Advancements in Hybrid Fixed Point Results and F-Contractive Operators

Abstract

The aim of this manuscript is to introduce a novel concept called Jaggi-type hybrid $(\varphi$ -$F)$-contraction and establish some fixed point results for this class of contractions in the framework of G-metric space. The validity of the main result is shown by a suitable example and the realized improvements with respect to the corresponding literature are highlighted. By using the constructed example, it is observed that the results established herein cannot be deduced from their analogs in previously announced results in the literature. As an application, the existence and uniqueness of solutions to certain nonlinear Volterra integral equations are investigated to illustrate the utility of our obtained results.

1. Introduction

It is well-recognized that the theory of fixed points (FPs) is extremely helpful in proving the existence and uniqueness theorems for differential and integral equations and related problems in nonlinear functional analysis. In FP theory, the contractive constraints or conditions on the underlying mappings are crucial to solving FP problems. Banach [1], in 1922, presented a significant fixed point result (also called the Banach contraction principle). This finding served as a springboard for researchers and unlocked a new avenue for metric FP theory, leading to its generalizations in numerous directions. One of the most embraced extensions of the Banach contraction principle involving rational inequality was presented by Jaggi [2]. For some surveys on FP results of contraction mappings, see Cho [3].

One of the active research areas in the mathematical community is the study of new spaces. In this direction, Mustafa and Sims [4] introduced the concept of G-metric space (or simply, G-MS) as an extension of MS and examined the topological structure of such spaces. The motivations for putting forward a G-MS are two-fold, viz. the form of a G-MS is to adequately visualize the geometry of three points instead of two points through the perimeter of a triangle. Furthermore, the aim is to correct some fundamental flaws in the idea of generalized metric spaces, as proposed by Dhage [5]. Based on the notion of G-MS, Mustafa et al. [6] proved some FP theorems for mappings satisfying different contractive conditions. On the earliest notion of a G-MS, Samet et al. [7], as well as Jleli et al. [8], observed that some FP results in the context of G-MS can be obtained directly by some existing results in the setting of a metric or quasi-MS. As a result of these observations, the authors in [9] remarked that unless the contraction condition in the statement of the theorem can be reduced to two variables, the techniques used in [8] are inapplicable. However, there is a relation between the conventional metric and every G-metric since the latter is developed from the former. Further details in this direction are contained in [10]. Meanwhile, a lot of useful results in the domain of G-MS have been presented. In 2023, Hammad et al. [11] defined a new class of control functions and discussed fixed point results which properly refined some known ones in the literature. The idea of [11] is to take care of some peculiarities of a G-MS. For a recent survey of the advancements in fixed point results in G-MS, the reader can consult Jiddah et al. [12].

Along the line, Wardowski [13] provided an intriguing but distinct generalization of the Banach contraction theorem, via a new contractive inequality called F-contraction. In 2014, Piri and Kumam [14] expanded the work of Wardowski by imposing weaker auxiliary conditions on the self-map of a complete metric space (MS) on the mapping F. Ahmad et al. [15] in 2015 defined two classes of functions based on the idea of F-contraction and proved some FP results in a complete MS. In 2016, Singh et al. [16] studied a new form of the Hardy–Roger type in G-metric spaces and improved the main results of [14]. Furthermore, Vujaković et al. [17] in 2020 initiated the idea of $(\varphi$ -$F)$-weak contraction and proved the corresponding FP results in metric space. Recently, Hammad and De la Sen [18] established some variants of F-contractions using Jaggi and Suzuki contractions in b-metric-like spaces. For some important developments in F-contraction-type fixed point results, we can refer to Fabiano et al. [19] and Joshi and Jain [20].

The study of hybrid contractions is one of the key aspects of FP theory that is grabbing the interest of researchers. This concept has been interpreted in two ways: firstly, hybrid contraction addresses contractions incorporating both single-valued and multi-valued mappings and the second perspective as a combination of linear and nonlinear contractions. The idea of Jaggi-type hybrid contraction was introduced by Karapinar and Fulga [21]. Alansari and Mohammed [22] presented a multivalued hybrid fixed point theorem which extended the main idea of [21] to a set-valued domain. Along the line, Mohammed et al. [23] examined new conditions for the existence of a class of nonlinear hybrid-type fixed point theorem and improved a few corresponding results in the literature.

Following this direction of research, we realize that hybrid FP results in G-MS vis-a-vis F-contractive-type mappings are not sufficiently explored. Jiddah et al. [24] established hybrid fixed point results in G-metric space and refined several corresponding results. However, [24] inherited a drawback from Karapinar and Fulga [21], who were only able to discuss the uniqueness of a fixed point under certain finite iterations of the considered mappings. We noticed that by using F-contractive operators, the former problem can be circumvented. To this effect, we present in this manuscript a new notion, called Jaggi-type hybrid $(\varphi$ -$F)$- contraction in G-MS, and established some new FP results. An illustrative, non-trivial example is built to show that our result is substantial and an improvement over previous findings. It is important to note that the primary ideas established herein are not reducible to any existing results. It is shown by several consequences presented that the idea proposed herein is a generalization of some well-known FP results in the domain of F-contractive operators in G-MS. Furthermore, one of our obtained corollaries is applied to prove the existence and uniqueness of a solution to a class of nonlinear Volterra integral equations.

The paper is structured as follows: In Section 1, the introduction and overview of related literature are presented. The basic concepts needed in this work are compiled in Section 2. In Section 3, the primary results and some consequences of the obtained FP results are discussed. With the help of one of the obtained results herein, the existence and uniqueness of a solution to a nonlinear integral equation of Volterra type is explored in Section 4. In Section 5, deductions, recommendations, and conclusions are given.

2. Preliminaries

Contained in this section, some basic notations and findings that will be applied subsequently are outlined. Throughout this paper, every set $\mathsf{\Omega }$ is considered non-empty. We denote by $\mathbb{R}$, ${\mathbb{R}}_{+}$, and $\mathbb{N}$, the set of real numbers, the set of non-negative real numbers, and the set of natural numbers, respectively.

Definition 1 

([4]). Let Ω be a non-empty set and let $G:\mathsf{\Omega }\times \mathsf{\Omega }\times \mathsf{\Omega }⟶{\mathbb{R}}_{+}$ be a function satisfying:

$\left(G_1\right)$

$G(h,j,w)=0$ if $h=j=w$;

$\left(G_2\right)$

$0<G(h,h,j)$ for all $h,j\in \mathsf{\Omega }$ with $h\ne j$;

$\left(G_3\right)$

$G(h,h,j)\le G(h,j,w)$, for all $h,j,w\in \mathsf{\Omega }$ with $w\ne j$;

$\left(G_4\right)$

$G(h,j,w)=G(h,w,j)=G(j,h,w)=\dots$ (symmetry in all three variables);

$\left(G_5\right)$

$G(h,j,w)\le G(h,a,a)+G(a,j,w)$, for all $h,j,w,a\in \mathsf{\Omega }$ (rectangle inequality).

Then, the function G is called a generalized metric, or more specifically, a G-metric on Ω, and the pair $(\mathsf{\Omega },G)$ is called a G-MS.

Going forward, we shall denote a G-MS by $(\mathsf{\Omega },G)$.

Example 1 

([6]). Given a usual MS $(\mathsf{\Omega },d)$, then $(\mathsf{\Omega },G_m)$ and $(\mathsf{\Omega },G_n)$ are G-MS given by:

$$\begin{array}{cc}\hfill G_m(h,j,w)& =d(h,j)+d(j,w)+d(h,w)\forall h,j,w\in \mathsf{\Omega }.\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill G_n(h,j,w)& =max\left\{d\right(h,j),d(j,w),d(h,w\left)\right\}\forall h,j,w\in \mathsf{\Omega }.\hfill \end{array}$$

(2)

Definition 2 

([6]). On $(\mathsf{\Omega },G)$, define a sequence of points ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ of Ω. Then, ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is said to be G-convergent to h if ${\displaystyle \underset{n,m\to \infty }{lim}G(h,h_n,h_m)=0}$; that is, for every $ϵ>0$, we can find $n_0\in \mathbb{N}$ such that $G(h,h_n,h_m)<ϵ$, $\forall n,m\ge n_0$. We call h the limit of the sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$.

Proposition 1 

([6]). In $(\mathsf{\Omega },G)$, the following are equivalent:

(i)

${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is G-convergent to h.

(ii)

$G(h,h_n,h_m)⟶0$, as $n,m\to \infty$.

(iii)

$G(h_n,h,h)⟶0$, as $n\to \infty$.

(iv)

$G(h_n,h_n,h)⟶0$, as $n\to \infty$.

Definition 3 

([6]). On $(\mathsf{\Omega },G)$, a sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is said to be G-Cauchy if for any $ϵ>0$, we can find $n_0\in \mathbb{N}$, such that $G(h_n,h_m,h_z)<ϵ$, $\forall n,m,z\ge n_0$. That is, $G(h_n,h_m,h_z)⟶0$, as $n,m,z\to \infty$.

Proposition 2 

([6]). In $(\mathsf{\Omega },G)$, the following are equivalent:

(i)

The sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is G-Cauchy.

(ii)

For any $ϵ>0$, we can find $n_0\in \mathbb{N}$ such that $G(h_n,h_m,h_m)<ϵ$, $\forall n,m\ge n_0$.

Definition 4 

([6]). Given $(\mathsf{\Omega },G)$ and $({\mathsf{\Omega }}^{\prime },G^{\prime })$, define a function $f:(\mathsf{\Omega },G)⟶({\mathsf{\Omega }}^{\prime },G^{\prime })$. We say that f is said to be G-continuous at a point $v\in \mathsf{\Omega }$ if and only if for any $ϵ>0$, we can find $\delta >0$ such that $h,j\in \mathsf{\Omega }$, and $G(v,h,j)<\delta$ implies $G^{\prime }(f\left(v\right),f\left(h\right),f\left(j\right))<ϵ$. A function f is G-continuous on Ω if and only if it is G-continuous at all $v\in \mathsf{\Omega }$.

Proposition 3 

([6]). Given $(\mathsf{\Omega },G)$ and $({\mathsf{\Omega }}^{\prime },G^{\prime })$, a function $f:(\mathsf{\Omega },G)⟶({\mathsf{\Omega }}^{\prime },G^{\prime })$ is defined to be G-continuous at a point $h\in \mathsf{\Omega }$ if and only if it is G-sequentially continuous at h. That is, whenever ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is G-convergent to h, $\left\{f{h}_n\right\}$ is G-convergent to $fh$.

Definition 5 

([6]). $(\mathsf{\Omega },G)$ is symmetric if:

$$\begin{array}{c}\hfill G(h,h,j)=G(j,h,h)\forall h,j\in \mathsf{\Omega }.\end{array}$$

Proposition 4 

([6]). A function $G(h,j,w)$ in $(\mathsf{\Omega },G)$ is jointly continuous in all three of its variables.

Proposition 5 

([6]). Every $(\mathsf{\Omega },G)$ defines a MS $(\mathsf{\Omega },d_G)$ by:

$$\begin{array}{cc}\hfill d_G(h,j)& =G(h,j,j)+G(j,h,h)\forall h,j\in \mathsf{\Omega }.\hfill \end{array}$$

(3)

If $(\mathsf{\Omega },G)$ is symmetric, then:

$$\begin{array}{cc}\hfill (\mathsf{\Omega },d_G)& =2G(h,j,j)\forall h,j\in \mathsf{\Omega }.\hfill \end{array}$$

(4)

Nevertheless, if $(\mathsf{\Omega },G)$ is not symmetric, then by the G-metric properties, the following holds:

$$\begin{array}{cc}\hfill \frac{3}{2}G(h,j,j)& \le d_G(h,j)\le 3G(h,j,j)\forall h,j\in \mathsf{\Omega },\hfill \end{array}$$

(5)

and hence, in general, these inequalities are sharp.

Definition 6 

([6]). $(\mathsf{\Omega },G)$ is said to be G-complete (or complete G-metric) if every G-Cauchy sequence in $(\mathsf{\Omega },G)$ is G-convergent in $(\mathsf{\Omega },G)$.

Proposition 6 

([6]). $(\mathsf{\Omega },G)$ is G-complete if and only if $(\mathsf{\Omega },d_G)$ is a complete MS.

Mustafa [25] proposed the following result in the setting of G-MS.

Theorem 1 

([25]). Given a complete $(\mathsf{\Omega },G)$, define a mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ satisfying the following constraint:

$$\begin{array}{cc}\hfill G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)& \le kG(h,j,w),\hfill \end{array}$$

(6)

for all $h,j,w\in \mathsf{\Omega }$ where $0\le k<1$, then Y has a unique FP (say u, i.e., $\mathrm{Y}u=u$), and Y is G-continuous at u.

Following in the direction of [13], the idea of F-contraction is defined as follows:

Definition 7 

([13]). Let ${\Delta }_f$ denote the family of functions $F:[0,\infty )⟶\mathbb{R}$, satisfying the following auxiliary conditions:

(F1) 

F is strictly increasing; that is, for all a, b $\in {\mathbb{R}}_{+}$, if $a<b$, then $F\left(a\right)<F\left(b\right)$;

(F2) 

for every sequence ${\left\{{\alpha }_n\right\}}_{n\in \mathbb{N}}$$\subseteq {\mathbb{R}}_{+}$, ${\displaystyle \underset{n\to \infty }{lim}{\alpha }_n=0}$ if and only if ${\displaystyle \underset{n\to \infty }{lim}F\left({\alpha }_n\right)=-\infty }$;

(F3) 

we can find $0<k<1$ such that ${\displaystyle \underset{\alpha \to 0^{+}}{lim}{\alpha }^{k}F\left(\alpha \right)=0}$.

Definition 8 

([13]). Let $(\mathsf{\Omega },d)$ be an MS. A self-mapping Y on Ω is called an F-contraction if we can find $\sigma >0$ and $F\in {\Delta }_f$ so that for all $h,j\in \mathsf{\Omega }$,

$$\begin{array}{c}\hfill d(\mathrm{Y}h,\mathrm{Y}j)>0⟹\sigma +F\left(d\right(\mathrm{Y}h,\mathrm{Y}j\left)\right)\le F\left(d\right(h,j\left)\right).\end{array}$$

(7)

Remark 1. 

From (F1) and (7), it is clear that if Y is an F-contraction, then $d(\mathrm{Y}h,\mathrm{Y}j)<d(h,j)$, for all $h,j\in \mathsf{\Omega }$ such that $\mathrm{Y}h\ne \mathrm{Y}j$. That is, Y is a contractive mapping and thus, every F-contraction is a continuous mapping.

Example 2 

([26]). Let $F\left(a\right)=\frac{-1}{\sqrt[\omega ]{a}}$, where $\omega >1$ and $a>0$. Then, $F\in {\Delta }_f$.

For related examples of the mappings in ${\Delta }_f$, see [13]. Wardowski [13] presented a variation in the Banach FP theorem as follows:

Theorem 2. 

Given a complete MS $(\mathsf{\Omega },d)$ and $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be an F-contraction. Thus, Y has a unique FP ($h\in \mathsf{\Omega }$) and for every $h_0\in \mathsf{\Omega }$, a sequence ${\left\{{\mathrm{Y}}^n{h}_0\right\}}_{n\in \mathbb{N}}$ is convergent to h.

In line with [27], a mapping $\varphi :{\mathbb{R}}_{+}⟶{\mathbb{R}}_{+}$ is called a comparison function if the following conditions are satisfied:

(1)

$\varphi$ is monotonic increasing;

(2)

${\displaystyle \underset{n\to \infty }{lim}{\varphi }^n\left(r\right)=0,\forall r>0.}$

Definition 9 

([27]). A mapping $\varphi :[0,\infty )⟶[0,\infty )$ is a $\left(c\right)$-comparison function if:

(1)

ϕ is monotonic increasing;

(2)

${\displaystyle \sum _{n=1}^{\infty }{\varphi }^n\left(r\right)<\infty ,\forall r>0.}$

It is obvious that every $\left(c\right)$-comparison function is a comparison function.

Lemma 1 

([28]). Let Φ denote the family of $\left(c\right)$-comparison functions and $\varphi \in \mathsf{\Phi }$; then, the following hold:

(i)

${\left\{{\varphi }^n\left(r\right)\right\}}_{n\in \mathbb{N}}⟶0$ as $n\to \infty$ for $r\ge 0$;

(ii)

$\varphi \left(r\right)<r$ for all $r\in {\mathbb{R}}_{+}$;

(iii)

ϕ is continuous;

(iv)

$\varphi \left(r\right)$ = 0 if and only if $r=0$;

(v)

the series ${\sum }_{n=1}^{\infty }{\varphi }^n\left(r\right)$ is convergent for $r\ge 0$.

Jiddah et al. [24] presented the following definition of Jaggi-type hybrid $(G$-$\varphi )$-contraction in G-MS.

Definition 10 

([24]). Given $(\mathsf{\Omega },G)$, a mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ is called a Jaggi-type hybrid $(G$-$\varphi )$-contraction, if we can find $\varphi \in \mathsf{\Phi }$ such that:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\le \varphi \left(M(h,j,\mathrm{Y}j)\right),\end{array}$$

(8)

for all $h,j\in \mathsf{\Omega }\backslash Fix\left(\mathrm{Y}\right)$, where:

$$M(h,j,\mathrm{Y}m)=\left\{\begin{array}{c}{\left[{\lambda }_1{\left(\frac{G(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)·G(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}{G(h,j,\mathrm{Y}j)}\right)}^q+{\lambda }_{2}G{(h,j,\mathrm{Y}j)}^q\right]}^{\frac{1}{q}},\hfill \\ forq>0;\hfill \\ \\ G{(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)}^{{\lambda }_1}·G{(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}^{{\lambda }_2},forq=0,\hfill \end{array}\right.$$

(9)

${\lambda }_1,{\lambda }_2\ge 0$ with ${\lambda }_1+{\lambda }_2=1$ and $Fix\left(\mathrm{Y}\right)=\{h\in \mathsf{\Omega }:\mathrm{Y}h=h\}$.

3. Main Results

In this section, we introduce the concept of Jaggi-type hybrid $(\varphi -F)$-contraction in the framework of G-MS and examine constraints for the existence of invariant points for such operators.

Definition 11. 

A mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ defined on $(\mathsf{\Omega },G)$ is called a Jaggi-type hybrid $(\varphi -F)$-contraction if we can find $\varphi \in \mathsf{\Phi }$, $F\in {\Delta }_f$ and $\sigma >0$, such that $G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)>0$ implies:

$$\begin{array}{c}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)\le F\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right),\end{array}$$

(10)

for all $h,j\in \mathsf{\Omega }\backslash Fix\left(\mathrm{Y}\right)$, where:

$$M(h,j,\mathrm{Y}m)=\left\{\begin{array}{c}{\left[{\lambda }_1{\left(\frac{G(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)·G(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}{G(h,j,\mathrm{Y}j)}\right)}^q+{\lambda }_{2}G{(h,j,\mathrm{Y}j)}^q\right]}^{\frac{1}{q}},\hfill \\ forq>0;\hfill \\ \\ G{(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)}^{{\lambda }_1}·G{(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}^{{\lambda }_2},forq=0,\hfill \end{array}\right.$$

(11)

${\lambda }_1,{\lambda }_2\ge 0$ with ${\lambda }_1+{\lambda }_2=1$ and $Fix\left(\mathrm{Y}\right)=\{h\in \mathsf{\Omega }:\mathrm{Y}h=h\}$.

If we consider variants of the mapping $F\in {\Delta }_f$, we obtain some examples of Jaggi-type hybrid $(\varphi -F)$-contractions. We list some of the special cases in the following examples.

Example 3. 

Define $F:[0,\infty )⟶\mathbb{R}$ by $F\left(t\right)=ln\left(t\right),t>0$. It is easy to see that $F\in {\Delta }_f$. Each mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ satisfying (10) is a Jaggi-type hybrid $(\varphi -F)$-contraction such that:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\le e^{-\sigma }\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right),\end{array}$$

(12)

for all $h,j\in \mathsf{\Omega }$. Note that for $h,j\in \mathsf{\Omega }$, such that $\mathrm{Y}h=\mathrm{Y}j={\mathrm{Y}}^{2}j$, the inequality (10) is still valid. That is, Y is a Jaggi-type hybrid $(\varphi -F)$-contraction.

Example 4. 

Suppose $F\left(t\right)=t+ln\left(t\right),t>0$ then $F\in {\Delta }_f$; thus, the inequality (10) takes the version:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)e^{G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)-\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right)}\le e^{-\sigma }\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right),\end{array}$$

(13)

for all $h,j\in \mathsf{\Omega }$.

Example 5. 

Let $F\left(t\right)=ln(t^2+t),t>0$. Clearly, $F\in {\Delta }_f$ and the inequality (10) takes the form:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)+1)\le e^{-\sigma }(\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right)\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right)+1),\end{array}$$

(14)

for all $h,j\in \mathsf{\Omega }$.

The following is one of our main results.

Theorem 3. 

Suppose that $(\mathsf{\Omega },G)$ is complete and the mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ is a continuous Jaggi-type hybrid $(\varphi -F)$-contraction, then Y has a unique FP u in Ω, and for any $u_0\in \mathsf{\Omega }$, the sequence ${\left\{{\mathrm{Y}}^n{u}_0\right\}}_{n\in \mathbb{N}}$ converges to u.

Proof. 

For arbitrary $h_0\in \mathsf{\Omega }$, define a sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ in $\mathsf{\Omega }$ by $h_n={\mathrm{Y}}^n{h}_0$. If we can find some $n_0\in \mathbb{N}$ for which $h_{n_0+1}=h_{n_0}$, then $\mathrm{Y}{h}_{n_0}=h_{n_0}$ and the proof is complete. Assume on the contrary that $h_n\ne h_{n-1}$ for all $n\in \mathbb{N}$. Since $\mathrm{Y}$ is a Jaggi-type hybrid $(\varphi -F)$-contraction, then we have from (10) that:

$$\begin{array}{c}\hfill 0<G(h_n,h_{n+1},h_{n+2})⟹\sigma +F\left(G(\mathrm{Y}{h}_{n-1},\mathrm{Y}{h}_n,{\mathrm{Y}}^2{h}_n)\right)\le F\left(\varphi \left(M(h_{n-1},h_n,\mathrm{Y}{h}_n)\right)\right).\end{array}$$

We then consider the following cases of (11). Case 1: for $q>0$, taking $h=h_{n-1}$ and $m=h_n$ we have:

$$\begin{array}{cc}\hfill M(h_{n-1},h_n,\mathrm{Y}{h}_n)=& {\left[{\lambda }_1{\left(\frac{G(h_{n-1},\mathrm{Y}{h}_{n-1},{\mathrm{Y}}^2{h}_{n-1})G(h_n,\mathrm{Y}{h}_n,{\mathrm{Y}}^2{h}_n)}{G(h_{n-1},h_n,\mathrm{Y}{h}_n)}\right)}^q+{\lambda }_{2}G{(h_{n-1},h_n,\mathrm{Y}{h}_n)}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill =& {\left[{\lambda }_1{\left(\frac{G(h_{n-1},h_n,h_{n+1})G(h_n,h_{n+1},h_{n+2})}{G(h_{n-1},h_n,h_{n+1})}\right)}^q+{\lambda }_{2}G{(h_{n-1},h_n,h_{n+1})}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill =& {\left[{\lambda }_{1}G{(h_n,h_{n+1},h_{n+2})}^q+{\lambda }_{2}G{(h_{n-1},h_n,h_{n+1})}^q\right]}^{\frac{1}{q}}.\hfill \end{array}$$

(15)

Assume that

$$\begin{array}{c}\hfill G(h_{n-1},h_n,h_{n+1})\le G(h_n,h_{n+1},h_{n+2}).\end{array}$$

Then, from (10) and (15), we have

$$\begin{array}{cc}\hfill & F\left(G(\mathrm{Y}{h}_{n-1},\mathrm{Y}{h}_n,{\mathrm{Y}}^2{h}_n)\right)\hfill \\ \hfill & \le F\left(\varphi \left(M(h_{n-1},h_n,\mathrm{Y}{h}_n)\right)\right)-\sigma \hfill \\ \hfill & =F\left(\varphi {\left[{\lambda }_{1}G{(h_n,h_{n+1},h_{n+2})}^q+{\lambda }_{2}G{(h_{n-1},h_n,h_{n+1})}^q\right]}^{\frac{1}{q}}\right)-\sigma \hfill \\ \hfill & \le F\left(\varphi {\left[{\lambda }_{1}G{(h_n,h_{n+1},h_{n+2})}^q+{\lambda }_{2}G{(h_n,h_{n+1},h_{n+2})}^q\right]}^{\frac{1}{q}}\right)-\sigma \hfill \\ \hfill & \le F\left(\varphi \left[{({\lambda }_1+{\lambda }_2)}^{\frac{1}{q}}G(h_n,h_{n+1},h_{n+2})\right]\right)-\sigma \hfill \\ \hfill & \le F\left(\varphi \left[G(h_n,h_{n+1},h_{n+2})\right]\right)-\sigma \hfill \\ \hfill & <F\left(G(h_n,h_{n+1},h_{n+2})\right).\hfill \end{array}$$

That is, $F\left(G(h_n,h_{n+1},h_{n+2})\right)<F\left(G(h_n,h_{n+1},h_{n+2})\right)$, which is a contradiction. Therefore, $max\{G(h_{n-1},h_n,h_{n+1}),G(h_n,h_{n+1},h_{n+2})\}=G(h_{n-1},h_n,h_{n+1})$. So (10) becomes:

$$\begin{array}{cc}\hfill F\left(G(h_n,h_{n+1},h_{n+2})\right)& \le F\left(\varphi {\left[{\lambda }_{1}G{(h_n,h_{n+1},h_{n+2})}^q+{\lambda }_{2}G{(h_{n-1},h_n,h_{n+1})}^q\right]}^{\frac{1}{q}}\right)-\sigma \hfill \\ \hfill & \le F\left(\varphi \left({({\lambda }_1+{\lambda }_2)}^{\frac{1}{q}}G(h_{n-1},h_n,h_{n+1})\right)\right)-\sigma \hfill \\ \hfill & \le F(\varphi \left(G(h_{n-1},h_n,h_{n+1})\right)-\sigma .\hfill \end{array}$$

(16)

Let ${\zeta }_n=G(h_n,h_{n+1},h_{n+2})$, then,

$$\begin{array}{c}\hfill F\left({\zeta }_n\right)\le F(\varphi \left({\zeta }_{n-1}\right)-\sigma \le F({\varphi }^2\left({\zeta }_{n-2}\right)-2\sigma \le \cdots \le F\left({\varphi }^n\left({\zeta }_0\right)\right)-n\sigma \end{array}$$

(17)

for all $n\in \mathbb{N}$ with $h_{n+1}\ne h_{n+2}$. From (17), we have

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}F\left({\zeta }_n\right)}& {\displaystyle \le \underset{n\to \infty }{lim}F\left({\varphi }^n\left({\zeta }_0\right)\right)-\underset{n\to \infty }{lim}n\sigma }\hfill \\ \hfill & {\displaystyle =F\underset{n\to \infty }{lim}\left({\varphi }^n\left({\zeta }_0\right)\right)-\underset{n\to \infty }{lim}n\sigma }\hfill \\ \hfill & =-\infty .\hfill \end{array}$$

(18)

Together with $\left(F2\right)$ gives:

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}{\zeta }_n=0.}\end{array}$$

(19)

From $\left(F3\right)$ we can find $k\in (0,1)$ such that

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}{{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_n\right)\right)=0.}\end{array}$$

(20)

By (17), the following holds for all $n\in \mathbb{N}$:

$$\begin{array}{cc}\hfill {{\zeta }_n}^{k}F\left({\zeta }_n\right)-{{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_0\right)\right)& \le {{\zeta }_n}^k(F\left({\varphi }^n\left({\zeta }_0\right)\right)-n\sigma )-{{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_0\right)\right)\hfill \\ \hfill & =-{{\zeta }_n}^{k}n\sigma \le 0\hfill \end{array}$$

$$\begin{array}{c}\hfill n{{\zeta }_n}^k=\frac{{{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_0\right)\right)-{{\zeta }_n}^{k}F\left({\zeta }_n\right)}{\sigma }\end{array}$$

(21)

Letting $n\to \infty$ in (21) and using (19) and (20),

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}n{{\zeta }_n}^k=}& {\displaystyle \underset{n\to \infty }{lim}\left[\frac{{{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_0\right)\right)-{{\zeta }_n}^{k}F\left({\zeta }_n\right)}{\sigma }\right]}\hfill \\ \hfill =& {\displaystyle \frac{1}{\sigma }\left[\underset{n\to \infty }{lim}\left({{\zeta }_n}^{k}F\left({\varphi }^n\left({\zeta }_0\right)\right)\right)-{\displaystyle \underset{n\to \infty }{lim}{{\zeta }_n}^{k}F\left({\zeta }_n\right)}\right]}\hfill \\ \hfill =& 0.\hfill \end{array}$$

(22)

Note that from (22), we can find $n_1\in \mathbb{N}$ such that $n{{\zeta }_n}^k\le 1$ for all $n\ge n_1$. Consequently, we have

$$\begin{array}{c}\hfill {\zeta }_n\le \frac{1}{n^{\frac{1}{k}}},\forall n\ge n_1\end{array}$$

(23)

To show that ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is a G-Cauchy sequence, consider $m,n\in \mathbb{N}$, such that $m>n\ge n_1$. From the definition of the metric and from (23), we have:

$$\begin{array}{cc}\hfill G(h_n,h_n,h_m)& \le G(h_n,h_{n-1},h_{n-1})+G(h_{n-1},h_{n-2},h_{n-2})+\cdots +G(h_{m-1},h_{m-1},h_m)\hfill \\ \hfill & ={\zeta }_n+{\zeta }_{n-1}+{\zeta }_{n-2}+\cdots +{\zeta }_{m-1}=\sum _{i=n}^{m-1}{\zeta }_i\le \sum _{i=n}^{\infty }{\zeta }_i\hfill \\ \hfill & \le \sum _{i=n}^{\infty }\frac{1}{i^{\frac{1}{k}}}.\hfill \end{array}$$

(24)

From (24), we see that the series ${\displaystyle \sum _{i=n}^{\infty }\frac{1}{i^{\frac{1}{k}}}}$ is G-convergent. Then, the sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ is a G-Cauchy sequence in $(\mathsf{\Omega },G)$ and so by the completeness of $(\mathsf{\Omega },G)$, we can find $u\in \mathsf{\Omega }$ such that $\left\{h_n\right\}$ converges to u. Hence, ${\displaystyle \underset{n\to \infty }{lim}G(h_n,h_n,u)=0}$. Next, to show that u is an FP of $\mathrm{Y}$. Using the continuity of $\mathrm{Y}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}G(u,u,\mathrm{Y}u)=}& {\displaystyle \underset{n\to \infty }{lim}G(h_{n+1},h_{n+1},\mathrm{Y}u)}\hfill \\ \hfill =& {\displaystyle \underset{n\to \infty }{lim}G(\mathrm{Y}{h}_n,\mathrm{Y}{h}_n,\mathrm{Y}u)}\hfill \\ \hfill =& {\displaystyle \underset{n\to \infty }{lim}G(\mathrm{Y}{h}_n,\mathrm{Y}{h}_n,\mathrm{Y}{h}_n)}\hfill \\ \hfill =& 0.\hfill \end{array}$$

So we obtain $\mathrm{Y}u=u$; that is, u is a FP of $\mathrm{Y}$.

Case 2: for $q=0$, we have

$$\begin{array}{cc}\hfill M(h_{n-1},h_n,\mathrm{Y}{h}_n)& =G{(h_{n-1},\mathrm{Y}{h}_{n-1},{\mathrm{Y}}^2{h}_{n-1})}^{{\lambda }_1}G{(h_n,\mathrm{Y}{h}_n,{\mathrm{Y}}^2{h}_n)}^{{\lambda }_2}\hfill \\ \hfill & =G{(h_{n-1},h_n,h_{n+1})}^{{\lambda }_1}G{(h_n,h_{n+1},h_{n+2})}^{{\lambda }_2}.\hfill \end{array}$$

(25)

If $G(h_{n-1},h_n,h_{n+1})\le G(h_n,h_{n+1},h_{n+2})$, from (10) and (25) we have

$$\begin{array}{cc}\hfill F\left(G(\mathrm{Y}{h}_{n-1},\mathrm{Y}{h}_n,{\mathrm{Y}}^2{h}_n)\right)& \le F\left(\varphi \left(M(h_{n-1},h_n,\mathrm{Y}{h}_n)\right)\right)-\sigma \hfill \\ \hfill & =F\left(\varphi \left[G{(h_{n-1},h_n,h_{n+1})}^{{\lambda }_1}G{(h_n,h_{n+1},h_{n+2})}^{{\lambda }_2}\right]\right)-\sigma \hfill \\ \hfill & \le F\left(\varphi \left[G{(h_n,h_{n+1},h_{n+2})}^{{\lambda }_1}G{(h_n,h_{n+1},h_{n+2})}^{{\lambda }_2}\right]\right)-\sigma \hfill \\ \hfill & =F\left(\varphi \left[G{(h_n,h_{n+1},h_{n+2})}^{{\lambda }_1+{\lambda }_2}\right]\right)-\sigma \hfill \\ \hfill & =F\left(\varphi \left[G(h_n,h_{n+1},h_{n+2})\right]\right)-\sigma \hfill \\ \hfill & <F\left(G(h_n,h_{n+1},h_{n+2})\right).\hfill \end{array}$$

i.e., $F\left(G(h_n,h_{n+1},h_{n+2})\right)<F\left(G(h_n,h_{n+1},h_{n+2})\right)$, which is a contradiction. Therefore,

$$\begin{array}{c}\hfill G(h_n,h_{n+1},h_{n+2})<G(h_{n-1},h_n,h_{n+1}),\forall n.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill F\left(G(h_n,h_{n+1},h_{n+2})\right)& <F\left(\varphi \left(G(h_{n-1},h_n,h_{n+1})\right)\right)-\sigma <F\left({\varphi }^2\left(G(h_{n-2},h_{n-1},h_n)\right)\right)-2\sigma \hfill \\ \hfill & <\cdots <F\left({\varphi }^n\left(G(h_0,h_1,h_2)\right)\right)-n\sigma .\hfill \end{array}$$

Using the same justification as in the case of $q>0$, we can establish the existence of a G-Cauchy sequence $\left\{h_n\right\}$ in $(\mathsf{\Omega },G)$ and a point $u\in \mathsf{\Omega }$ such that ${\displaystyle \underset{n\to \infty }{lim}\left\{h_n\right\}=u}$. In a similar manner, under the hypothesis that $\mathrm{Y}$ is continuous and supported by the uniqueness of limit, we have $\mathrm{Y}u=u$. That is, u is a FP of $\mathrm{Y}$.

Assume that $\mathrm{Y}$ has more than one FP. Indeed, if $u_1,u_2\in \mathsf{\Omega }$ such that $\mathrm{Y}{u}_1=u_1\ne u_2=\mathrm{Y}{u}_2$, then we obtain

$$\begin{array}{cc}\hfill & \sigma +F\left(G(\mathrm{Y}{u}_1,\mathrm{Y}{u}_2,{\mathrm{Y}}^2{u}_2)\right)\le F\left(\varphi \left(M(u_1,u_2,\mathrm{Y}{u}_2)\right)\right)\hfill \\ \hfill \sigma & \le F\left(\varphi \left(M(u_1,u_2,\mathrm{Y}{u}_2)\right)\right)-F\left(G(\mathrm{Y}{u}_1,\mathrm{Y}{u}_2,{\mathrm{Y}}^2{u}_2)\right)\hfill \\ \hfill & \le F\left(\varphi \left(M(u_1,u_2,\mathrm{Y}{u}_2)\right)\right)-F\left(G(u_1,u_2,u_2)\right)\hfill \\ \hfill & <F\left(M(u_1,u_2,\mathrm{Y}{u}_2)\right)-F\left(G(u_1,u_2,u_2)\right).\hfill \end{array}$$

(26)

Case 1: For $q>0$,

$$\begin{array}{cc}\hfill M(u_1,u_2,\mathrm{Y}{u}_2)=& {\left[{\lambda }_1{\left(\frac{G(u_1,\mathrm{Y}{u}_1,{\mathrm{Y}}^2{u}_1)G(u_2,\mathrm{Y}{u}_2,{\mathrm{Y}}^2{u}_2)}{G(u_1,u_2,\mathrm{Y}{u}_2)}\right)}^q+{\lambda }_2{\left(G(u_1,u_2,\mathrm{Y}{u}_2)\right)}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill =& {\left[{\lambda }_1{\left(\frac{G(u_1,u_1,u_1)G(u_2,u_2,u_2)}{G(u_1,u_2,u_2)}\right)}^q+{\lambda }_2{\left(G(u_1,u_2,u_2)\right)}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill =& {\left[{\lambda }_2{\left(G(u_1,u_2,u_2)\right)}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill =& {\lambda }_2^{\frac{1}{q}}G(u_1,u_2,u_2)\le G(u_1,u_2,u_2).\hfill \end{array}$$

Hence, (26) becomes

$$\begin{array}{c}\hfill \sigma <F\left(G(u_1,u_2,u_2)\right)-F\left(G(u_1,u_2,u_2)\right)=0.\end{array}$$

That is, $\sigma <0$, a contradiction. Case 2: for $q=0$

$$\begin{array}{cc}\hfill M(u_1,u_2,\mathrm{Y}{u}_2)=& G{(u_1,\mathrm{Y}{u}_1,{\mathrm{Y}}^2{u}_1)}^{{\lambda }_1}.G{(u_2,\mathrm{Y}{u}_2,{\mathrm{Y}}^2{u}_2)}^{{\lambda }_2}\hfill \\ \hfill =& G{(u_1,u_1,u_1)}^{{\lambda }_1}.G{(u_2,u_2,u_2)}^{{\lambda }_2}=0.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \sigma \le & F\left(\varphi \left(M(u_1,u_2,\mathrm{Y}{u}_2)\right)\right)-F\left(G(u_1,u_2,u_2)\right)\hfill \\ \hfill \le & F\left(\varphi \left(0\right)\right)-F\left(G(u_1,u_2,u_2)\right)\hfill \\ \hfill <& F\left(0\right)-F\left(G(u_1,u_2,u_2)\right).\hfill \end{array}$$

From $\left(G2\right)$ and $\left(F1\right)$, we have that:

$$\begin{array}{c}\hfill 0<G(u_1,u_2,u_2)\mathrm{implies}F\left(0\right)<F\left(G(u_1,u_2,u_2)\right).\end{array}$$

It follows that:

$$\begin{array}{c}\hfill \sigma <F\left(0\right)-F\left(G(u_1,u_2,u_2)\right)<0,\end{array}$$

which is a contradiction. Hence, we conclude that $\mathrm{Y}$ has a unique FP in $(\mathsf{\Omega },G)$. □

Theorem 4. 

Let $(\mathsf{\Omega },G)$ be complete and $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be a Jaggi-type hybrid $(\varphi -F)$-contraction. If Y is a continuous mapping, we can find $\eta \in [0,1)$ such that for all $h,j\in \mathsf{\Omega }$,

$$\begin{array}{c}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)\le F\left(\eta \left(M(h,j,\mathrm{Y}j)\right)\right),\end{array}$$

(27)

where $\sigma >0$, $F\in {\Delta }_f$ and $M(h,j,\mathrm{Y}j)$ is as given in (11). Then, Y has a unique FP in Ω.

Proof. 

Define a mapping $\varphi :{\mathbb{R}}_{+}⟶\mathbb{R}$ by $\varphi \left(r\right)=\eta r,\forall r\ge 0and\eta \in [0,1)$. Clearly, $\varphi$ is a $\left(c\right)$-comparison function. Consequently, by Therorem 3, $\mathrm{Y}$ has a unique FP in $\mathsf{\Omega }$. □

In what follows, we construct an example to support the hypothesis of Theorems 3 and 4.

Example 6. 

Let $\mathsf{\Omega }=[0,\infty )$ and $G(h,j,\mathrm{Y}m)=|h-j|+|h-\mathrm{Y}j|+|j-\mathrm{Y}j|$, $\forall h,j\in \mathsf{\Omega }$. Then $(\mathsf{\Omega },G)$ is complete. Take $\sigma >0$, and consider the mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ defined by:

$$\begin{array}{c}\hfill \mathrm{Y}h=\left\{\begin{array}{cc}\frac{2}{5}h{e}^{-\sigma },\hfill & ifh\in [0,1];\hfill \\ \frac{2}{5}{e}^{-\sigma },\hfill & ifh>1\hfill \end{array}\right.\end{array}$$

for all $h\in \mathsf{\Omega }$. Then, Y is continuous.

Let $\varphi \left(r\right)=\frac{2}{5}r,forallr>0$, and $F\left(t\right)=ln(t^2+t),t\in (0,\infty )$. Obviously, ϕ is a $\left(c\right)$-comparison function and $F\in {\Delta }_f$.

Notice that for all $h,j\in (1,\infty )$, there is nothing to show. So for all $h,j\in [0,1]$, let ${\lambda }_1=0,{\lambda }_2=1$. Then, to show that the mapping Y is a Jaggi-type hybrid $(\varphi -F)$ contraction, we examine the following two cases.

Case 1: For $q>0$, consider $q=1$. Then,

$$\begin{array}{cc}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)& =\left|\mathrm{Y}h-\mathrm{Y}j\right|+|\mathrm{Y}h-{\mathrm{Y}}^{2}j|+\left|\mathrm{Y}h-{\mathrm{Y}}^{2}j\right|\hfill \\ \hfill & =|\frac{2}{5}h{e}^{-\sigma }-\frac{2}{5}j{e}^{-\sigma }|+|\frac{2}{5}h{e}^{-\sigma }|-\frac{4}{25}j{e}^{-2\sigma }+\left|\frac{2}{5}j{e}^{-\sigma }-\frac{4}{25}j{e}^{-2\sigma }\right|\hfill \\ \hfill & =\frac{2}{5}{e}^{-\sigma }\left[\left|h-j\right|+\left|h-\frac{2}{5}j{e}^{-\sigma }\right|+\left|j-\frac{2}{5}j{e}^{-\sigma }\right|\right]\hfill \\ \hfill & =e^{-\sigma }\varphi \left(G\right(h,j,\mathrm{Y}j\left)\right)\hfill \\ \hfill & \le e^{-\sigma }\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right).\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill [G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)+1]& =\left[\frac{2}{5}{e}^{-\sigma }\left(\left|h-j\right|+\left|h-\frac{2}{5}j{e}^{-\sigma }\right|+\left|j-\frac{2}{5}j{e}^{-\sigma }\right|\right)+1\right]\hfill \\ \hfill & \le \left[e^{-\sigma }\varphi \left(G\right(h,j,\mathrm{Y}j\left)\right)+1\right]\hfill \\ \hfill & =\left[e^{-\sigma }\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)+1\right].\hfill \end{array}$$

(29)

From (28) and (29),

$$\begin{array}{cc}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j).[G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)+1]& =e^{-\sigma }\varphi \left(G\right(h,j,\mathrm{Y}j\left)\right).\left[e^{-\sigma }\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)+1\right]\hfill \\ \hfill & \le e^{-\sigma }\left[\varphi \left(G\right(h,j,\mathrm{Y}j\left)\right).\left(\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)+1\right)\right].\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)& =\sigma +ln(G{(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}^2+G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j))\hfill \\ \hfill & \le \sigma +ln\left[e^{-\sigma }({\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)}^2+\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right))\right]\hfill \\ \hfill & =ln\left[{\left(\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)\right)}^2+\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)\right]\hfill \\ \hfill & =F\left(\varphi \right(M(h,j,\mathrm{Y}j)\left)\right).\hfill \end{array}$$

Case 2: Similarly, for $q=0$, ${\lambda }_1=0$, ${\lambda }_2=1$ and $h={\mathrm{Y}}^{2}j,\forall h,j\in \mathsf{\Omega }$ we obtain:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\le e^{-\sigma }\varphi \left(M(h,j,\mathrm{Y}j)\right).\end{array}$$

(30)

Similarly to Case 1, the inequality (30) gives

$$\begin{array}{c}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)\le F\left(\varphi \left(M(h,j,\mathrm{Y}j)\right)\right).\end{array}$$

In Figure 1, we demonstrate the validity of the contractive inequality (10) using Example 6.

Thus, all the assumptions of Theorem (3) are satisfied. Consequently, we see that $h=0$ is the unique FP of Y.

On the other hand, it is easy to check that the main result of Jiddah et al. [24] is not applicable to this example. In fact, suppose that the mapping Y is a Jaggi-type hybrid $(G$-$\varphi )$-contraction; that is, for all $h,j\in \mathsf{\Omega }/Fix\left(\mathrm{Y}\right)$,

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\le \varphi \left(M(h,j,\mathrm{Y}j)\right)\end{array}$$

(31)

where

$$M(h,j,\mathrm{Y}m)=\left\{\begin{array}{c}{\left[{\lambda }_1{\left(\frac{G(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)·G(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}{G(h,j,\mathrm{Y}j)}\right)}^q+{\lambda }_{2}G{(h,j,\mathrm{Y}j)}^q\right]}^{\frac{1}{q}},\hfill \\ forq>0;\hfill \\ \\ G{(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)}^{{\lambda }_1}·G{(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}^{{\lambda }_2},forq=0,\hfill \end{array}\right.$$

${\lambda }_1,{\lambda }_2\ge 0$ with ${\lambda }_1+{\lambda }_2=1$.

Then, for the chosen parameters ${\lambda }_1=0$, ${\lambda }_2=1$, $q=1$, we take $h=\frac{2}{5}{e}^{-\sigma }$ and $j=\frac{4}{5}{e}^{-\sigma }$, for $\sigma >0$. By direct calculation, we have

$$\begin{array}{cc}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)& =\left|\mathrm{Y}h-\mathrm{Y}j\right|+\left|\mathrm{Y}h-{\mathrm{Y}}^{2}j\right|+\left|\mathrm{Y}j-{\mathrm{Y}}^{2}j\right|\hfill \\ \hfill & =\left|\frac{4}{25}{e}^{-2\sigma }-\frac{8}{25}{e}^{-2\sigma }\right|+\left|\frac{4}{25}{e}^{-2\sigma }-\frac{16}{125}{e}^{-3\sigma }\right|\hfill \\ \hfill & +\left|\frac{8}{25}{e}^{-2\sigma }-\frac{16}{125}{e}^{-3\sigma }\right|\hfill \\ \hfill & =\frac{16}{25}{e}^{-2\sigma }-\frac{32}{125}{e}^{-3\sigma }\hfill \\ \hfill & =\frac{80e^{-2\sigma }-32e^{-3\sigma }}{125}.\hfill \end{array}$$

(32)

Similarly,

$$\begin{array}{cc}\hfill M(h,j,\mathrm{Y}m)=G(h,j,\mathrm{Y}j)& =|h-j|+\left|h-\mathrm{Y}j\right|+\left|j-\mathrm{Y}j\right|\hfill \\ \hfill & =\left|\frac{2}{5}{e}^{-\sigma }-\frac{4}{5}{e}^{-\sigma }\right|+\left|\frac{2}{5}{e}^{-\sigma }-\frac{8}{25}{e}^{-2\sigma }\right|\hfill \\ \hfill & +\left|\frac{4}{5}{e}^{-\sigma }-\frac{8}{25}{e}^{-2\sigma }\right|\hfill \\ \hfill & =\frac{8}{5}{e}^{-\sigma }-\frac{16}{25}{e}^{-2\sigma }\hfill \\ \hfill & =\frac{40e^{-\sigma }-16e^{-2\sigma }}{25}.\hfill \end{array}$$

(33)

$$\begin{array}{c}\hfill \varphi \left(M(h,j,\mathrm{Y}j)\right)=\frac{16}{125}\left[5e^{-\sigma }-2e^{-2\sigma }\right]\end{array}$$

(34)

By (31),

$$\begin{array}{c}\hfill \frac{G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}{\varphi \left(M\right(h,j,\mathrm{Y}j\left)\right)}\le 1.\end{array}$$

From (32)–(34), we have

$$\begin{array}{c}\hfill \frac{80e^{-2\sigma }-32e^{-3\sigma }}{16\left[5e^{-\sigma }-2e^{-2\sigma }\right]}\le 1.\end{array}$$

(35)

Letting $\sigma ⟶\infty$ in (35), gives $\infty \le 1$ a contradiction.

The following are some immediate consequences of our results.

Corollary 1. 

Let $(\mathsf{\Omega },G)$ be complete and $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be a continuous mapping. If we can find $\varphi \in \mathsf{\Phi }$ and $F\in {\Delta }_f$ such that:

$$\begin{array}{cc}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)& \le F\left(\varphi \left(M_1(h,j,\mathrm{Y}m)\right)\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill M_1(h,j,\mathrm{Y}m)={\left[{\lambda }_1{\left(\frac{G(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)G(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}{G(h,j,\mathrm{Y}j)}\right)}^q+{\lambda }_{2}G{(h,j,\mathrm{Y}j)}^q\right]}^{\frac{1}{q}},\\ \hfill for\mathrm{some}q>0;\end{array}$$

for all $h,j\in \mathsf{\Omega }$ and $q>0$, then Y has a unique FP (say u) in Ω.

Proof. 

Consider Case 1 of Theorem 3. □

Corollary 2. 

Suppose $(\mathsf{\Omega },G)$ is complete and the mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ is continuous. If we can find $\varphi \in \mathsf{\Phi }$ and $F\in {\Delta }_f$ such that:

$$\begin{array}{cc}\hfill \sigma +F\left(G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\right)& \le F\left(\varphi \left(M_2(h,j,\mathrm{Y}m)\right)\right),\hfill \end{array}$$

where

$$M_2(h,j,\mathrm{Y}m)=\left[G{(h,\mathrm{Y}h,{\mathrm{Y}}^{2}h)}^{{\lambda }_1}G{(j,\mathrm{Y}j,{\mathrm{Y}}^{2}j)}^{{\lambda }_2}\right],$$

for all $h,j\in \mathsf{\Omega }$ and $q=0$, then Y has a unique FP (say u) in Ω.

Proof. 

See Case 2 of Theorem 3. □

Corollary 3. 

Given $(\mathsf{\Omega },G)$ to be complete and $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be a continuous mapping satisfying the following condition:

$$\begin{array}{c}\hfill \sigma +F\left(G\right(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w\left)\right)\le F\left(\eta \right(G(h,j,w)\left)\right),\end{array}$$

$\forall h,j,w\in \mathsf{\Omega },\sigma >0andF\in {\Delta }_f$.

Proof. 

Consider (10) for $q>0$. Let $\mathrm{Y}j=w,{\lambda }_1=0,{\lambda }_2=1,\varphi \left(r\right)=\eta rforall$

$r\ge 0$ and for some $\eta \in [0,1)$. Then, the conclusion follows. □

We point out a few more consequences of our principal findings as follows.

Corollary 4 

([24]). Let $(\mathsf{\Omega },G)$ be complete and let $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be a continuous Jaggi-type hybrid (G $- \varphi )$ contraction. Then Y has a FP in Ω.

Proof. 

It is enough to take $F\left(t\right)=ln\left(t\right),t>0$ in Theorem 3. □

Corollary 5 

([24]). Suppose $(\mathsf{\Omega },G)$ is complete and $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ be a continuous mapping satisfying the following condition:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\le \eta \left(G\right(h,j,w\left)\right),\end{array}$$

for all $h,j,w\in \mathsf{\Omega }where\eta \in [0,1)$, then Y has a unique FP in Ω.

Proof. 

Take $F\left(t\right)=ln\left(t\right),t>0$ in Corollary 3. □

Corollary 6 

([24]). Let $(\mathsf{\Omega },G)$ be complete and if we can find $\eta \in (0,1)$ such that for all $h,j\in \mathsf{\Omega }$ the continuous mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ satisfying:

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,{\mathrm{Y}}^{2}j)\le \eta \left(G(h,j,\mathrm{Y}j)\right),\end{array}$$

then Y has a unique FP in Ω.

Proof. 

Take $F\left(t\right)=ln\left(t\right),t>0andw=\mathrm{Y}j$ in Corollary 3. □

4. Applications to a Nonlinear Volterra Integral Equation

One major significance of FP theory is seen in the provision of essential tools for solving problems arising in proving the existence of solutions of integral and differential equations. Aydi et al. [29] presented a typical application of FP methods to the study of the existence of solutions for integral equations. Furthermore, Mohammed et al. [23] discussed the criteria for analysing the solutions to certain functional equations arising in dynamic programming, and integral equations of Volterra-type exist. In this section, we examine the existence and uniqueness criteria for a solution to a class of integral equations via the aid of Corollary 3.

Consider the following integral equation of Volterra-type:

$$\begin{array}{c}\hfill h\left(t\right)=f\left(t\right)+{\int }_0^t\mathcal{L}(t,s,h\left(s\right))ds,t\in [0,b]=\mathcal{W},\end{array}$$

(36)

where $b>0$, $\mathcal{L}:\mathcal{W}\times \mathcal{W}\times \mathbb{R}⟶\mathbb{R}$ and $f:\mathcal{W}⟶\mathbb{R}$.

Let $\mathsf{\Omega }=C(\mathcal{W},\mathbb{R})$ be the space of all continuous real-valued functions defined on $\mathcal{W}$. For arbitrary $h\in \mathsf{\Omega }$, define ${\displaystyle {\parallel h\parallel }_{\sigma }=\underset{t\in \mathcal{W}}{sup}\left\{\left|h\right(t)e^{-\sigma t}\right|\}}$, where $\sigma >0$. ${\parallel h\parallel }_{\sigma }$ is a norm equivalent to the supremum norm and $\mathsf{\Omega }$ equipped with the metric G defined by:

$$\begin{array}{c}\hfill {\displaystyle G(h,j,w)=\underset{t\in \mathcal{W}}{sup}\left\{\left(\left|h\right(t)-j(t\left)\right|+\left|h\right(t)-w(t\left)\right|+\left|j\right(t)-w(t\left)\right|\right)e^{-\sigma t}\right\}}\\ \hfill \forall h,j,w\in \mathsf{\Omega },t\in \mathcal{W},\end{array}$$

(37)

is a complete G-MS. Define a mapping $\mathrm{Y}:\mathsf{\Omega }⟶\mathsf{\Omega }$ as follows.

$$\begin{array}{c}\hfill \mathrm{Y}h\left(t\right)=f\left(t\right)+{\int }_0^t\mathcal{L}(t,s,h\left(s\right))ds,t\in \mathcal{W},h\in \mathsf{\Omega }.\end{array}$$

(38)

Then, a point u is said to be an FP of $\mathrm{Y}$ if and only if u is a solution to (36).

Now, we study the existence conditions of the integral Equation (36) under the following assumptions.

Theorem 5. 

Suppose that the following conditions are satisfied:

(i)

$\mathcal{L}:\mathcal{W}\times \mathcal{W}\times \mathbb{R}⟶\mathbb{R}$ and $f:\mathcal{W}⟶\mathbb{R}$ are continuous;

(ii)

we can find $\sigma >0$ such that for all $s,t\in \mathcal{W}$ and $h,j\in \mathbb{R}$,

$$\begin{array}{c}\hfill |\mathcal{L}(t,s,h\left(s\right))-\mathcal{L}(t,s,j\left(s\right))|\le \frac{\sigma }{k}{e}^{-\sigma }|h\left(s\right)-j\left(s\right)|\end{array}$$

where k > 1.

Then, the integral Equation (36) has a unique solution u in Ω.

Proof. 

Let $h,j,w\in \mathsf{\Omega }$ such that $\mathrm{Y}h\ne \mathrm{Y}j\ne \mathrm{Y}w$. Then, using (38) and the above assumptions, we obtain:

$$\begin{array}{cc}\hfill \left|\mathrm{Y}h\right(t)-\mathrm{Y}j(t\left)\right|=& \left|{\int }_0^t\left[\mathcal{L}(t,s,h(s\left)\right)-\mathcal{L}(t,s,j(s\left)\right)\right]ds\right|\hfill \\ \hfill \le & {\int }_0^t\left|\mathcal{L}(t,s,h(s\left)\right)-\mathcal{L}(t,s,j(s\left)\right)\right|ds\hfill \\ \hfill \le & {\int }_0^t\frac{\sigma }{k}{e}^{-\sigma }\left|h\left(s\right)-j\left(s\right)\right|ds\hfill \\ \hfill \le & {\int }_0^t\frac{\sigma }{k}{e}^{-\sigma }\left|h\left(s\right)-j\left(s\right)\right|e^{-\sigma s}.e^{\sigma s}ds\hfill \\ \hfill \le & {\int }_0^t\frac{\sigma }{k}{e}^{-\sigma }{∥h-j∥}_{\sigma }{e}^{\sigma s}ds\hfill \\ \hfill =& {\int }_0^t\frac{\sigma }{k}{e}^{-\sigma }{e}^{\sigma s}{∥h-z∥}_{\sigma }ds\hfill \\ \hfill \le & \frac{\sigma }{k}{e}^{-\sigma }{∥h-j∥}_{\sigma }{\int }_0^t{e}^{\sigma s}ds\hfill \\ \hfill \le & \frac{1}{k}{e}^{-\sigma }{e}^{\sigma t}{∥h-j∥}_{\sigma }.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \left(\right|\mathrm{Y}h\left(t\right)-\mathrm{Y}j\left(t\right)|+|\mathrm{Y}h\left(t\right)-\mathrm{Y}w\left(t\right)|+|\mathrm{Y}j\left(t\right)-\mathrm{Y}w\left(t\right)\left|\right)e^{\sigma t}\hfill \\ \hfill & \le \frac{1}{k}{e}^{-\sigma }({∥h-j∥}_{\sigma }+{∥h-w∥}_{\sigma }+{∥j-w∥}_{\sigma }).\hfill \end{array}$$

(39)

Taking the supremum over all $r\in \mathcal{W}$ in (39) gives

$$\begin{array}{c}\hfill G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\le \frac{1}{k}{e}^{-\sigma }\left(G(h,j,w)\right).\end{array}$$

Let $\eta =\frac{1}{k},k>1$, then $\eta \in (0,1)$. Hence, we have

$$G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\le \eta e^{-\sigma }\left(G(h,j,w)\right),$$

which gives

$$e^{\sigma }G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\le \eta \left(G(h,j,w)\right).$$

(40)

Taking the log of both sides in (40), yields $ln\left(e^{\sigma }G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\right)\le ln\left(\eta \left(G(h,j,w)\right)\right)$, from which we have $ln\left(e^{\sigma }\right)+ln\left(G(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w)\right)\le ln\left(\eta \left(G(h,j,w)\right)\right)$, and hence

$$\begin{array}{cc}\hfill \sigma +ln\left(G\right(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w\left)\right)& \le ln\left(\eta \right(G(h,j,w)\left)\right).\hfill \end{array}$$

(41)

By defining $F:{\mathbb{R}}_{+}⟶\mathbb{R}$ as $F\left(t\right)=ln\left(t\right)$ for all $r>0$, (41) can be rewritten as

$$\begin{array}{c}\hfill \sigma +F\left(G\right(\mathrm{Y}h,\mathrm{Y}j,\mathrm{Y}w\left)\right)\le F\left(\eta \right(G(h,j,w)\left)\right).\end{array}$$

As a result, all the hypotheses of Corollary 3 are satisfied. Hence, $\mathrm{Y}$ has a unique FP in $\mathsf{\Omega }$, which is the unique solution (say u) of the integral Equation (36).

Conversely, if u is a solution of (36), then u is a solution of (38) so that $\mathrm{Y}u=u$. □

Remark 2. 

(i)

It is clear that by fixing the constants ${\lambda }_i(i=1,2)$ and q, we can obtain several more consequences of Theorems 3 and 4.

(ii)

None of the findings proposed in this work can be written in the form of $G(h,j,j)$ or $G(h,h,j)$. Therefore, they cannot be deduced from their analogs in MS.

5. Conclusions

The concept of F-contraction was introduced by Wardowski [13] to demonstrate the improvement in the Banach contraction principle regarding the existence of FP in complete MS. In this work, a new concept called Jaggi-type hybrid $(\varphi$-$F)$-contraction has been introduced, and some FP results for such mappings in the context of complete G-MS were presented (see Theorems 3 and 4). Following Theorems 3 and 4, Corollaries 1–3 are presented as special cases. The proposed notion in this work is an extension of F-contraction in MS and some recent corresponding findings in G-MS. It is noted that the established FP results cannot be inferred from their comparable ones in symmetric or asymmetric spaces. In support of the latter observation, a comparative example is given to demonstrate the veracity of our obtained results. From application points of consideration, the existence and uniqueness of solutions to nonlinear Volterra integral equations were established via the aid of one of the deduced corollaries. This work is limited in scope by the fact that the mathematical formulation, analysis, and results presented are purely theoretical. The application to the integral equation has been developed analytically and the conclusion is deduced based on the abstract assumptions of our theorems. However, it is worth noting that the idea of this paper points researchers toward further investigations and applications. It will be intriguing to apply these concepts in the structure of various spaces and the concerned mapping can also be extended to multi-valued mappings.