Perturbed Mixed Variational-like Inequalities and Auxiliary Principle Pertaining to a Fuzzy Environment

Abstract

Convex and non-convex fuzzy mappings are well known to be important in the research of fuzzy optimization. Symmetry and the idea of convexity are closely related. Therefore, the concept of symmetry and convexity is important in the discussion of inequalities because of how its definition behaves. This study aims to consider new class of generalized fuzzy variational-like inequality for fuzzy mapping which is known as perturbed fuzzy mixed variational-like inequality. We also introduce strongly fuzzy mixed variational inequality, as a particular case of perturbed fuzzy mixed variational-like inequality which is also a new one. Furthermore, by using the generalized auxiliary principle technique and some new analytic techniques, some existence results and efficient numerical techniques of perturbed fuzzy mixed variational-like inequality are established. As exceptional cases, some known and new results are obtained. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

1. Introduction

In [1], Zadeh introduced fuzzy set theory, which has been extensively used to simulate the uncertainty found in practical applications. The expansion of fuzzy set theory and its applications have drawn the interest of numerous academics, see [2]. With the passage of time, these have been expanded and generalized to analyze a vast class of problems like in optimization and control system, mechanics, economics and transportation, physics and so forth. The idea of variational inequality was initiated by Hartman and Stampacchia [3] in 1964. A useful generalization of variational inequality is generalized mixed variational-like inequality (in short, GMVLI). The “GMVLI” have capability and noteworthy applications in different fields like structural analysis [4], optimization theory [5,6,7,8,9,10,11], and economics [12,13]. Motivated and inspired by going on research work, many authors discussed fuzzy variational-like inequalities and its generalizations, and its applications in different fields. For more informations related to fuzzy theory and inequalities, see [14,15,16,17]. In fuzzy optimization, Noor [18,19] studied the characterization of minimum of convex fuzzy mapping through fuzzy variational inequality and fuzzy mixed variational inequality, and obtained some iterative algorithms. It is worthy to mention one of the most considered generalization of convex fuzzy mapping is preinvex fuzzy mapping. The idea of fuzzy preinvex mapping on the invex set was introduced and studied by Noor [20], Moreover, any local minimum of a preinvex fuzzy mapping is a global minimum on invex set and necessary and sufficient condition for fuzzy mapping is to be preinvex if its epigraph is an invex set. Furthermore, it has been verified that a fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities. Then many another’s extended this concept and discussed the applications of fuzzy variational-like inequalities. Chang [21], Chang and Zhu [22], and Chang et al. [23] and Kumam and Petrot [24], studied the idea of “GMVLI” and complementarity problems for fuzzy mappings in different context. By using the Berge maximum theorem in finite and infinite dimensional space, some particular cases of “GMVLI” are studied by Tian [25], Parida and Sen [26], and Yao [27]. It is convenient to mention that their methods are not productive. Therefore, the enlargement of a well-organized and applicable technique for solving variational-like inequality is one of the impotent and engaging problem. Although, there exist substantial numbers of numerical methods as well as projection method and its alternative forms, Newton’s methods, linear approximation and descent. But the projection type method can be used to recommend iterative method for variational-like inequalities. To overcome this drawback of above mentioned method, Glowinski, Lions and Tremolieres [28], was suggested auxiliary principle technique. Then many authors expended the auxiliary principle technique to study the existence and uniqueness of a solution of “GMVLI” for set valued mappings with compact and non-compact values and single valued mappings, see [29,30,31,32,33,34,35,36,37,38]. By using auxiliary principle technique, “GMVLI” can easily be handled as well as its particular cases.

Motivated and inspired by going on research work in this interesting and fascinating field, the objective of this article to introduce new class of “GMVLI”, which is known as perturbed fuzzy mixed variational-like inequality (PFMVLI) and to study some existence theorems. We also prove the existence of auxiliary problem for “PFMVLI”. By utilizing the theorems, we construct an iterative algorithms auxiliary problem for “PFMVLI”. The results given in this paper are up to date and they generalize, refine and consolidate a number of recent results in [3,21,28,31].

2. Preliminaries

Let $\mathcal{H}$ be a real Hilbert space and $\varnothing \ne K\subset \mathcal{H}$ be a convex set. we denote the collection CB$\left(\mathcal{H}\right)$ of all nonempty bounded and closed subsets of $\mathcal{H}$ and $D\left(.,.\right)$ is the Hausdorff metric on CB$\left(\mathcal{H}\right)$ defined by

$$D\left(\mathcal{A},\mathcal{C}\right)=\max \left\{\underset{u\in \mathcal{A}}{\sup }d\left(u,\mathcal{C}\right), \underset{\vartheta \in \mathcal{C}}{\sup }d\left(\vartheta ,\mathcal{A}\right)\right\}, \mathrm{where} \mathcal{A},\mathcal{C}\in CB\left(\mathcal{H}\right).$$

A fuzzy set on $\mathcal{H}$ is a mapping $\psi :\mathcal{H}\to \left[0,1\right]$, for each fuzzy set and $\alpha \in \left(0, 1\right]$, then $\alpha$-level sets of $\psi$ is denoted and defined as follows ${\psi }_{\alpha }=\left\{u\in \mathcal{H}| \psi \left(u\right)\ge \alpha \right\}$. If $\alpha =0$, then $\mathrm{supp}\left(\psi \right)=\left\{u\in \mathcal{H}\left| \psi \left(u\right)\right.>0\right\}$ is called support of $\psi$. By ${\left[\psi \right]}^0$ we define the closure of $\mathrm{supp}\left(\psi \right)$.

In what follows, $\mathcal{F}\left(\mathcal{H}\right)=\left\{\mathfrak{A}:\mathcal{H}\to I=\left[0,1\right]\right\}$ denote the family of all fuzzy sets on $\mathcal{H}$. A mapping $\mathcal{T}$ from $\mathcal{H}$ to $\mathcal{F}\left(\mathcal{H}\right)$ is called a fuzzy mapping. If $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is a fuzzy mapping, then the set $\mathcal{T}\left(u\right)$, for $u\in \mathcal{H}$ is a fuzzy set in $\mathcal{F}\left(\mathcal{H}\right)$ (in the sequel we denote $\mathcal{T}\left(u\right)$ by ${\mathcal{T}}_u$) and ${\mathcal{T}}_u\left(\vartheta \right)$ , $\vartheta \in \mathcal{H}$ is the degree of membership of $\vartheta$ in ${\mathcal{T}}_u$.

Definition 1.

(i) If for each $u\in \mathcal{H},$ the function $\vartheta \to {\mathcal{T}}_u\left(\vartheta \right)$ is upper semicontinuous, then fuzzy mapping  $\mathcal{T}$ is called closed. If  $\left\{{\vartheta }_{\alpha }\right\}\subset \mathcal{H}$ is a net and satisfying ${\vartheta }_{\alpha }$ → ${\vartheta }_0\in \mathcal{H}$, then ${\mathcal{T}}_u$ have following property, we have

$$\lim {\sup }_{\alpha }{\mathcal{T}}_u\left({\vartheta }_{\alpha }\right)\le {\mathcal{T}}_u\left({\vartheta }_0\right).$$

(ii) A closed fuzzy mapping $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is said to satisfy the condition (∗), if there exists a function $\alpha :\mathcal{H}\to \left[0, 1\right]$ such that for each $u\in \mathcal{H}$ the set

$${[{\mathcal{T}}_u]}_{\alpha \left(u\right)}=\left\{\vartheta \in \mathcal{H}:{\mathcal{T}}_u\left(\vartheta \right)\ge \alpha \left(u\right)\right\}\ne \varnothing ,$$

(1)

is a bounded subset of  $\mathcal{H}$.

Remark 1.

It is important to mention that, from Definition 1, (i and ii), it can easily be seen that ${[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\in$ CB  $\left(\mathcal{H}\right)$. Indeed, let ${\left\{{\vartheta }_{\alpha }\right\}}_{\alpha \in \Gamma }\subset {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ be a net and ${\vartheta }_{\alpha }$ → ${\vartheta }_0\in \mathcal{H}.$ Then, from (1) and (2), we have  $\lim su{p}_{\alpha }{\mathcal{T}}_u\left({\vartheta }_{\alpha }\right)\le {\mathcal{T}}_u\left({\vartheta }_0\right)$ and ${\mathcal{T}}_u\left({\vartheta }_{\alpha }\right)\ge \alpha \left(u\right)$ for each $\alpha \in \Gamma .$ Since $\mathcal{T}$ is closed with condition (∗), we have

$${\mathcal{T}}_u\left({\vartheta }_0\right)\ge \lim su{p}_{\alpha }{\mathcal{T}}_u\left({\vartheta }_{\alpha }\right)\ge \alpha \left(u\right).$$

(2)

This implies that  ${\vartheta }_0\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ and so ${[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\in CB\left(\mathcal{H}\right)$.

Problem 1.

Let $\mathcal{H}$ be a real Hilbert space. Then, for given nonlinear mappings $\xi \left(.,.\right):\mathcal{H}\times \mathcal{H}\to \mathcal{H}$, $\mathcal{M}\left(.\right):\mathcal{H}\to \mathcal{H},$ we consider the problem of finding $u\in \mathcal{H}$ and constant $\omega >0$ , such that

$$\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega {\Vert \xi \left(\vartheta ,u\right)\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}.$$

(3)

where $\mathcal{J}:\mathcal{H}\to R\cup \left\{+\infty \right\}$ is a nondifferentiable function and $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is a fuzzy mapping satisfying condition (∗) with function $\alpha :\mathcal{H}\to \left[0, 1\right]$ , such that

$${\mathcal{T}}_u\left(p\right)\ge \alpha \left(u\right), \mathrm{i}.\mathrm{e}., p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}.$$

The inequality (3) is known as perturbed fuzzy mixed variational-like inequality (in short, PFMVLI). It can easily be seen that; this inequality is more general than fuzzy mixed variational-like inequality (in short, FMVLI) and include classical mixed variational-like inequality (in short, MVLI) and associated fuzzy optimizations problems are particular cases. For applications, see [3,17,21,22,26] and the references therein.

Now we study some certain cases of Problem (1).

(1) 

Let $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ be an ordinary multivalued mapping and $\mathcal{J},$ $\xi$ be the mapping in problem (1).

Now we define a fuzzy mapping $\tilde{\mathcal{T}}\left(.\right):\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ as follows

$${\tilde{\mathcal{T}}}_u=\mathcal{X}{\mathcal{T}}_u$$

(4)

where $\mathcal{X}{\mathcal{T}}_u$ is the characteristic functions of the $\mathcal{T}\left(u\right).$ From (4), it can straightforwardly be noticed that the $\tilde{\mathcal{T}}$ is a closed fuzzy mapping fulfilling condition (∗) with constant function $\alpha \left(u\right)=1,$ for all $u\in \mathcal{H}.$ Also

$${\left[{\tilde{\mathcal{T}}}_u\right]}_{\alpha \left(u\right)}={\left[\mathcal{X}{\mathcal{T}}_u\right]}_1=\left\{\vartheta \in \mathcal{H}:{\mathcal{T}}_u\left(\vartheta \right)=1\right\}=\mathcal{T}\left(u\right).$$

• then problem (1), is parallel to finding $u\in \mathcal{H}$ , such that

$$\begin{gathered} \left({\tilde{\mathcal{T}}}_u\right)\left(p\right)=1, \mathrm{i}.\mathrm{e}., p\in \mathcal{T}\left(u\right). \\ \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega {\Vert \xi \left(\vartheta ,u\right)\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}. \end{gathered}$$

(5)

This kind of problem is called the set valued “PMVLI”. This inequality is also new one.

If $\mathcal{M}=I$ (identity mapping), $p\in \mathcal{H}$ and $\mathcal{T}:\mathcal{H}\to \mathcal{H}$ is single valued, then inequality (5) is parallel to finding a $u\in \mathcal{H}$ such that

$$\langle \mathcal{T}\left(u\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega {\Vert \xi \left(\vartheta ,u\right)\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}.$$

(6)

This is known as “PMVLI” and studied by Noor et al. [31].

• If $\xi \left(\vartheta ,u\right)=\vartheta -u,$ then problem (1) is called strongly fuzzy mixed variational inequality and is parallel to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

  $$\langle \mathcal{M}\left(p\right), \vartheta -u\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega {\Vert \vartheta -u\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}.$$

  (7)

This class of “FMVI” is also new one. This inequality is more general as well as classical “FMVI” and related fuzzy optimization problems as particular cases. In case of ordinary set-valued mapping as flourished by Noor [9].

• If $\omega =0,$ then problem (1), is parallel to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

$$\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)\ge 0, \forall \vartheta \in \mathcal{H}.$$

(8)

which is known as “FMVLI”. In generalized form as flourished by Chang et al. [23], and Kumam and Petrot [24]. In case of ordinary set-valued mapping as flourished by Noor [6]. For the development of numerical methods and applications of (8), see [7,26,33,34] and the references therein.

• For $\omega =0,$ inequality (7) reduces to

$$\begin{gathered} u\in \mathcal{H}, p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)} \\ \langle \mathcal{M}\left(p\right), \vartheta -u\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)\ge 0, \forall \vartheta \in \mathcal{H}. \end{gathered}$$

(9)

which is known as called “FMVI”. Chang et al. [23] and Kumam and Petrot [24], studied as a special case. In case of ordinary set-valued mapping as developed by Lions and Stampacchia [5], for applications, see [6,7,28,33] and the references therein.

• When $\mathcal{M}=I$ and $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is a fuzzy mapping, then inequality (9) is parallel to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$

$$\langle p, \vartheta -u\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)\ge 0, \forall \vartheta \in \mathcal{H}.$$

This is known as “FMVI”, see [18].

• If $\mathcal{J}\left(.\right)$ is an indicator mapping of a closed invex set $K_{\xi }$ in $\mathcal{H},$ that is

$$I_{\mathcal{J}}\left(u\right)=\left\{\begin{array}{c}0, u\in K_{\xi }\\ \infty , otherwise\end{array}\right.$$

then inequality (8) is analogous to finding $p\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

$$\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle \ge 0, \forall \vartheta \in \mathcal{H}.$$

(10)

which is known as “FVLI”.

• when $\mathcal{M}=I,$ then inequality (10), is analogous to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

$$\langle p, \xi \left(\vartheta ,u\right)\rangle \ge 0, \forall \vartheta \in \mathcal{H}.$$

(11)

which is also known as “FVLI”, see [20]. For the applications “FVLI” and fuzzy optimization problem, see [6,7,26,34] and the references therein.

• If $\xi \left(\vartheta ,u\right)=\vartheta -u,$ then inequality (10), is analogous to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

$$\langle \mathcal{M}\left(p\right),\vartheta -u\rangle \ge 0, \forall \vartheta \in \mathcal{H}.$$

(12)

which is known as “FVI”.

• If $\mathcal{M}=I$ , then inequality (12), is analogous to finding $u\in \mathcal{H}$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ such that

$$\langle p,\vartheta -u\rangle \ge 0, \forall \vartheta \in \mathcal{H}.$$

(13)

This is also known as “FVI”, see [10,22]. The “VI” is mainly due to Stampacchia and Guido [13], and Lions and Stampacchia [5], as developed and studied by Noor [10]. For the applications of inequality (13), see [5,9,26,28] and the references therein.

From above discussion, it can easily be seen that inequalities (7)–(13) are particular cases of “PFMVLI” (3). In fact, “PFMVLI” is more generalize and unifying one, which is main motivation of our work. For a proper and suitable choice of $\mathcal{T}$, $\xi$ and $\mathcal{J}$, we can choose a number of known and unknown “FVLI” and complementary problems.

Next, we will use mathematical terminologies S-monotone and L-continuous for strongly monotone and Lipschitz continuous, respectively.

Definition 2.

If fuzzy mapping $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is closed and fulfil the condition (∗) with function $\alpha :\mathcal{H}\to \left[0, 1\right],$ then nonlinear mapping  $\mathcal{M}:\mathcal{H}\to \mathcal{H}$ is said to be:

(a)

$\beta$-S-Monotone: if there exist a constant $\beta >0,$such that

$$\langle \mathcal{M}\left(q\right)-\mathcal{M}\left(p\right), \vartheta -u\rangle \ge \beta {\Vert \vartheta -u\Vert }^2, for all p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)} and q\in {[{\mathcal{T}}_{\vartheta }]}_{\alpha \left(\vartheta \right)}.$$

(b)

γ-L-continuous: if there exist a constant  $\gamma >0,$ such that

$$\Vert \left(\mathcal{M}\left(q\right)-\mathcal{M}\left(p\right)\right)\Vert ⪯\gamma \Vert q-p\Vert , for all p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)} and q\in {[{\mathcal{T}}_{\vartheta }]}_{\alpha \left(\vartheta \right)}.$$

Definition 3.

A fuzzy mapping $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ is said to be:

(c)

$\mathcal{T}$-L-continuous: if there exist a function $\alpha :K\to \left[0, 1\right]$ and constant $\gamma >0,$ such that 

$$D\left({[{\mathcal{T}}_u]}_{\alpha \left(u\right)}, {[{\mathcal{T}}_{\vartheta }]}_{\alpha \left(\vartheta \right)}\right)⪯\lambda \Vert \vartheta -u\Vert ,$$

where $D\left(.,.\right)$ is the Housdroff metric on $\mathcal{F}\left(\mathcal{H}\right).$

In particular, from (a) and (b), we have

$$\begin{gathered} \beta {\Vert \vartheta -u\Vert }^2⪯\mathcal{M}\left(q\right)-\mathcal{M}\left(p\right), \vartheta -u, \\ ⪯\Vert \mathcal{M}\left(q\right)-\mathcal{M}\left(p\right)\Vert \Vert \vartheta -u\Vert , \\ ⪯D\left({[{\mathcal{T}}_u]}_{\alpha \left(u\right)}, {[{\mathcal{T}}_{\vartheta }]}_{\alpha \left(\vartheta \right)}\right)\Vert \vartheta -u\Vert , \end{gathered}$$

From (c), we have

$$⪯\gamma \lambda {\Vert \vartheta -u\Vert }^2$$

which implies that $\beta ⪯\gamma \lambda .$

Definition 4.

The bi-function $\xi \left(.,.\right):\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is said to be:

(d)

S-monotone: if there exist constant $\mu >0,$

 such that

$$\langle \xi \left(\vartheta ,u\right), \vartheta -u\rangle \ge \mu {\Vert \vartheta -u\Vert }^2, for all u,\vartheta \in \mathcal{H}.$$

(e)

L-continuous: if there exist constant $\delta >0,$

  such that

$$\Vert \xi \left(\vartheta ,u\right)\Vert ⪯\delta \Vert \vartheta -u\Vert , for all u,\vartheta \in \mathcal{H}.$$

From (d) and (e), we can observe that $\mu ⪯\delta .$

Definition 5.

A mapping $\mathcal{J}:\mathcal{H}\to \mathcal{H}$ is said to be $\partial$ -L-continuous if there exist constant $\partial >0,$ such that

$$\Vert \mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)\Vert ⪯\partial \Vert \vartheta -u\Vert , for all u,\vartheta \in \mathcal{H}.$$

For any $\mathcal{C}\subseteq \mathcal{H},$ we denote the conv(  $\mathcal{C}$ ), the convex hull of $\mathcal{C}$. A set valued mapping $T:\mathcal{H}\to 2^{\mathcal{H}}$ is called a KKM mapping if, for every finite subset $\left\{{\vartheta }_1, {\vartheta }_2, {\vartheta }_3, {\vartheta }_4\dots \dots \dots \dots , {\vartheta }_n\right\}$ of $\mathcal{H},$

$$conv\left\{{\vartheta }_1, {\vartheta }_2, {\vartheta }_3, {\vartheta }_4\dots \dots \dots \dots , {\vartheta }_n\right\}\subseteq {{\displaystyle \cup }}_{i=1}^{n}T\left({\vartheta }_i\right).$$

Lemma 1.

 [8] Let $\mathcal{C}$ be an arbitrary nonempty in a topological vector space $G,$ and let $T:\mathcal{C}\to 2^{\mathcal{H}}$ is a KKM mapping. If $T\left(\vartheta \right)$ is closed for all $\vartheta \in \mathcal{C}$ and is compact for at least one  $\vartheta \in \mathcal{C}$ , then

$${{\displaystyle \cap }}_{\vartheta \in \mathcal{C}}T\left(\vartheta \right)\ne \varnothing .$$

Theorem 1.

[21] Let  $G$ be a locally convex Hausdorff topologically vector space and  $g:K\to R\cup \left\{+\infty \right\}$ be a properly convex functional. Then  $g$ is lower semicontinuous on   $G$ if and only if,  $g$ is weakly lower semicontinuous on  $G.$

Assumption 1.

Let  $\mathcal{M}:\mathcal{H}\to \mathcal{H}$ and $\xi \left(.,.\right):\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ be two mappings satisfying the following condition

(a)

$\xi \left(\vartheta ,u\right)+\xi \left(u,\vartheta \right)=0$ (and so  $\xi \left(u,u\right)=0,$ for all   $u\in \mathcal{H})$ , for all  $u,\vartheta \in \mathcal{H}.$

(b)

For any given  $u\in \mathcal{H},$ the mapping  $\vartheta \to \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle$ is concave, where  $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}.$

(c)

For any given  $u\in \mathcal{H},$ the mapping $\vartheta \to \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle$ is lower semicontinuous, where  $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}.$

$$u_n\to u, p_n\to p, imply \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle \le \underset{n\to \infty }{liminf}\langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_n\right)\rangle .$$

Lemma 2.

[11] Let $\left(X, d\right)$ be a complete metric space and  ${\mathcal{C}}_1, {\mathcal{C}}_2\in$ CB(  $X$ ) and $r\ge 1$ be any real number. Then, for every $c_1\in {\mathcal{C}}_1$ then there exist  $c_2\in {\mathcal{C}}_2$ such that $d\left(c_1,c_2\right)\le rD\left({\mathcal{C}}_1, {\mathcal{C}}_2\right).$

In next sections, we will use the above results.

3. Auxiliary Principle and Algorithm

In this portion, we explore the auxiliary principle technique which is mainly due to Glowinski, Lions and Tremolieres [28], as flourished and improved by Noor [32], and Huang and Deng [17], to examine “PFMVLI”. We give an existence result of the solution of auxiliary problem for the “PFMVLI” (3). Furthermore, based on this existence result, we suggest an iterative algorithm for the “PFMVLI” (3).

For a given $u\in K, p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$satisfying the problem (1), we consider the problem of finding $𝔃\in \mathcal{H},$ such that

$$\langle 𝔃, \vartheta -𝔃\rangle \ge \langle u, \vartheta -𝔃\rangle -\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,𝔃\right)\rangle +\rho \mathcal{J}\left(\vartheta \right)-\rho \mathcal{J}\left(𝔃\right)-\omega \rho {\Vert \xi \left(\vartheta ,𝔃\right)\Vert }^2,$$

(14)

$\forall \vartheta \in \mathcal{H},$ where $\rho >0$ is a constant. The inequality (14) is also called auxiliary “PFMVLI”.

Theorem 2.

Let $\mathcal{H}$ be real Hilbert space and  $K\ne \varnothing$ be a closed bounded subset of  $\mathcal{H}$ . Let $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$ be a closed continuous fuzzy mapping satisfying condition (∗) with function $\alpha :\mathcal{H}\to \left[0, 1\right]$. Let $\mathcal{J}:K\to R\cup \left\{+\infty \right\}$ be a properly convex functional (i.e., A functional $\mathcal{J}\left(.\right)$ is called proper, if  $\mathcal{J}\left(u\right)>-\infty$ for all $u\in K$ and $\mathcal{J}\left(u\right)\equiv +\infty$ ) and $\partial$ -L-continuous with constant  $\partial$ . If the Assumption 1 is satisfied and bi-function $\xi \left(.,.\right):K\times K\to \mathcal{H}$ is L-continuous with constant  $\delta >0$ . Then auxiliary problem (14), has a unique solution.

Proof. 

Let given $u\in K$, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)},$ we consider the mapping $T:\mathcal{H}\to 2^{\mathcal{H}}$ define by

$$T\left( \vartheta \right)=\left\{w\in K:\langle w-u, \vartheta -w\rangle \ge -\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,w\right)\rangle +\rho \mathcal{J}\left(\vartheta \right)-\rho \mathcal{J}\left(w\right)-\omega \rho {\Vert \xi \left(\vartheta ,w\right)\Vert }^2\right\},$$

(15)

for all $\vartheta \in \mathcal{H}$.

From (15), it can easily be seen that, for each $\vartheta \in K$, $T\left( \vartheta \right)\ne \varnothing ,$ since $\vartheta \in T\left( \vartheta \right).$ To obtain the solution, firstly we show that $T:\mathcal{H}\to 2^{\mathcal{H}}$ is a KKM mapping. Suppose the contrary, that is, $T:\mathcal{H}\to 2^{\mathcal{H}}$ is not KKM mapping, then there exist a finite subset $\left\{{\vartheta }_1, {\vartheta }_2, {\vartheta }_3, {\vartheta }_4,\dots \dots \dots \dots , {\vartheta }_n\right\}$ of $K$ and constant ${\tau }_i\ge 0$, $i=1, 2, 3, 4,\dots \dots \dots \dots k,$ with $\sum _{i=1}^k{\tau }_i=1,$ such that”

$$w_{*}={\sum }_{i=1}^k{\tau }_i{\vartheta }_i\notin {\cup }_{i=1}^{k}T\left({\vartheta }_i\right), \mathrm{for} \mathrm{all} 1, 2, 3, 4,\dots \dots \dots \dots .k,$$

(16)

Then, we have

$$\langle w_{*}-u, {\vartheta }_i-w_{*}\rangle +\rho \langle \mathcal{M}\left(p\right), \xi \left({\vartheta }_i,w_{*}\right)\rangle -\rho \mathcal{J}\left({\vartheta }_i\right)+\rho \mathcal{J}\left(w_{*}\right)+\omega \rho {\Vert \xi \left({\vartheta }_i,w_{*}\right)\Vert }^2<0,$$

From Assumption 1 and by the convexity of $\mathcal{J}$, so the above inequality yield

$$\begin{gathered} 0>{\displaystyle \sum }_{i=1}^k{\tau }_i\langle w_{*}-u, {\vartheta }_i-w_{*}\rangle +\rho {\displaystyle \sum }_{i=1}^k{\tau }_i\langle \mathcal{M}\left(p\right), \xi \left({\vartheta }_i,w_{*}\right)\rangle -\rho \mathcal{J}\left({\vartheta }_i\right) \\ +\rho \mathcal{J}\left(w_{*}\right)+\omega \rho {\displaystyle \sum }_{i=1}^k{\tau }_i{\Vert \xi \left({\vartheta }_i,w_{*}\right)\Vert }^2, \\ \ge \langle w_{*}-u, w_{*}-w_{*}\rangle +\rho \langle \mathcal{M}\left(p\right), \xi \left(w_{*},w_{*}\right)\rangle -\rho \mathcal{J}\left(w_{*}\right) \\ +\rho \mathcal{J}\left(w_{*}\right)+\omega \rho {\Vert \xi \left(w_{*},w_{*}\right)\Vert }^2=0, \end{gathered}$$

This is a contradiction. Hence, $T:\mathcal{H}\to 2^{\mathcal{H}}$ is a KKM mapping.

Since ${\overline{T\left(\vartheta \right)}}^p,$ the weak closure of $T\left(\vartheta \right)$] is weakly closed subset of a bounded set $K\subseteq \mathcal{H},$ so it is weakly compact. Hence by Lemma 1

$${{\displaystyle \cap }}_{\vartheta \in K}{\overline{T\left(\vartheta \right)}}^p\ne \varnothing .$$

Let

$$𝔃\in {{\displaystyle \cap }}_{\vartheta \in K}{\overline{T\left(\vartheta \right)}}^p.$$

Then, there exists a sequence ${𝔃}_n$ in $T\left(\vartheta \right)$ (fix $\vartheta$), such that ${𝔃}_n\to 𝔃.$ Then

$$\langle {𝔃}_n-u, \vartheta -{𝔃}_n\rangle +\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,{𝔃}_n\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left({𝔃}_n\right)+\omega \rho {\Vert \xi \left(\vartheta ,{𝔃}_n\right)\Vert }^2\ge 0.$$

(17)

Now, by using the property of inner product

$$\begin{gathered} \langle {𝔃}_n-u, \vartheta -{𝔃}_n\rangle =\langle {𝔃}_n-u, \vartheta \rangle +\langle u, {𝔃}_n\rangle -{𝔃}_n, {𝔃}_n, \\ =\langle {𝔃}_n-u, \vartheta \rangle +\langle u, {𝔃}_n\rangle -{{𝔃}_n}^2. \end{gathered}$$

Since $\Vert .\Vert$ is weakly lower semicontinuous, we have

$$\begin{gathered} \underset{n\to \infty }{\mathrm{limsup}}\langle {𝔃}_n-u, \vartheta -{𝔃}_n\rangle =\underset{n\to \infty }{\mathrm{limsup}}\left\{\langle {𝔃}_n-u, \vartheta \rangle +\langle u, {𝔃}_n\rangle -{\Vert {𝔃}_n\Vert }^2\right\} \\ =\underset{n\to \infty }{\mathrm{limsup}}\langle {𝔃}_n-u, \vartheta \rangle +\underset{n\to \infty }{\mathrm{limsup}}\langle u, {𝔃}_n\rangle -\underset{n\to \infty }{\mathrm{liminf}}{\Vert {𝔃}_n\Vert }^2, \\ \le \langle 𝔃-u, \vartheta -𝔃\rangle , \end{gathered}$$

(18)

By using Lipchitz continuity and convexity of $\mathcal{J}$ and Assumption 1, we get

$$\begin{array}{l}0\le \underset{n\to \infty }{\mathrm{limsup}} \left\{\langle {𝔃}_n-u, \vartheta -{𝔃}_n\rangle +\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,{𝔃}_n\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left({𝔃}_n\right)+\omega \rho {\Vert \left(\xi ,\vartheta ,{𝔃}_n\right)\Vert }^2\right\}\\ =\underset{n\to \infty }{\mathrm{limsup}} \left\{\langle {𝔃}_n-u, \vartheta -{𝔃}_n\rangle +\rho \underset{n\to \infty }{\mathrm{limsup}}\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,{𝔃}_n\right)\rangle -\rho \underset{n\to \infty }{\mathrm{liminf}}\mathcal{J}\left(\vartheta \right)+\rho \underset{n\to \infty }{\mathrm{limsup}}\mathcal{J}\left({𝔃}_n\right)+\omega \rho \underset{n\to \infty }{\mathrm{limsup}}\Vert \xi {\left(\vartheta ,{𝔃}_n\right)\Vert }^2\right\},\\ \le \langle 𝔃-u, \vartheta -𝔃\rangle +\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,𝔃\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(𝔃\right)+\omega \rho {\Vert \xi \left(\vartheta ,𝔃\right)\Vert }^2.\end{array}$$

This implies that

$$\langle 𝔃, \vartheta -𝔃\rangle \ge \langle u, \vartheta -𝔃\rangle -\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,𝔃\right)\rangle +\rho \mathcal{J}\left(\vartheta \right)-\rho \mathcal{J}\left(𝔃\right)-\omega \rho {\Vert \xi \left(\vartheta ,𝔃\right)\Vert }^2,$$

(19)

for all $\vartheta \in \mathcal{H}.$

Hence $𝔃$ is solution of auxiliary problem (14).

Uniqueness: Let ${𝔃}_1\in K$ be also a solution of auxiliary problem. Then we have

$$\langle {𝔃}_1, \vartheta -{𝔃}_1\rangle \ge \langle u, \vartheta -{𝔃}_1\rangle -\rho \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,{𝔃}_1\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left({𝔃}_1\right)-\rho \omega {\Vert \xi \left(\vartheta ,{𝔃}_1\right)\Vert }^2,$$

(20)

for all $\vartheta \in \mathcal{H}.$

Replacing $\vartheta$ by ${𝔃}_1$ in (19) and $\vartheta$ by $𝔃$ in (20), we have

$$\langle 𝔃, {𝔃}_1-𝔃\rangle \ge \langle u,{𝔃}_1-𝔃\rangle -\rho \langle \mathcal{M}\left(p\right), \xi \left({𝔃}_1,𝔃\right)\rangle +\rho \mathcal{J}\left({𝔃}_1\right)-\rho \mathcal{J}\left(𝔃\right)-\omega \rho {\Vert \xi \left({𝔃}_1,𝔃\right)\Vert }^2,$$

(21)

and

$$\langle {𝔃}_1, 𝔃-{𝔃}_1\rangle \ge \langle u, 𝔃-{𝔃}_1\rangle -\rho \langle \mathcal{M}\left(p\right), \xi \left(𝔃,{𝔃}_1\right)\rangle -\rho \mathcal{J}\left(𝔃\right)+\rho \mathcal{J}\left({𝔃}_1\right)-\rho \omega {\Vert \xi \left(𝔃,{𝔃}_1\right)\Vert }^2,$$

(22)

using the Assumption 1, that $\xi \left(u_1,u_2\right)=-\xi \left(u_2,u_1\right)$ and then adding the (21) and (22), we have

$$\langle {𝔃}_1-𝔃, 𝔃-{𝔃}_1\rangle \ge 0,$$

which implies that ${𝔃}_1=𝔃$ is the uniqueness of the solution of auxiliary problem (14).This complete the proof of Theorem 2. □

Algorithm 1.

At $n=0,$ start with initial value  $u_0\in \mathcal{H},$  $p_0\in {[{\mathcal{T}}_{u_0}]}_{\alpha \left(u_0\right)},$ from Theorem 2, the auxiliary problem (14) has a unique solution $u_1\in \mathcal{H}$ , such that

$$\begin{gathered} \langle u_1, \vartheta -u_1\rangle \ge \langle u_0, \vartheta -u_1\rangle -\rho \langle \mathcal{M}\left(p_0\right), \xi \left(\vartheta ,u_1\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_1\right)-\rho \omega \Vert \xi {\left(\vartheta ,u_1\right)\Vert }^2, \\ \forall \vartheta \in \mathcal{H}. \end{gathered}$$

Since  $p_0\in {[{\mathcal{T}}_{u_0}]}_{\alpha \left(u_0\right)},$  then by Nadler’s Lemma 2, there exist  $p_1\in {[{\mathcal{T}}_{u_1}]}_{\alpha \left(u_1\right)}$ , such that

$$\Vert p_0-p_1\Vert \le \left(1+1\right)D\left({[{\mathcal{T}}_{u_0}]}_{\alpha \left(u_0\right)}, {[{\mathcal{T}}_{u_1}]}_{\alpha \left(u_1\right)}\right),$$

For  $n=1,$  $u_1\in \mathcal{H},$ $p_1\in {[{\mathcal{T}}_{u_1}]}_{\alpha \left(u_1\right)},$ again from Theorem 2, the auxiliary problem (14) has a unique solution  $u_2\in \mathcal{H}$ , such that

$$\langle u_2, \vartheta -u_2\rangle \ge \langle u_1, \vartheta -u_2\rangle -\rho \langle \mathcal{M}\left(p_1\right), \xi \left(\vartheta ,u_2\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_2\right)-\rho \omega {\Vert \xi \left(\vartheta ,u_2\right)\Vert }^2, \forall \vartheta \in \mathcal{H},$$

 Since  $p_1\in {[{\mathcal{T}}_{u_1}]}_{\alpha \left(u_1\right)},$  then by Nadler’s Lemma 2, there exist  $p_2\in {[{\mathcal{T}}_{u_2}]}_{\alpha \left(u_2\right)}$ , such that

$$\Vert p_1-p_2\Vert \le \left(1+\frac{1}{2}\right)D\left({[{\mathcal{T}}_{u_1}]}_{\alpha \left(u_1\right)}, {[{\mathcal{T}}_{u_2}]}_{\alpha \left(u_2\right)}\right),$$

  At step  $n,$  we can obtain sequences  $u_n\in \mathcal{H},$  $p_n\in {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}\in CB\left(\mathcal{H}\right),$  such that

(i)

$\Vert p_n-p_{n+1}\Vert \le \left(1+\frac{1}{1+n}\right)D\left({[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_{u_{n+1}}]}_{\alpha \left(u_{n+1}\right)}\right),$

(ii)

$\langle u_{n+1}, \vartheta -u_{n+1}\rangle \ge \langle u_n, \vartheta -u_{n+1}\rangle -\rho \langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_{n+1}\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_{n+1}\right)$

$$-\rho \omega {\Vert \xi \left(\vartheta ,u_{n+1}\right)\Vert }^2, \forall \vartheta \in \mathcal{H}, \forall n\ge 0$$

(23)

.

4. Existence and Convergence

In this section, we discuss the existence and convergence of perturbed fuzzy-mixed variational-like inequality with the help of auxilary prencilple.

Theorem 3.

Let statement of Theorem 2 hold and let $\mathcal{T}$ be a $\mathcal{T}$ -L-continuous with constant  $\lambda >0$. Let nonlinear continuous mapping  $M:\mathcal{H}\to \mathcal{H}$ be $\beta$ -S-monotone and $\gamma$ -L-continuous with constants  $\beta >0$ and $\gamma >0,$ respectively and $\mathcal{J}\left(.\right)$ is nondifferentiable. If the bi-function $\xi \left(.,.\right):\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is L-continuous with constant  $\delta >0$ , respectively, then for constant  $\rho >0$, such that

$$0<\rho <\frac{2\left(\beta -\left(\mathsf{\eta }+2\omega {\delta }^2\right)\right)}{{\left(\gamma \mathsf{\lambda }\right)}^2-\left(\mathsf{\eta }+2\omega {\delta }^2\right)}, \rho <\frac{1}{\left(\mathsf{\eta }+2\omega {\delta }^2\right)}, \beta >\mathsf{\eta }+2\omega {\delta }^2,$$

(24)

where

$$\mathsf{\eta }=\gamma \mathsf{\lambda }\sqrt{1-2\mu +{\delta }^2}.$$

Then there exist  $u\in K,$  $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ satisfying the “PFMVLI” (3) and the sequences $\left\{u_n\right\}$ and $\left\{p_n\right\}$ , generated by (23) converge strongly to $u$ and $p$, respectively.

Proof .

From Algorithm 1 and auxiliary problem (14), for any $\vartheta \in \mathcal{H},$ we have

$$\begin{gathered} \langle u_n, \vartheta -u_n\rangle \ge \langle u_{n-1}, \vartheta -u_n\rangle -\rho \langle \mathcal{M}\left(p_{n-1}\right), \xi \left(\vartheta ,u_n\right)\rangle -\rho \mathcal{J}\left(\vartheta \right) \\ +\rho \mathcal{J}\left(u_n\right)-\rho \omega \Vert \xi {\vartheta ,u_n\Vert }^2, \end{gathered}$$

(25)

and

$$\begin{gathered} \langle u_{n+1}, \vartheta -u_{n+1}\rangle \ge \langle u_n, \vartheta -u_{n+1}\rangle -\rho \langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_{n+1}\right)\rangle -\rho \mathcal{J}\left(\vartheta \right) \\ +\rho \mathcal{J}\left(u_{n+1}\right)-\rho \omega {\Vert \xi \left(\vartheta ,u_{n+1}\right)\Vert }^2, \end{gathered}$$

(26)

By taking $\vartheta =u_{n+1}$ in (25) and $\vartheta =u_n$ in (26), we get

$$\begin{gathered} \langle u_n, u_{n+1}-u_n\rangle \ge \langle u_{n-1}, u_{n+1}-u_n\rangle -\rho \langle \mathcal{M}\left(p_{n-1}\right), \xi \left(u_{n+1},u_n\right)\rangle -\rho \mathcal{J}\left(u_{n+1}\right) \\ +\rho \mathcal{J}\left(u_n\right)-\rho \omega {\Vert \xi \left(u_{n+1},u_n\right)\Vert }^2, \end{gathered}$$

(27)

and

$$\begin{gathered} \langle u_{n+1}, u_n-u_{n+1}\rangle \ge \langle u_n, u_n-u_{n+1}\rangle -\rho \langle \mathcal{M}\left(p_n\right), \xi \left(u_n,u_{n+1}\right)\rangle -\rho \mathcal{J}\left(u_n\right) \\ +\rho \mathcal{J}\left(u_{n+1}\right)-\rho \omega {\Vert \xi \left(u_n,u_{n+1}\right)\Vert }^2, \end{gathered}$$

(28)

Adding (27) and (28) and using $\xi \left(\vartheta ,u\right)+\xi \left(u,\vartheta \right)=0,$ we have

$$\begin{gathered} \langle u_n-u_{n+1}, u_n-u_{n+1}\rangle \le \langle u_{n-1}-u_n, u_n-u_{n+1}\rangle -\rho \mathcal{M}\left(p_n\right)-\mathcal{M}\left(p_{n-1}\right), \xi \left(u_n,u_{n+1}\right) \\ +2\rho \omega {\Vert \xi \left(u_n,u_{n+1}\right)\Vert }^2, \end{gathered}$$

It follows that

$$\begin{gathered} \Vert u_n-u_{n+1}\Vert {}^2\le \Vert u_{n-1}-u_n-\rho \left(\mathcal{M}\left(p_{n-1}\right)-\mathcal{M}\left(p_n\right)\right)\Vert \Vert u_n-u_{n+1}\Vert + \\ \rho \Vert \mathcal{M}\left(p_{n-1}\right)-\mathcal{M}\left(p_n\right)\Vert \Vert u_n-u_{n+1}-\xi \left(u_n,u_{n+1}\right)\Vert +2\omega \rho {\Vert \xi \left(u_n,u_{n+1}\right)\Vert }^2. \end{gathered}$$

(29)

By the $\beta$-S-monotonicity and $\gamma$-L-continuity of $\mathcal{T},$ $\mathcal{M}$ and $\mathcal{T}$-L-continuity of $\mathcal{T},$ we have

$$\begin{gathered} \Vert u_{n-1}-u_n-\rho {\left(\mathcal{M}\left(p_{n-1}\right)-\mathcal{M}\left(p_n\right)\right)\Vert }^2\le \Vert {u_{n-1}-u_n\Vert }^2-2\rho \langle \mathcal{M}\left(p_{n-1}\right)-\mathcal{M}\left(p_n\right), u_{n-1}-u_n\rangle \\ +{\rho }^2\Vert \mathcal{M}\left(p_{n-1}\right)-\mathcal{M}{\left(p_n\right)\Vert }^2, \\ \le \Vert u_{n-1}-u_n\Vert {}^2-2\rho \beta \Vert u_{n-1}-u_n\Vert {}^2 \\ +{\rho }^2{\gamma }^2\Vert p_{n-1}-p_n\Vert {}^2 \\ \le \Vert u_{n-1}-u_n\Vert {}^2-2\rho \beta \Vert u_{n-1}-u_n\Vert {}^2 \\ +{\rho }^2{\gamma }^2{\left(D\left({[{\mathcal{T}}_{u_{n-1}}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}\right)\right)}^2 \\ \le \left(1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2\right){\left(1+\frac{1}{n}\right)}^2\Vert u_{n-1}-u_n\Vert {}^2, \end{gathered}$$

(30)

and

$$\Vert p_{n-1}-p_n\Vert \le \left(1+\frac{1}{n}\right)D\left({[{\mathcal{T}}_{u_{n-1}}]}_{\alpha \left(u_{n-1}\right)}, {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}\right).$$

Similarly, by the S-monotonicity and L-continuity of bifunction $\xi$, we have

$$\begin{gathered} \Vert u_n-u_{n+1}-\xi {\left(u_n,u_{n+1}\right)\Vert }^2\le \Vert {u_n-u_{n+1}\Vert }^2-2\langle u_n-u_{n+1}, \xi \left(u_n,u_{n+1}\right)\rangle +{\Vert \xi \left(u_n,u_{n+1}\right)\Vert }^2, \\ \le \Vert u_n-u_{n+1}\Vert {}^2-2\Vert u_n-u_{n+1}\Vert {}^2+\Vert u_n-u_{n+1}\Vert {}^2, \\ \le \left(1-2\mu +{\delta }^2\right)\Vert u_n-u_{n+1}\Vert {}^2. \end{gathered}$$

(31)

Combining (29), (30) and (31) and using the $\gamma$-L-continuity of $\mathcal{M}$ and $\mathcal{T}$-L-continuity of $\mathcal{T}$ and $\xi$, respectively, we have

$$\begin{gathered} {\Vert u_n-u_{n+1}\Vert }^2\le \\ \left\{\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2{\left(1+\frac{1}{n}\right)}^2}+\rho \gamma \mathsf{\lambda }\left(1+\frac{1}{n}\right)\sqrt{1-2\mu +{\delta }^2}\right\}\Vert u_{n-1}-u_n\Vert \Vert u_n-u_{n+1}\Vert \\ +2\omega \rho {\delta }^2{\Vert u_n-u_{n+1}\Vert }^2, \end{gathered}$$

which implies that

$$\begin{gathered} \Vert u_n-u_{n+1}\Vert \le \left\{\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2{\left(1+\frac{1}{n}\right)}^2}+\rho \gamma \mathsf{\lambda }\left(1+\frac{1}{n}\right)\sqrt{1-2\mu +{\delta }^2}\right\}\Vert u_{n-1}-u_n\Vert \\ +2\omega \rho {\delta }^2\Vert u_n-u_{n+1}\Vert , \end{gathered}$$

It follows that,

$$\begin{gathered} \Vert u_n-u_{n+1}\Vert \le \frac{\left\{\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2{\left(1+\frac{1}{n}\right)}^2}+\rho \gamma \mathsf{\lambda }\left(1+\frac{1}{n}\right)\sqrt{1-2\mu +{\delta }^2}\right\}}{1-2\omega \rho {\delta }^2}\Vert u_{n-1}-u_n\Vert , \\ ={\phi }_n\Vert u_{n-1}-u_n\Vert . \end{gathered}$$

(32)

where

$$\begin{gathered} {\phi }_n=\frac{\left\{\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2{\left(1+\frac{1}{n}\right)}^2}+\rho \gamma \mathsf{\lambda }\left(1+\frac{1}{n}\right)\sqrt{1-2\mu +{\delta }^2}\right\}}{1-2\omega \rho {\delta }^2}, \\ \phi =\frac{\left\{\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2}+\rho \gamma \mathsf{\lambda }\sqrt{1-2\mu +{\delta }^2}\right\}}{1-2\omega \rho {\delta }^2} \end{gathered}$$

Clearly

$${\phi }_n \to \phi =\frac{k\left(\rho \right)+\rho \mathsf{\eta }}{1-2\omega \rho {\delta }^2}$$

where

$$k\left(\rho \right)=\sqrt{1-2\rho \beta +{\rho }^2{\left(\gamma \mathsf{\lambda }\right)}^2},\mathsf{\eta }=\gamma \mathsf{\lambda }\sqrt{1-2\mu +{\delta }^2},$$

(33)

From (24), it follows that $\phi <1$. Hence, it follows from (32) and (33) that {$u_n$} is Cauchy sequence in $K$. Hence it converges to some point. Since $K$ is closed convex set in $K$, then there exist a $u$ in $K$ such that $u_n\to u,$ which satisfy the “PFMVLI” (3). On the other hand, from algorithm 1, we have

$$\begin{gathered} \Vert p_n-p_{n+1}\Vert \le \left(1+\frac{1}{1+n}\right)D\left({[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_{u_{n+1}}]}_{\alpha \left(u_{n+1}\right)}\right), \\ \le \left(1+\frac{1}{1+n}\right)\gamma \Vert u_n-u_{n+1}\Vert {}^2 \end{gathered}$$

This implies that {$p_n$} is Cauchy sequence in $\mathcal{H}$, since {$u_n$} is converging sequence. Then, we consider that $p_n\to p,$ when$n\to \infty$. Since $p_n\in {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)},$ so we have

$$\begin{gathered} d\left(p, {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}\right)\le \Vert p-p_n\Vert +d\left(p_n, {[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}\right)+\left({[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\right), \\ \le \Vert p-p_n\Vert +0+\left({[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\right)+\gamma \Vert u_n-{u\Vert }^2 \end{gathered}$$

When $n\to \infty$, we have

$$d\left(p, {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\right) \le \Vert p-p_n\Vert +0+\left({[{\mathcal{T}}_{u_n}]}_{\alpha \left(u_n\right)}, {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}\right)+\gamma \Vert u_n-{u\Vert }^2\to 0,$$

Hence, $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}.$

Now finally we show that

$$\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega \Vert \xi {\left(\vartheta ,u\right)\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}.$$

We again study (23), as follows

$$\begin{gathered} \langle u_{n+1}, \vartheta -u_{n+1}\rangle \ge \langle u_n, \vartheta -u_{n+1}\rangle -\rho \langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_{n+1}\right)\rangle -\rho \mathcal{J}\left(\vartheta \right) \\ +\rho \mathcal{J}\left(u_{n+1}\right)-\rho \omega \Vert \xi {\left(\vartheta ,u_{n+1}\right)\Vert }^2, \forall \vartheta \in \mathcal{H}, \forall n\ge 0 \end{gathered}$$

Now from Lipchitz continuity and convexity of $\mathcal{J}$ and by Assumption 1, we get

$$\begin{gathered} 0\le \underset{n\to \infty }{\mathrm{limsup}}[\langle u_{n+1}, \vartheta -u_{n+1}\rangle -\langle u_n, \vartheta -u_{n+1}\rangle +\rho \langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_{n+1}\right)\rangle +\rho \mathcal{J}\left(\vartheta \right) \\ -\rho \mathcal{J}\left(u_{n+1}\right)+\rho \omega \Vert \xi {\left(\vartheta ,u_{n+1}\right)\Vert }^2], \\ \le \rho [\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega \Vert \xi {\left(\vartheta ,u\right)\Vert }^2, \end{gathered}$$

This implies that

$$0\le [\langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega \Vert \xi {\left(\vartheta ,u\right)\Vert }^2.$$

This completes the proof. □

Special Cases:

Theorem 4.

Let the operator $\mathcal{T}:\mathcal{H}\to CB\left(\mathcal{H}\right)$ be $\mathcal{T}$ -S-monotone with constant  $\beta >0$ and $\mathcal{T}$ -L-continuous with constant  $\lambda >0$ respectively. Let nonlinear continuous mapping $M:\mathcal{H}\to \mathcal{H}$ be $\beta$ -S-monotone and $\gamma$ -L-continuous with constants  $\beta >0$ and $\gamma >0,$ respectively. Let  $\mathcal{J}:K\to R\cup \left\{+\infty \right\}$ be a properly convex functional and $\partial$ -L-continuous with constant  $\partial$ and $\mathcal{J}\left(.\right)$ is nondifferentiable. If the bi-function $\xi \left(.,.\right)$ is S-monotone with constant  $\mu >0$ and L-continuous with constant  $\delta >0$ , respectively. If Assumption 1 hold, then for constant $\rho >0$ , such that

$$0<\rho <\frac{2\left(\beta -\left(\mathsf{\eta }+2\omega {\delta }^2\right)\right)}{{\left(\gamma \mathsf{\lambda }\right)}^2-\left(\mathsf{\eta }+2\omega {\delta }^2\right)}, \rho <\frac{1}{\left(\mathsf{\eta }+2\omega {\delta }^2\right)>}, \beta \mathsf{\eta }+2\omega {\delta }^2,$$

(34)

where

$$\mathsf{\eta }=\gamma \mathsf{\lambda }\sqrt{1-2\mu +{\delta }^2}.$$

Then there exist  $u\in K,$   $p\in {[{\mathcal{T}}_u]}_{\alpha \left(u\right)}$ satisfying the set-valued “PMVLI” (5) and the sequences $\left\{u_n\right\}$ and $\left\{p_n\right\}$ , generated by (23) converge strongly to $u$ and $p$, respectively.

Proof .

By using the set valued mapping $F:\mathcal{H}\to CB\left(\mathcal{H}\right),$ we define the fuzzy mapping $\tilde{\mathcal{T}}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$, as follows

$${\tilde{\mathcal{T}}}_u=\mathcal{X}\mathcal{T}\left(u\right),$$

where $\mathcal{X}\mathcal{T}\left(u\right)$ is a characteristic function of the sets $\mathcal{T}\left(u\right).$ It is easy to see that $\tilde{\mathcal{T}}$ is a closed fuzzy mapping satisfying condition (∗) with constant functions $\alpha \left(u\right)=1,$ for all $u\in \mathcal{H}.$ Also

$${\left[{\tilde{\mathcal{T}}}_u\right]}_{\alpha \left(u\right)}={\left[\mathcal{X}{\mathcal{T}}_u\right]}_1=\left\{\vartheta \in \mathcal{H} :{\mathcal{T}}_u\left(\vartheta \right)=1\right\}=\mathcal{T}\left(u\right).$$

(35)

• then Problem (1), is analogous to finding $u\in \mathcal{H}$, such that

$$\begin{gathered} \left({\tilde{\mathcal{T}}}_u\right)\left(p\right)=1 , \mathrm{i}.\mathrm{e}., p\in \mathcal{T}\left(u\right). \\ \langle \mathcal{M}\left(p\right), \xi \left(\vartheta ,u\right)\rangle +\mathcal{J}\left(\vartheta \right)-\mathcal{J}\left(u\right)+\omega \Vert \xi {\left(\vartheta ,u\right)\Vert }^2\ge 0, \forall \vartheta \in \mathcal{H}. \end{gathered}$$

(36)

Therefore, the conclusion of Theorem 4 can be obtained from Theorem 2 and Theorem 3 immediately. □

Algorithm 2.

At $n=0,$ start with initial value  $u_0\in \mathcal{H},$   $p_0\in \mathcal{T}\left(u_0\right),$ from Theorem 2, the auxiliary problem (14) has a unique solution $u_1\in \mathcal{H}$ , such that

$$\langle u_1, \vartheta -u_1\rangle \ge \langle u_0, \vartheta -u_1\rangle -\rho \langle \mathcal{M}\left(p_0\right), \xi \left(\vartheta ,u_1\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_1\right)-\rho \omega \Vert \xi {\left(\vartheta ,u_1\right)\Vert }^2, \forall \vartheta \in \mathcal{H}$$

Since $p_0\in \mathcal{T}\left(u_0\right),$ then by Nadler’s Lemma 2, there exist $p_1 \in \mathcal{T}\left(u_1\right)$ , such that

$$\Vert p_0-p_1\Vert \le \left(1+1\right)D\left(\mathcal{T}\left(u_0\right), \mathcal{T}\left(u_1\right)\right),$$

For $n=1,$   $u_1\in \mathcal{H},$  $p_1\in \mathcal{T}\left(u_1\right),$ again from Theorem 2, the auxiliary problem (14) has a unique solution $u_2\in \mathcal{H}$ , such that

$$\langle u_2, \vartheta -u_2\rangle \ge \langle u_1, \vartheta -u_2\rangle -\rho \langle \mathcal{M}\left(p_1\right), \xi \left(\vartheta ,u_2\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_2\right)-\rho \omega \Vert \xi {\left(\vartheta ,u_2\right)\Vert }^2, \forall \vartheta \in \mathcal{H},$$

Since $p_1\in \mathcal{T}\left(u_1\right),$ then by Nadler’s Lemma 2, there exist $p_2\in \mathcal{T}\left(u_2\right)$ , such that

$$\Vert p_1-p_2\Vert \le \left(1+\frac{1}{2}\right)D\left(\mathcal{T}\left(u_1\right), \mathcal{T}\left(u_2\right)\right),$$

At step $n,$ we can obtain sequences  $u_n\in \mathcal{H},$  $p_n\in \mathcal{T}\left(u_n\right)\in CB\left(\mathcal{H}\right),$ such that

(i)

$\Vert p_n-p_{n+1}\Vert \le \left(1+\frac{1}{1+n}\right)D\left(\mathcal{T}\left(u_n\right), \mathcal{T}\left(u_{n+1}\right)\right),$

(ii)

$\langle u_{n+1}, \vartheta -u_{n+1}\rangle \ge \langle u_n, \vartheta -u_{n+1}\rangle -\rho \langle \mathcal{M}\left(p_n\right), \xi \left(\vartheta ,u_{n+1}\right)\rangle -\rho \mathcal{J}\left(\vartheta \right)+\rho \mathcal{J}\left(u_{n+1}\right)$

$$-\rho \omega {\Vert \xi \left(\vartheta ,u_{n+1}\right)\Vert }^2, \forall \vartheta \in \mathcal{H}, \forall n\ge 0$$

(37)

If $\mathcal{M}=I$ (identity mapping),  $p\in \mathcal{H}$ and $\mathcal{T}:\mathcal{H}\to \mathcal{H}$ is single valued, then Theorem 4, reduces to:

Theorem 5.

Let $\mathcal{T}:\mathcal{H}\to H$ be $\mathcal{T}$ >-S-monotone with constant  $\beta >0$ and $\mathcal{T}$ -L-continuous with constant  $\lambda >0$ respectively. Let  $\mathcal{J}:H\to R$ be a nondifferentiable mapping and the bi-function $\xi \left(.,.\right)$ is S-monotone with constant  $\mu >0$ and L-continuous with constant  $\delta >0$ , respectively. If  $\xi \left(.,.\right)$ satisfies following condition

$$\xi \left(\vartheta ,u\right)+\xi \left(u,\vartheta \right)=0 (\mathrm{and} \mathrm{so} \xi \left(u,u\right)=0, \mathrm{for} \mathrm{all} u\in \mathcal{H}), \mathrm{for} \mathrm{all} u,\vartheta \in \mathcal{H}.$$

then for constant $\rho >0$ , such that

$$0<\rho <\frac{2\left(\beta -\left(\mathsf{\eta }+2\omega {\delta }^2\right)\right)}{{\left(\gamma \right)}^2-\left(\mathsf{\eta }+2\omega {\delta }^2\right)}, \rho <\frac{1}{\left(\mathsf{\eta }+2\omega {\delta }^2\right)}, \beta >\mathsf{\eta }+2\omega {\delta }^2,$$

(38)

where

$$\mathsf{\eta }=\gamma \sqrt{1-2\mu +{\delta }^2}.$$

Then there exist $u\in K,$ satisfying “PVLI” (6) and the sequences $\left\{u_n\right\}$ generated by (23) converge strongly to $u$ , see [31].

Remark 2.

At the end, we would like to mention that many earlier defined familiar methods as well as decent, projection techniques and its mixed forms, relaxation, and Newton’s methods that can be obtained form auxiliary “PFMVLI”. Similarly, for suitable choice of the fuzzy mapping  $\mathcal{T}:\mathcal{H}\to \mathcal{F}\left(\mathcal{H}\right)$  and bi-function  $\xi \left(.,.\right):\mathcal{H}\times \mathcal{H}\to \mathcal{H},$  many old of (fuzzy) “VI” and (fuzzy) variational inclusion and corresponding (fuzzy) optimization problems from the “PFMVLI” (3) can be obtain. In future, we will try to find “PFMVLI” for higher order strongly-preinvex fuzzy mappings.

5. Conclusions

In this paper, we have proposed the idea of “PFMVLI”. As a particular case of “PFMVLI”, strongly fuzzy mixed variational inequality are also introduced. With the help of generalized auxiliary principle technique and some new analytic techniques, some existence theorems of auxiliary “PFMVLI” are studied for “PFMVLI” and some iterative methods are obtained for the solution of “PFMVLI”. Then we have obtained some known and new results. There is much room for further study to explore this concept because for suitable choice of the fuzzy mapping $\mathcal{T}\left(.\right)$, $\xi \left(.,.\right)$, $\mathcal{J}\left(.\right)$ and $K$ many new classes of (fuzzy) “VI” and (fuzzy) variational inclusion and corresponding (fuzzy) optimization problems can be obtain from the “PFMVLI” (3).