alpha }_n)\parallel r_n-c\parallel +{\alpha }_n\underset{t\in [0,T]}{max}|\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,r_n,K)ds\hfill \\ \hfill & -\mathcal{K}\left({\mathsf{\Psi }}_0\right)-\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & \le (1-{\alpha }_n)\parallel r_n-c\parallel +{\alpha }_n\underset{t\in [0,T]}{max}|\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,r_n\left(s\right),K)ds\hfill \\ \hfill & -\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & =(1-{\alpha }_n)\parallel r_n-c\parallel +\frac{{\alpha }_n}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}|{\int }_0^t\mathcal{H}(s,r_n\left(s\right),K)-\mathcal{H}(s,c\left(s\right),K)|ds\hfill \\ \hfill & \le (1-{\alpha }_n)\parallel r_n-c\parallel +\frac{{\alpha }_n{L}_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|r_n\left(s\right)-c\left(s\right)|ds\hfill \\ \hfill & \le (1={\alpha }_n)\parallel r_n-c\parallel +\frac{{\alpha }_n{L}_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\parallel r_n-c\parallel \hfill \\ \hfill & =[1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\alpha }_n]\parallel r_n-c\parallel \hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \parallel t_n-c\parallel & = \parallel \mathcal{T}{s}_n-c\parallel \le \parallel \mathcal{T}{s}_n-\mathcal{T}c\parallel \hfill \\ \hfill & \le \underset{t\in [0,T]}{max}|\mathcal{T}{s}_n\left(t\right)-\mathcal{T}c\left(t\right)|\hfill \\ \hfill & \le \underset{t\in [0,T]}{max}|\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,s_n\left(s\right),K)ds\hfill \\ \hfill & -\mathcal{K}\left({\mathsf{\Psi }}_0\right)-\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & =\frac{1}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}|{\int }_0^t\mathcal{H}(s,s_n\left(s\right),K)ds-{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & \le \frac{1}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|\mathcal{H}(s,s_n\left(s\right),K)-\mathcal{H}(s,c\left(s\right),K)|ds\hfill \\ \hfill & \le \frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|s_n\left(s\right)-c\left(s\right)|ds\hfill \\ \hfill & \le \frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}\parallel s_n-c\parallel \hfill \end{array}$$

(67)

Putting (66) in (67), we have

$$\parallel t_n-c\parallel \le \frac{l_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T[1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\alpha }_n]\parallel r_n-c\parallel$$

$$\begin{array}{cc}\hfill \parallel u_n-c\parallel & = \parallel (1-{\beta }_n)t_n+{\beta }_n\mathcal{T}{t}_n-c\parallel \hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +{\beta }_n\parallel \mathcal{T}{t}_n-c\parallel \hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +{\beta }_n\parallel \mathcal{T}{t}_n-\mathcal{T}c\parallel \hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +{\beta }_n\underset{t\in [0,T]}{max}|\mathcal{T}{t}_n\left(s\right)-\mathcal{T}c\left(s\right)|\hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +{\beta }_n\underset{t\in [0,T]}{max}|\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,t_n\left(s\right),K)ds\hfill \\ \hfill & -\mathcal{K}\left({\mathsf{\Psi }}_0\right)-\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & =(1-{\beta }_n)\parallel t_n-c\parallel +\frac{{\beta }_n}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}|{\int }_0^t\mathcal{H}(s,t_n\left(s\right),K)ds-{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +\frac{{\beta }_n}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|\mathcal{H}(s,t_n\left(s\right),K)-\mathcal{H}(s,c\left(s\right),K)|ds\hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +\frac{{\beta }_n}{\mathsf{\Gamma }\left(\xi \right)}{L}_{\mathcal{H}}\underset{t\in [0,T]}{max}{\int }_0^t|t_n\left(s\right)-c\left(s\right)|ds\hfill \\ \hfill & \le (1-{\beta }_n)\parallel t_n-c\parallel +\frac{{\beta }_n{L}_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\parallel t_n-c\parallel \hfill \\ \hfill & =[1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}){\beta }_n]\parallel t_n-c\parallel \hfill \\ \hfill & \le \frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}[1-(1-\frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}){\alpha }_n][1-(1-\frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}){\beta }_n]\parallel r_n-c\parallel \hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill \parallel v_{n+1}-c\parallel & = \parallel (1-{\gamma }_n)u_n+\gamma \mathcal{T}{u}_n-c\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +{\gamma }_n\parallel \mathcal{T}{u}_n-c\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +{\gamma }_n\parallel \mathcal{T}{u}_n-\mathcal{T}c\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +{\gamma }_n\underset{t\in [0,T]}{max}|\mathcal{T}{u}_n-\mathcal{T}c|\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +{\gamma }_n\underset{t\in [0,T]}{max}|\mathcal{K}\left({\mathsf{\Psi }}_0\right)+\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,u_n\left(s\right),K)ds\hfill \\ & -\mathcal{K}\left({\mathsf{\Psi }}_0\right)-\frac{1}{\mathsf{\Gamma }\left(\xi \right)}{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +\frac{{\gamma }_n}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}|{\int }_0^t\mathcal{H}(s,u_n\left(s\right),K)ds-{\int }_0^t\mathcal{H}(s,c\left(s\right),K)ds|\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +\frac{{\gamma }_n}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|\mathcal{H}(s,u_n\left(s\right),K)-\mathcal{H}(s,c\left(s\right),K)|ds\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +\frac{{\gamma }_n\mathcal{H}}{\mathsf{\Gamma }\left(\xi \right)}\underset{t\in [0,T]}{max}{\int }_0^t|u_n\left(s\right)-c\left(s\right)|ds\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +\frac{{\gamma }_n{L}_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\underset{t\in [0,T]}{max}|u_n-c|\hfill \\ & \le (1-{\gamma }_n)\parallel u_n-c\parallel +\frac{{\gamma }_n{L}_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\parallel u_n-c\parallel \hfill \\ & =[1-(1-\frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}){\gamma }_n]\parallel u_n-c\parallel \hfill \end{array}$$

(69)

Combining (65), (68) and (69), we have

$$\parallel v_{n+1}-c\parallel \le \frac{L_{\mathcal{H}}^2{T}^2}{{\mathsf{\Gamma }}^2\left(\xi \right)}\left[1-\left(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\right){\alpha }_n\right]\left[1-\left(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\right){\beta }_n\right]\left[1-\left(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T\right){\gamma }_n\right]\parallel v_n-c\parallel$$

(70)

From assumption $\left(E_2\right)$ and the fact that $[1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\beta }_n][1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\gamma }_n]<1$, (70) reduces to

$$\parallel v_{n+1}-c\parallel \le [1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\alpha }_n]\parallel v_n-c\parallel .$$

Via induction,

$$\parallel v_{n+1}-c\parallel \le \parallel v_0-c\parallel \prod _{m=0}^n[1-(1-\frac{L_{\mathcal{H}}}{\mathsf{\Gamma }\left(\xi \right)}T){\alpha }_m].$$

(71)

From classical analysis, $1-x\le e^{-x}$ for $x\in [0,1]$.

$$\parallel v_{n+1}-c\parallel \le \parallel v_0-c\parallel e^{-(1-\frac{L_{\mathcal{H}}T}{\mathsf{\Gamma }\left(\xi \right)}){\sum }_{m=0}^n{\alpha }_m}$$

Taking the limit as $n\to \infty$, we have $\underset{n\to \infty }{lim}\parallel v_n-c\parallel =0$. Hence, the proof is complete.    □

5. Application to Boundary Value Problem via Green’s Function

5.1. Construction of Green’s Function

To construct the Green’s function, we consider a third-order boundary value problem (BVP),

$$L[g]\equiv p_1\left(t\right)g^{\prime \prime \prime }\left(t\right)+p_2\left(t\right)g^{″}\left(t\right)+p_3\left(t\right)g^{\prime }\left(t\right)+p_4\left(t\right)g\left(t\right)=M\left(t\right)$$

(72)

where $t\in [a,b]$, with the corresponding boundary conditions (BCs)

$$\begin{array}{cc}\hfill B_{k_1}[g]& ={\phi }_{1}g\left(k_1\right)+{\phi }_2{g}^{\prime }\left(k_1\right)+{\phi }_3{g}^{″}\left(k_1\right)=\phi \hfill \\ \hfill B_{k_2}[g]& ={\varpi }_{1}g\left(k_2\right)+{\varpi }_2{g}^{\prime }\left(k_2\right)+{\varpi }_3{g}^{″}\left(k_2\right)=\varpi \hfill \\ \hfill B_{k_3}[g]& ={\vartheta }_{1}g\left(k_3\right)+{\vartheta }_2{g}^{\prime }\left(k_3\right)+{\vartheta }_3{g}^{″}\left(k_3\right)=\vartheta \hfill \end{array}$$

(73)

for $k_3=k_2$ or $k_3=k_1$. From (72), $L[g]$ is linear and the righthand side can be written as $M(t,g\left(t\right),g^{\prime }\left(t\right),g^{″}\left(t\right))$. The righthand side could be linear or nonlinear; $\phi$, $\varpi$, $\vartheta$ are constants.

The homogeneous part $L[g]=0$ of (72) can be solved to obtain three linearly independent complementary solutions, $g_1,g_2$ and $g_3$, and will be used to obtain the Green’s function, which is a piecewise function expressed as a linear combination of the linearly independent complementary solutions $g_1,g_2$ and $g_3$; thus,

$$G(t,s)=\left\{\begin{array}{c}{d}_1{g}_1+d_2{g}_2+d_3{g}_3,a<t<s\hfill \\ e_1{g}_1+e_2{g}_2+e_3{g}_3,s<t<b,\hfill \end{array}\right.$$

(74)

where $d_1, d_2, d_3, e_1, e_2, e_3$ are constants that can be determined accordingly through the hypotheses of the following axioms;

$\left(A_1\right)$

G satisfies the associated boundary conditions:

$$B_{k_1}[G(t,s)]=B_{k_2}[G(t,s)]=B_{k_3}[G(t,s)]=0$$

$\left(A_2\right)$

G is continuous at $t=s$:

$$d_1{g}_1\left(s\right)+d_2{g}_2\left(s\right)+d_3{g}_3\left(s\right)=e_1{g}_1\left(s\right)+e_2{g}_2\left(s\right)+e_3{g}_3\left(s\right)$$

$\left(A_3\right)$

$G^{\prime }$ is continuous at $t=s$:

$$d_1{g}_1^{\prime }\left(s\right)+d_2{g}_2^{\prime }\left(s\right)+d_3{g}_3^{\prime }\left(s\right)=e_1{g}_1^{\prime }\left(s\right)+e_2{g}_2^{\prime }\left(s\right)+e_3{g}_3^{\prime }\left(s\right)$$

$\left(A_4\right)$

$G^{″}$ has jump discontinuity at $t=s$:

$$d_1{g}_1^{″}\left(s\right)+d_2{g}_2^{″}\left(s\right)+d_3{g}_3^{″}\left(s\right)+\frac{1}{h\left(s\right)}=e_1{g}_1^{″}\left(s\right)+e_2{g}_2^{″}\left(s\right)+e_3{g}_3^{″}\left(s\right)$$

If the Green’s function $G(t,s)$ is the solution to the BVP (72), then it will satisfy the equation

$$-L[G(t,s)]=\delta (t-s)$$

(75)

where $\delta$ is the Kronecker Delta that is subject to the homogeneous boundary conditions

$$B_{k_1}[G(t,s)]=B_{k_2}[G(t,s)]=B_{k_3}[G(t,s)]=0.$$

As a matter of fact, the righthand side of (75) will be $-\delta (t-s)$ for self-adjoint operators. The Green’s function in (73) will satisfy the homogeneous equation $L[G(t,s)]=0$ for $t\ne s$.

5.2. UO–Green Iterative Scheme

Our aim here is to embed the Green’s function obtained from the preceding section in the iterative scheme (3). This aim can be achieved by considering the nonlinear boundary value problem

$$L[g]+N[g]=M(t,g),$$

(76)

where $L[g]$ is linear in g, $N[g]$ is nonlinear in g and $M(t,g)$ is a function in g that could be linear or nonlinear. The general solution to (76) is given as $g=g_c+g_p$ where $g_c$ is the complementary solution obtained from the homogeneous part $L[g]=0$ subject to the boundary conditions as expressed in axiom $\left(A_1\right)$.

Assume $g_p$ is the particular solution to the nonhomogeneous part of (76). We define an integral operator in terms of the Green’s function, $G(t,s)$, and the particular solution, $g_p$.

$$\mathsf{\Omega }[g_p]={\int }_a^{b}G(t,s)L[g_p]ds.$$

(77)

Setting $g_p$ to g for convenience so that (77) becomes

$$\mathsf{\Omega }[g]={\int }_a^{b}G(t,s)L[g]ds.$$

(78)

Obviously, g is a fixed point if and only if g is the solution to (76). Suppose $g_p-g={\int }_a^{b}G(t,s)[M(t,g)-N[g]]ds$,

$$\begin{array}{cc}\hfill \mathsf{\Omega }[g]& ={\int }_a^{b}G(t,s)[L[g]+N[g]-M(t,g)-N[g]+M(t,g)]ds\hfill \\ & \le {\int }_a^{b}G(t,s)[L[g]+N[g]-M(t,g)]ds\hfill \\ & +{\int }_a^{b}G(t,s)[M(t,g)-N[g]]ds\hfill \\ & =g+{\int }_a^{b}G(t,s)[L[g]+N[g]-M(t,g)]ds.\hfill \end{array}$$

Applying the UO iterative scheme (3), we have

$$\left\{\begin{array}{c}{r}_n=\mathsf{\Omega }[v_n]\hfill \\ s_n=(1-{\alpha }_n)r_n+{\alpha }_n\mathsf{\Omega }[r_n]\hfill \\ t_n=\mathsf{\Omega }[s_n]\hfill \\ u_n=(1-{\beta }_n)t_n+{\beta }_n\mathsf{\Omega }[t_n]\hfill \\ v_{n+1}=(1-{\gamma }_n)u_n+{\gamma }_n\mathsf{\Omega }[u_n],\hfill \end{array}\right.$$

(79)

where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}$ and $\left\{{\gamma }_n\right\}$ are real sequences in $(0,1)$ for all $n\in \mathbb{N}$. In an expanded form, (79) can be expressed as

$$\begin{array}{cc}\hfill r_n& =v_n+{\int }_a^{b}G(t,s)[L[v_n]+N[v_n]-M(t,v_n)]ds\hfill \\ \hfill s_n& =(1-{\alpha }_n)r_n+{\alpha }_n[r_n+{\int }_a^{b}G(t,s)[L[r_n]+N[r_n]-M(t,r_n)]ds]\hfill \\ \hfill t_n& =s_n+{\int }_a^{b}G(t,s)[L[s_n]+N[s_n]-M(t,s_n)]ds\hfill \\ \hfill u_n& =(1-{\beta }_n)t_n+{\beta }_n\left\{t_n+{\int }_a^{b}G(t,s)[L[t_n]+N[t_n]-M(t,t_n)]ds\right\}\hfill \\ \hfill v_{n+1}& =(1-{\gamma }_n)u_n+{\gamma }_n\left\{u_n+{\int }_a^{b}G(t,s)[L[u_n]+N[u_n]-M(t,u_n)]ds\right\}\hfill \end{array}$$

which reduces to

$$\begin{array}{cc}\hfill r_n& =v_n+{\int }_a^{b}G(t,s)[L[v_n]+N[v_n]-M(t,v_n)]ds\hfill \\ \hfill s_n& =r_n+{\alpha }_n{\int }_a^{b}G(t,s)[L[r_n]+N[r_n]-M(t,r_n)]ds\hfill \\ \hfill t_n& =s_n+{\int }_a^{b}G(t,s)[L[s_n]+N[s_n]-M(t,s_n)]ds\hfill \\ \hfill u_n& =t_n+{\beta }_n{\int }_a^{b}G(t,s)[L[t_n]+N[t_n]-M(t,t_n)]ds\hfill \\ \hfill v_{n+1}& =u_n+{\gamma }_n{\int }_a^{b}G(t,s)[L[u_n]+N[u_n]-M(t,u_n)]ds.\hfill \end{array}$$

(80)

5.3. Convergence Analysis

It is our aim to show the convergence of our iterative scheme (3) to a solution to the BVP via Green’s function. To achieve our aim, we consider the following BVP:

$$-g^{\prime \prime \prime }\left(t\right)=M(t,g\left(t\right),g^{\prime }\left(t\right),g^{″}\left(t\right))$$

with boundary conditions

$$g\left(1\right)=C_1,g^{″}\left(1\right)=C_2,g\left(2\right)=C_3$$

When the homogeneous equation $g^{\prime \prime \prime }\left(t\right)=0$ is solved, the following Green’s function is obtained

$$G(t,s)=\left\{\begin{array}{c}{d}_1{t}^2+d_{2}t+d_3,1\le t\le s\le 2\hfill \\ e_t^2+e_{2}t+e_3,1\le s\le t\le 2\hfill \end{array}\right.$$

(81)

If axioms $\left(A_1\right)$–$\left(A_4\right)$ are invoked, then real values for the constants $d_i,e_i(i=1,2,3)$ are obtained. Hence, (81) becomes

$$G(t,s)=\left\{\begin{array}{c}-\frac{s^2}{2}+2s-2+(\frac{s^2}{2}-2s+2)t,1\le t\le s\le 2\hfill \\ -s^2+2s-2+(\frac{s^2}{2}-2s+2)t,1\le s\le t\le 2.\hfill \end{array}\right.$$

(82)

Furthermore, the UO–Green iterative scheme (80) is given as

$$\left\{\begin{array}{c}{r}_n={\mathcal{T}}_G{v}_n\hfill \\ s_n=(1-{\alpha }_n)r_n+{\alpha }_n{\mathcal{T}}_G{r}_n\hfill \\ t_n={\mathcal{T}}_G{s}_n\hfill \\ u_n=(1-{\beta }_n)t_n+{\beta }_n{\mathcal{T}}_G{t}_n\hfill \\ v_{n+1}=(1-{\gamma }_n)u_n+{\mathcal{T}}_G{u}_n\hfill \end{array}\right.$$

(83)

where the operator ${\mathcal{T}}_G:C^2\left([1,2]\right)\to C^2\left([1,2]\right)$ is defined as

$${\mathcal{T}}_G\left(v\right)=v+{\int }_1^{2}G(t,s)(v^{\prime \prime \prime }-M(s,v,v^{\prime },v^{″}))ds$$

(84)

The initial iterate $v_0$ to (83) satisfies the homogeneous equation $v_0^{\prime \prime \prime }=0$ and the BCs: $v_0\left(1\right)=C_1,v_0^{″}\left(1\right)=C_2$ and $v_0\left(2\right)=C_3$.

Suppose we use integration by parts for ${\int }_1^{2}G(t,s)v^{\prime \prime \prime }ds$ in (84) and, noting that ${\int }_1^2\frac{{\partial }^{3}G(t,s)}{\partial s^3}g\left(s\right)ds={\int }_1^2\delta (t-s)g\left(s\right)ds$, we then have

$${\mathcal{T}}_G\left(v\right)=(2-t)C_1+\frac{1}{2}(t^2-3t+2)C_2+(t-1)C_3-{\int }_1^{2}G(t,s)M(s,v,v^{\prime },v^{″})ds.$$

Our next aim is to prove that the operator ${\mathcal{T}}_G$ is a contraction on the Banach space $C^2\left([1,2]\right)$ for the norm

$${\parallel v\parallel }_{C^2}=\sum _{i=0}^2\underset{s\in [1,2]}{sup}\left|v^{\left(i\right)}\left(s\right)\right|$$

under certain conditions on M. Moreover, we shall show that ${\mathcal{T}}_G$ is a Zamfirescu operator under certain hypotheses on M.

Theorem 9. 

Let M, which appears in ${\mathcal{T}}_G$, satisfy the following Lipschitz condition:

$$|M(s,v,v^{\prime },v^{″})-M(s,\ell ,{\ell }^{\prime },{\ell }^{″})| \le {\mu }_1|v\left(s\right)-\ell \left(s\right)|+{\mu }_2|v^{\prime }\left(s\right)-{\ell }^{\prime }\left(s\right)| + {\mu }_3|v^{″}\left(s\right)-{\ell }^{″}\left(s\right)|$$

(85)

where ${\mu }_1,{\mu }_2$ and ${\mu }_3$ are positive constants such that

$$\frac{1}{8}max\{{\mu }_1,{\mu }_2,{\mu }_3\}\le 1.$$

The operator ${\mathcal{T}}_G$ is a contraction on the Banach space $C^2{([1,2],\parallel ·\parallel }_{C^2})$, and the sequence $\left\{v_n\right\}$ defined by the UO iterative scheme (3) converges to the fixed point of ${\mathcal{T}}_G$.

Proof. 

Assume $v_1,v_2\in C^2\left([1,2]\right)$, so that by (85), we have

$$\begin{array}{cc}\hfill |{\mathcal{T}}_G\left(v_1\right)& - {\mathcal{T}}_G\left(v_2\right)|\hfill \\ & = |{\int }_1^{2}G(t,s)M(s,v_1,v_1^{\prime },v_1^{″})ds-{\int }_1^{2}G(t,s)M(s,v_2,v_2^{\prime },v_2^{″})ds|\hfill \\ & = |{\int }_1^{2}G(t,s)(M(s,v_1,v_1^{\prime },v_1^{″})-M(s,v_2,v_2^{\prime },v_2^{″}))ds|\hfill \\ & \le {\int }_1^2\left|G(t,s)\right||M(s,v_1,v_1^{\prime },v_1^{″})-M(s,v_2,v_2^{\prime },v_2^{″})|ds\hfill \\ & \le \left(\underset{[1,2]\times [1,2]}{sup}\left|G(t,s)\right|\right){\int }_1^2|M(s,v_1,v_1^{\prime },v_1^{″})-M(s,v_2,v_2^{\prime },v_2^{″})|ds\hfill \\ & =G(\frac{3}{4},1){\int }_1^2|M(s,v_1,v_1^{\prime },v_1^{″})-M(s,v_2,v_2^{\prime },v_2^{″})|ds\hfill \\ & =\frac{1}{8}{\int }_1^2|M(s,v_1,v_1^{\prime },v_1^{″})-M(s,v_2,v_2^{\prime },v_2^{″})|ds\hfill \\ & \le \frac{1}{8}{\int }_1^2[{\mu }_1|v_1\left(s\right)-v_2\left(s\right)| + {\mu }_2|v_1^{\prime }\left(s\right)-v_2^{\prime }\left(s\right)| + {\mu }_3|v_1^{″}\left(s\right)-v_2^{″}\left(s\right)|]ds\hfill \\ & \le \frac{1}{8}max\{{\mu }_1,{\mu }_2,{\mu }_3\}{\int }_1^2\left(\sum _{i=1}^2|v_1^{\left(i\right)}\left(s\right)-v_2^{\left(i\right)}\left(s\right)|\right)ds\hfill \\ & \le \frac{1}{8}max\{{\mu }_1,{\mu }_2,{\mu }_3\}{\parallel v_1-v_2\parallel }_{C^2}\hfill \\ & < \parallel v_1-v_2{\parallel }_{C^2}.\hfill \end{array}$$

which shows that ${\mathcal{T}}_G$ is a contraction.

Next, we want to show the strong convergence of the sequence $\left\{v_n\right\}$ defined by the UO iterative scheme (3) to the fixed point of the operator ${\mathcal{T}}_G$.

Since ${\mathcal{T}}_G$ is a contraction, it is clear from the known Banach contraction principle that the existence of a unique fixed point, ${\tau }^{*}$, of ${\mathcal{T}}_G$ in the Banach space $C^2{([1,2],\parallel ·\parallel }_{C^2})$ is certain. That is, we shall prove that $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$.

From (83), we have

$$\begin{array}{cc}\hfill \parallel r_n-{\tau }^{*}\parallel & = \parallel {\mathcal{T}}_G{v}_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel {\mathcal{T}}_G{v}_n-{\mathcal{T}}_G{\tau }^{*}\parallel \hfill \\ & \le \delta \parallel v_n-{\tau }^{*}\parallel \hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill \parallel s_n-{\tau }^{*}\parallel & =(1-{\alpha }_n)r_n+{\alpha }_n{\mathcal{T}}_G{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\tau }^{*}\parallel + {\alpha }_n\parallel {\mathcal{T}}_G{r}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel r_n-{\tau }^{*}\parallel + {\alpha }_n\delta \parallel r_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\alpha }_n]\parallel r_n-ta{u}^{*}\parallel ,\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill \parallel t_n-{\tau }^{*}\parallel & = \parallel {\mathcal{T}}_G{s}_n-{\tau }^{*}\parallel \hfill \\ & \le \parallel {\mathcal{T}}_G{s}_n-{\mathcal{T}}_G{\tau }^{*}\parallel \hfill \\ & \le \delta \parallel s_n-{\tau }^{*}\parallel ,\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill \parallel u_n-{\tau }^{*}\parallel & = \parallel (1-{\beta }_n)t_n+{\beta }_n{\mathcal{T}}_G{t}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\parallel {\mathcal{T}}_G{t}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\beta }_n)\parallel t_n-{\tau }^{*}\parallel + {\beta }_n\parallel {\mathcal{T}}_G{t}_n-{\mathcal{T}}_G{\tau }^{*}\parallel \hfill \\ & =[(1-{\beta }_n)+{\beta }_n\delta ]\parallel t_n-{\tau }^{*}\parallel \hfill \\ & \le [1-(1-\delta ){\beta }_n]\parallel t_n-{\tau }^{*}\parallel \hfill \end{array}$$

(89)

and

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & = \parallel (1-{\gamma }_n)u_n+{\gamma }_n{\mathcal{T}}_G{u}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\gamma }_n)\parallel u_n-{\tau }^{*}\parallel + {\gamma }_n\parallel {\mathcal{T}}_G{u}_n-{\tau }^{*}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel u_n-{\tau }^8\parallel + {\gamma }_n\delta \parallel u_n-{\tau }^{*}\parallel \hfill \\ & =[1-(1-\delta ){\gamma }_n]\parallel u_n-{\tau }^{*}\parallel \hfill \end{array}$$

(90)

and, combining (86)–(90), we have

$$\parallel v_{n+1}-{\tau }^{*}\parallel \le {\delta }^2[1-(1-\delta ){\alpha }_n][1-(1-\delta ){\beta }_n][1-(1-\delta ){\beta }_n]\parallel v_n-{\tau }^{*}\parallel$$

Since $\delta \in [0,1)$ and ${\alpha }_n, {\beta }_n, {\gamma }_n\in (0,1)$, we can say that $[1-(1-\delta ){\beta }_n][1-(1-\delta ){\gamma }_n]<1$.

It follows that

$$\parallel v_{n+1}-{\tau }^{*}\parallel \le {\delta }^2[1-(1-\delta ){\alpha }_n]\parallel v_n-{\tau }^{*}\parallel .$$

Inductively,

$$\parallel v_{n+1}-{\tau }^{*}\parallel \le {\delta }^{2(n+1)}[1-(1-\delta ){\alpha }_n]\parallel v_0-{\tau }^{*}\parallel$$

$$v_{n+1}-{\tau }^{*}\parallel \le {\delta }^{2(n+1)}\parallel v_0-{\tau }^{*}\parallel \prod _{k=0}^n[1-(1-\delta ){\alpha }_k]$$

From elementary analysis, it is clear that $1-x\le e^{-x}$ for $0<x<1$, so that

$$\begin{array}{cc}\hfill \parallel v_{n+1}-{\tau }^{*}\parallel & \le {\delta }^{2(n+1)}{\parallel v_0-{\tau }^{*}\parallel }^{n+1}\prod _{k=0}^n{e}^{-(1-\delta ){\alpha }_k}\hfill \\ & le{\delta }^{2(n+1)}{\parallel v_0-{\tau }^{*}\parallel }^{n+1}{e}^{-(1-\delta ){\sum }_{k=0}^{\infty }{\alpha }_k}\hfill \end{array}$$

Clearly, if ${\sum }_{k=0}^{\infty }{\alpha }_k=\infty$, such that $e^{-(1-\delta ){\sum }_{k=0}^n{\alpha }_k}\to 0$ as $n\to \infty$, then $\underset{n\to \infty }{lim}\parallel v_n-{\tau }^{*}\parallel =0$, thereby completing the proof.    □

Example 2. 

Consider the BVP

$$g^{\prime \prime \prime }\left(t\right)=-t{g}^{″}\left(t\right)-2t^2+t-2$$

(91)

with BCs

$$g\left(0\right)=g^{\prime }\left(0\right)=g^{\prime }\left(1\right)=0$$

(92)

The exact solution is $g\left(t\right)=\frac{t^2}{2}-\frac{t^3}{3}$.

The corresponding Green’s function is given as follows:

$$G(t,s)=\left\{\begin{array}{c}\frac{(s-1)}{2}{t}^2,0<t<s\hfill \\ \frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2},s<t<1\hfill \end{array}\right.$$

Embedding the Green’s function in the UO–Green fixed point iterative scheme (80), we have

$$\begin{array}{cc}\hfill r_n& =v_n+{\int }_0^t\left[\frac{(s-1)}{2}{t}^2\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ & + {\int }_t^1\left[\frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2}\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ \hfill s_n& =r_n+{\alpha }_n{\int }_0^t\left[\frac{(s-1)}{2}{t}^2\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ & + {\alpha }_n{\int }_t^1\left[\frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2}\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ \hfill t_n& =s_n+{\int }_0^t\left[\frac{(s-1)}{2}{t}^2\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ & + {\int }_t^1\left[\frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2}\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ \hfill u_n& =t_n+{\beta }_n{\int }_0^t\left[\frac{(s-1)}{2}{t}^2\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ & + {\beta }_n{\int }_t^1\left[\frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2}\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ \hfill u_{n+1}& =u_n+{\gamma }_n{\int }_0^t\left[\frac{(s-1)}{2}{t}^2\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds\hfill \\ & + {\gamma }_n{\int }_t^1\left[\frac{s{(1-t)}^2}{2}+\frac{s^2-s}{2}\right][g^{\prime \prime \prime }\left(t\right)+t{g}^{″}\left(t\right)+2t^2-t+2]ds.\hfill \end{array}$$

With a better choice of ${\alpha }_n, {\beta }_n,$ and ${\gamma }_n\in (0,1)$, it is guaranteed that the UO–Green iterative scheme converges faster than the Picard–Green [26], Mann–Green [27], Khan–Green [28], Ishikawa–Green [29] and GA–Green [2].

Furthermore, the minimization of the $L^2$-norm of the residual error guarantees a perfect computation.

6. Conclusions

The UO iterative scheme generalizes and extends other existing iterative schemes in the literature as shown in Example 1, where our scheme converges to the fixed point 2 faster than all of the CR, $F^{*}$ Picard-S, Modified-SP Uddin et al. and Picard–Ishikawa iterative schemes with visualization in tables and graphs. Our newly developed UO iteration process is applied in solving a multi-term fractional diffusion equation for oxygen delivery via a capillary of tissues, as found in [25]. Embedding the Green’s function in the UO scheme (3) gives rise to the UO–Green iterative scheme, which is used to approximate the solution of a BVP.