A Study on Mixed Variational-like Inequality Under Generalized Quasimonotonicity in Banach Spaces

Abstract

This paper studies nonlinear mixed variational-like inequalities under generalized quasimonotone mappings in Banach spaces. The main objective of this work is to relax the well-known KKM (Knaster–Kuratowski–Mazurkiewicz) condition which is extensively used in the literature to prove the existence of solutions for variational inequalities and equilibrium problems, and to establish the existence of solutions for the nonlinear mixed variational-like inequalities, which uncovers another approach of solving variational inequalities. Further, we propose an iterative algorithm to find approximate solutions to our problem and to study its convergence criteria. Finally, as an application, we find a gap function for nonlinear mixed variational-like inequalities, which uncovers another way of solving our variational-like inequalities using the methods of solutions and algorithms of the optimization problems.

1. Introduction

Let $\mathbb{B}$ be a real reflexive Banach space, ${\mathbb{B}}^{\ast }$ be its dual, and K be a non-empty closed and convex subset of B. Let us denote the duality pairing between $u^{\ast }\in {\mathbb{B}}^{\ast }$ and $u\in \mathbb{B}$ by $\langle u^{\ast },u\rangle$. Let $\eta :K\times K\to \mathbb{B}$ and $\alpha :\mathbb{B}\to \mathbb{R}$ be the mappings. In this paper, we consider the following problem: find $w\in K$, such that, for all $y,v\in K$, we obtain the following:

$$\langle N(w,y),\eta (v,w)\rangle +b(w,v)-b(w,w)\ge 0,$$

(1)

where $N:\mathbb{B}\times \mathbb{B}\to {\mathbb{B}}^{\ast }$ is a weakly relaxed $\eta -\alpha$ quasimonotone operator which is $\eta -$hemicontinuous, and the bi-function $b:\mathbb{B}\times \mathbb{B}\to \mathbb{R}$ is a convex and lower semicontinuous function in the second variable. The problem is defined as a nonlinear mixed variational-like inequality problem (NMVLIP) because of the presence of the nonlinear bi-function b. Such classes of problems find important applications in fields like structural analysis [1] and optimization theory [2,3].

The monotonicity concept executes a crucial role for establishing the existence and uniqueness of solutions to variational inequalities and complementarity problems. Efforts are always made to use the weakest kind of monotonicity, so that the results obtained on the variational inequalities can be applied to a broader class of problems. Some generalizations of monotonicity have been studied in [4,5,6,7,8,9,10] and the references therein. Studies on the generalization of monotonicity are also important from the application point of view, because of its connections to generalized convexity in mathematical programming problems [9]. Out of the numerous research papers on generalized monotonicities and variational inequalities, we name a few which are mostly connected to this work. For the set-valued maps, strongly $\eta -\alpha$ monotonicity was studied by Huang and Deng [11] in order to explore strongly nonlinear mixed variational-like inequalities. Under relaxed $\eta -\alpha$ monotonicity assumptions, Fang and Huang [12] found the solutions for variational-like inequalities in the framework of reflexive Banach spaces. Later on, Bai et al. [13] carried out an extension of the work of Fang and Huang [12] by introducing relaxed $\eta -\alpha$ pseudomonotonicity. Kutbi and Sintunavarat [8], in 2013, proposed some results on the existence of variational-like inequalities involving a single-valued operator under weakly relaxed $\eta -\alpha$ monotonicity. However, in these works, the numerical aspect was not taken into consideration.

Further, all the aforementioned works are based on a common source, the so-called Knaster–Kuratowski–Mazurkiewicz (KKM) technique [14]. This technique involves formulating the variational inequality problem as a set-valued map, which is called a KKM map, and is, furthermore, closed for all values and compacted for some points in the domain. Then, it can be predicted that the map has a non-empty intersection over a certain set. However, showing a mapping as a KKM mapping is not always simple, as one can predict.

A large number of authors in the literature have developed various numerical methods to solve variational inequalities. Out of these, the projection method [15,16], the projection and contraction method [17], the auxiliary problem principle technique [18,19], and the subgradient extragradient methods [20] are noteworthy. It is noted that the projection type methods cannot be applied to the Problem (1), because of the presence of the nonlinear bi-function b. Therefore, in this regard, the auxiliary principle technique is worthwhile. The concept of an auxiliary problem principle technique was invented by Cohen [18], in the year 1988, in order to solve variational inequalities. Later on, a rigorous study was carried out on the auxiliary principle technique, including other works; in particular, the works of [19,21,22,23] are notable. Additionally, Pany et al. [10,24] provided some iterative algorithms that can be used in order to find some approximate solutions to mixed invex equilibrium problems and mixed variational-like inequalities and analyzed their convergence criteria.

On the other hand, various researchers in the literature have found the solutions of variational inequalities through the introduction of gap functions; see, for instance, [9,25,26]. The gap function approach is particularly useful for computational purposes, specifically for constructing new globally convergent algorithms, studying the rate of convergence of some iterative methods and finding the error bounds, a measure of the distance between solution set and an arbitrary point. The concept of gap function bridges the gap between optimization and variational inequality by converting the variational inequality to an optimization problem (see [27,28,29,30]).

Inspired and motivated by these works, we find the solutions of the Problem (1) using the auxiliary principle technique. Although the results use a similar procedure as in [10], the difference lies in the underlying operator for the variational-like inequality problem. The operator is a weakly relaxed $\eta -\alpha$ quasimonotone, in this context. In the next section, the novelty of this operator is explained with some examples. We then formulate the corresponding gap function for the nonlinear mixed variational-like inequality problem and show that the solution set for the NMVLIP (1) is equivalent to the optimization problem involving the gap function, which leads to another approach for solving the Problem (1). Motivated by the interesting aspects of gap functions, we formulate the gap function for the NMVLIP (1) studied in this paper. We then attempt to relax the KKM condition. In this context, we use the results from the study by Abbasi and Rezaie [31]. To the best of our knowledge, there have not been any attempts to relax the KKM technique in the context of mixed variational-like inequalities. Furthermore, we investigate the solution procedure for this problem. Our results extend and generalize the results of Cohen [18], Fang and Huang [12], Pany and Pani [10], and others.

The remainder of this paper is organized as set out below. Section 2 provides some definitions and preliminaries that we require, together with some examples showing the importance of our weakly relaxed $\eta -\alpha$ quasimonotone operator. Section 3 finds the existence of solutions for the nonlinear mixed variational-like inequalities. Section 4 focuses on the approximate solutions to our problem and its convergence criteria. As an application of our problem, we find the corresponding gap function for our nonlinear mixed variational-like inequalities in Section 5, which converts our variational inequalities into optimization problems.

2. Preliminaries

Let K be a non-empty closed and convex subset of a real reflexive Banach space $\mathbb{B}$ and ${\mathbb{B}}^{\ast }$ be its dual. For any subset $A\subset \mathbb{B}$, we shall denote the convex hull of A by $co\left(A\right)$ and the interior of A by $intA$.

Definition 1. 

Let K be a convex subset of $\mathbb{B}$ and $\eta :K\times K\to \mathbb{B}$ be a mapping. An operator T from K to ${\mathbb{B}}^{\ast }$ is said to be $\eta -$hemicontinuous if the function $f:[0,1]\to (-\infty ,+\infty )$, defined by $f\left(t\right)=\langle T(x+t(y-x\left)\right),\eta (y,x)\rangle$, is continuous at 0.

Definition 2 

([32]). Let K be a convex subset of $\mathbb{B}$ and let $f:K\times K\to \mathbb{R}$ be a mapping. $f(u,v)$ is said to be $0-$diagonally concave in v if, for any finite subset $\{v_1,v_2,...,v_n\}\subset K$ and any $w={\sum }_{i=1}^m{\lambda }_i{v}_i$, with ${\lambda }_i\ge 0$, for $i=1,2,...,m$ and ${\sum }_{i=1}^m{\lambda }_i=1$, such that the following is true:

$$\sum _{i=1}^m{\lambda }_{i}f(w,v_i)\le 0.$$

The $0-$diagonally concavity of $f(u,v)$ in v, given by Zhou and Chen in [32], was then extended by Pany and Pani [10] in 2015, as follows:

Definition 3 

([10]). Let K be a convex subset of $\mathbb{B}$ and let $N:K\times K\to {\mathbb{B}}^{\ast }$ and $\eta :K\times K\to \mathbb{B}$ be the mappings. N and η are said to be $0-$diagonally concave if the function $\varphi :K\times K\to \mathbb{R}$, as defined by the following:

$$\varphi (w,v)=\langle N(w,v),\eta (w,v)\rangle$$

where $0-$ is diagonally concave in the second argument. If $-N$ and η are $0-$diagonally concave, then N and η are said to be $0-$diagonally convex on K.

Definition 4. 

Let $N:K\times K\to {\mathbb{B}}^{\ast }$ and let $\eta :K\times K\to \mathbb{B}$ be the mappings. The operator N is said to be $\eta -$monotone in first argument if the following is true:

$$\langle N(w_1,v)-N(w_2,v),\eta (w,v)\rangle \ge 0,\forall w,v\in K.$$

Definition 5. 

Let $N:K\times K\to {\mathbb{B}}^{\ast }$ and let $\eta :K\times K\to \mathbb{B}$ be the mappings. The condition for the $\eta -$antimonotonicity of N in second argument is given by the following equation:

$$\langle N(w,v_1)-N(w,v_2),\eta (w,v)\rangle \le 0,forallw,v\in K.$$

Definition 6. 

A single-valued mapping $f:K\to (-\infty ,+\infty ]$ is said to be lower semicontinuous at $x_0$ if $f\left(x_0\right)\le \mathit{lim}{\mathit{inf}}_{x\to x_0}f\left(x\right)$.

Definition 7. 

An operator $T:K\to {\mathbb{B}}^{\ast }$ is Lipschitz continuous if there exists a positive constant $\alpha ,$ such that $\parallel Tx-Ty\parallel \le \alpha \parallel x-y\parallel ,$$\mathrm{for}\mathrm{all}x,y\in K.$

Definition 8. 

Let $T:K\to {\mathbb{B}}^{\ast }$, $\eta :K\times K\to \mathbb{B}$ and let $f:K\to (-\infty ,+\infty ]$ be the mappings. T is said to be $\eta -$coercive with respect to f, if there exists $x_0\in K$, such that the following is true:

$${\displaystyle \frac{\langle Tx-T{x}_0,\eta (x,x_0)\rangle +f\left(x\right)-f\left(x_0\right)}{\parallel \eta (x,x_0)\parallel }}\to \infty .$$

Definition 9. 

Let $T:K\to {\mathbb{B}}^{\ast }$ and $\eta :K\times K\to \mathbb{B}$ be the mappings, and let $\alpha :\mathbb{B}\to \mathbb{R}$ be the real valued function, such that $\alpha \left(tz\right)=t^p\alpha \left(z\right)$ for all $z\in \mathbb{B}$ and for all $t,p\in \mathbb{R}(t>0 and p>1)$. The operator T is said to be

(i) 

relaxed $\eta -\alpha$ monotone if the following holds:

$$\langle Tx-Ty,\eta (y,x)\rangle \ge \alpha (x-y),\forall x,y\in K,$$

(ii) 

relaxed $\eta -\alpha$ pseudomonotone if the following holds:

$$\langle Ty,\eta (x,y)\rangle \ge 0⟹\langle Tx,\eta (x,y)\rangle \ge \alpha (x-y),\forall x,y\in K,$$

(iii) 

relaxed $\eta -\alpha$ quasimonotone if the following holds:

$$\langle Ty,\eta (x,y)\rangle >0⟹\langle Tx,\eta (x,y)\rangle \ge \alpha (x-y),\forall x,y\in K.$$

Definition 10. 

Let $T:K\to {\mathbb{B}}^{\ast }$, let $\eta :K\times K\to \mathbb{B}$ be the mappings, and let $\alpha :\mathbb{B}\to \mathbb{R}$ be the real valued function, such that $\underset{t\to 0}{lim}{\displaystyle \frac{\alpha \left(tz\right)}{t}}=0\mathrm{and}\underset{t\to 0}{lim}{\displaystyle \frac{d}{dt}}\alpha \left(tz\right)=0$ for all $z\in \mathbb{B}$ and $t\in \mathbb{R}(t>0)$. The operator T is said to be

(i) 

strongly $\eta -$ monotone if there exists a constant $c\in \mathbb{R}(c>0)$, such that the following is true:

$$\langle Tx-Ty,\eta (y,x)\rangle \ge {c\parallel x-y\parallel }^2,\forall x,y\in K,$$

(ii) 

weakly relaxed $\eta -\alpha$ quasimonotone if the following is true:

$$\langle Ty,\eta (x,y)\rangle >0⟹\langle Tx,\eta (x,y)\rangle \ge \alpha (x-y),\forall x,y\in K.$$

Remark 1. 

The following may be observed from the above definitions:

1. 

If $\alpha (x-y)$ takes the form of ${c\parallel x-y\parallel }^2,$ then the relaxed $\eta -\alpha$ monotonicity reduces to strongly $\eta -$ monotonicity. In other words, the strongly $\eta -$ monotone operator comes under the particular category of a relaxed $\eta -\alpha$ monotone operator.

2. 

The definition of weakly relaxed $\eta -\alpha$ monotonicity involves two new conditions on $\alpha :\mathbb{B}\to \mathbb{R}.$ It may be noted that, if $\alpha \left(tz\right)=t^p\alpha \left(z\right),$ then $\underset{t\to 0}{lim}{\displaystyle \frac{\alpha \left(tz\right)}{t}}=0,\mathrm{and}\underset{t\to 0}{lim}{\displaystyle \frac{d}{dt}}\alpha \left(tz\right)=0,$ but the converse is not always true. A supporting example in this context has been provided in [10] (see Example 1). In this sense, weakly relaxed $\eta -\alpha$ monotonicity is a generalized version of relaxed $\eta -\alpha$ monotonicity.

3. 

Similarly, weakly relaxed $\eta -\alpha$ quasimonotonicity is a generalized version of weakly relaxed $\eta -\alpha$ monotonicity.

We illustrate the relationship among different kinds of monotonicity below.

However, the converse relationships in Figure 1 do not hold. For example, the class of weakly relaxed $\eta -\alpha$ monotone operators does not include the class of relaxed $\eta -\alpha$ monotone operators, which has been proven by means of examples in [10]. The following example shows that there is an operator which is a weakly relaxed $\eta -\alpha$ quasimonotone, but not a relaxed $\eta -\alpha$ quasimonotone.

Example 1. 

Consider $T\left(x\right)=x^2$ and the underlying space as $K=(-\infty ,+\infty ).$ The bi-functions $\eta :K\times K\to \mathbb{R}$ and $\alpha :\mathbb{B}\to \mathbb{R}$ are given by the following:

$$\eta (x,y)\left\{\begin{array}{cc}=3(x-y);\hfill & \hfill x>y\\ >2(x-y);\hfill & \hfill x\le y\end{array}\right.$$

$$\alpha \left(z\right)\left\{\begin{array}{cc}=-{sin}^{2}z;\hfill & \hfill z>0\\ >{sin}^{2}z;\hfill & \hfill z\le 0\end{array}\right.$$

It can be shown that T is a weakly relaxed $\eta -\alpha$ quasimonotone, but, since $\alpha \left(tz\right)\ne t^p\alpha \left(tz\right),$ T is not a relaxed $\eta -\alpha$ quasimonotone.

In the following example, we show that there is a weakly relaxed $\eta -\alpha$ quasimonotone operator which is not a weakly relaxed $\eta -\alpha$ monotone operator.

Example 2. 

In the previous example, if $x=0,y={\displaystyle \frac{-\pi }{2}},$ then T is a weakly relaxed $\eta -\alpha$ quasimonotone. Now, if $x=0$, and $y={\displaystyle \frac{-\pi }{2}},$ the following is true:

$$\begin{array}{c}\hfill \langle Tx-Ty,\eta (x,y)\rangle =3{(x-y)}^2(x+y).\end{array}$$

For T to be weakly relaxed $\eta -\alpha$ monotone, the following should hold:

$$\begin{array}{c}\hfill 3{(x-y)}^2(x+y)\ge -{sin}^2(x-y).\end{array}$$

The inequality finally leads to $\pi <2,$ which is impossible. Hence, T is not a weakly relaxed $\eta -\alpha$ monotone.

Let X be a Hausdorff topological vector space and K be a non-empty subset X. A multi-valued mapping $F:K\to 2^X$ is said to be a Knaster–Kuratowski–Mazurkiewicz (KKM) mapping if, for any finite set $\{x_1,x_2,\cdots ,x_n\}$ in K, $co\left(\{x_1,x_2,\cdots ,x_n\}\right)$ is a subset of ${\displaystyle \bigcup _{i=1}^{n}F\left(x_i\right)}$.

Lemma 1 

([14]). Suppose X is a Hausdorff topological vector space. Consider $K\subset X$ to be non-empty and the multi-valued mapping $F:K\to 2^X$ to be KKM. Let $F\left(x\right)$ be closed in X for all x in K and compact in X for some x in K; then, the following holds:

$$\bigcap _{x\in K}F\left(x\right)\ne Ø.$$

As we pointed out earlier, it may not be always easy to verify whether a map satisfies the KKM condition. In order to relax the KKM condition of the underlying map, we need to consider the following result of Abbasi and Rezaie [31].

Lemma 2 

([31]). Suppose X is a Hausdorff topological vector space and $A\subseteq X$ is a convex set. Let $F:A\to 2^X$ be a set-valued mapping, which is closed for all x in A, and let $F\left(x\right)$ be compact for some $x\in A.$ Furthermore, if F satisfies the following conditions:

(i) 

$x\in F\left(x\right)$ for each $x\in A;$

(ii) 

$F(\lambda x+(1-\lambda \left)u\right)\subseteq F\left(x\right)\cup F\left(u\right)$ for any $\lambda \in [0,1]$ and $x,u\in A,$

then ${\cap }_{x\in A}F\left(x\right)\ne Ø.$

In particular we make use of the following example in [31] to establish the existing results.

Example 3 

([31], Example 2.4). If G in the following form:

$$G\left(y\right)=\{u\in F(M):u\notin y+ϵ+intC\}$$

is a set valued mapping defined on $F\left(M\right),$ where $F:M\to 2^X,$ such that $F\left(M\right)$ is convex, closed and bounded, $F\left(M\right)={\cup }_{x\in M}F\left(x\right)$, and C is a pointed closed convex cone in X with a non-empty interior and $ϵ\in intC,$ then G satisfies the conditions of Lemma 2.

In order to establish the existence of solutions, we need to consider the following results from Ding and Tan [33].

Lemma 3 

([33]). Let X be a topological vector space and K be a non-empty convex subset of X, and $\psi :K\times K\to \mathbb{R}$ be a mapping, such that the following is true:

(i) 

for each $x\in K$, the mapping $y\mapsto \psi (x,y)$ is lower semicontinuous on every non-empty compact subset of K;

(ii) 

for each non-empty finite set $\{x_1,\cdots ,x_m\}\subset K$ and for each $y={\displaystyle \sum _{i=1}^m}{\lambda }_i{x}_i,({\lambda }_i\ge 0,$${\displaystyle \sum _{i=1}^m}{\lambda }_i=1)$ $satisfies\underset{1\le i\le m}{min}\psi (x_i,y)\le 0$;

(iii) 

there exists a non-empty compact convex subset $X_0$ of K and a non-empty compact subset D of K, such that, for each $y\in K\backslash D,$ there is an $x\in co(X_0\cup \left\{y\right\})$ with $\psi (x,y)>0$.

Then, there exists an $\widehat{y}\in D$, such that $\psi (x,\widehat{y})\le 0$ for all $x\in K$.

3. Existence of Solutions for Nonlinear Mixed Variational-like Inequalities

In this section, we shall find the solutions for the nonlinear mixed variational-like inequalities, with the aim to relax the KKM conditions.

Theorem 1. 

Let K be a non-empty, closed, convex, and bounded subset of $\mathbb{B}$, and let $\eta :K\times K\to \mathbb{B}$ and $\alpha :\mathbb{B}\to \mathbb{R}$ be the mappings. Suppose $N:K\times K\to {\mathbb{B}}^{\ast }$ is $\eta -$hemicontinuous and a weakly relaxed $\eta -\alpha$ quasimonotone operator, and suppose $b:\mathbb{B}\times \mathbb{B}\to \mathbb{R}$ is convex and a lower semicontinuous mapping in the second argument. In addition, let us assume the following are true:

(i) 

the bi-function η is antisymmetric on K;

(ii) 

N is η−convex with respect to the second argument, and lower semicontinuous for any $w,y\in K$;

(iii) 

$\left\{\alpha \left(v_{\beta }\right)\right\}$ is weakly lower semicontinuous;

(iv) 

$\alpha (w-v)\ge 0$, if $w\ne v$.

Then, the NMVLIP (1) has a solution.

Proof. 

Let us define the set-valued mappings $F,G:K\to 2^{\mathbb{B}}$ as follows:

$$F\left(z\right)=\{x\in K,\langle N(x,y),\eta (z,x)\rangle +b(x,z)-b(x,x)\ge 0\},$$

$$G\left(z\right)=\{x\in K,\langle N(z,y),\eta (z,x)\rangle +b(x,z)-b(x,x)\ge \alpha (x-z\left)\right\}.$$

For each $z\in K$, $G\left(z\right)$ is weakly closed, as $\alpha$ is weakly lower semicontinuous; also, the mappings $z\mapsto \langle N(x,y),\eta (z,x)\rangle$ and $z\mapsto b(x,z)$ are convex and lower semicontinuous and, thus, weakly lower semicontinuous. Since K is weakly compact, as it is bounded, closed, and convex in $\mathbb{B}$, which is reflexive, $G\left(z\right)$ is weakly compact. Now, when x is different from $z,$$F\left(z\right)$ may take the following form:

$$F\left(z\right)=\{x\in G(z):x\notin z+ϵ+intC\},$$

where C is a pointed closed convex cone in $\mathbb{B}$ with a non-empty interior and $ϵ\in intC$; hence, $F\left(z\right)$ satisfies the conditions of Lemma 2. So, the family $\left\{F\right(z\left)\right\}$ has a non-empty intersection over K, as follows:

$$\bigcap _{z\in K}F\left(z\right)\ne Ø.$$

Thus, we conclude that there exists $x\in K$, such that, for all $y,z\in K$, we obtain the following:

$$\langle N(x,y),\eta (z,x)\rangle +b(x,z)-b(x,x)\ge 0.$$

□

Remark 2.

1. 

If we fix $v\in K$, then the mapping $N:K\times K\to {\mathbb{B}}^{\ast }$ reduces to the the mapping $T:K\to {\mathbb{B}}^{\ast }$, defined by $T\left(u\right):=N(u,v)$; similarly, by fixing $u\in \mathbb{B}$, the mapping $b:\mathbb{B}\times \mathbb{B}\to \mathbb{R}$ reduces to the mapping $f:\mathbb{B}\to \mathbb{R}$, defined by $f\left(v\right):=b(u,v)$. Furthermore, we assume that N is a weakly relaxed $\eta -\alpha$ quasimonotone, which is a proper generalization of the assumption made by Fang and Huang [12] that T is a relaxed $\eta -\alpha$ monotone, as shown in Figure 1 and the explanation below it. Taking into account all of these, we conclude that Theorem 1 is a proper generalization of Theorem 2.2 from Fang and Huang [12].

2. 

In the proof of Theorem 2 from Pany and Pani [10], the authors used the KKM technique, which is not an easy task, as we remarked in Section 1. We have proven Theorem 1 by relaxing the KKM condition and, thus, we have given here an alternative and simple proof of Theorem 2 from Pany and Pani [10]. Furthermore, we have weakened the stronger assumption that N is a weakly relaxed $\eta -\alpha$ to a weakly relaxed $\eta -\alpha$ quasimonotone. Therefore, Theorem 1 is a generalization of Theorem 2 from Pany and Pani [10].

The existence of solution to (1) can be extended when the underlying set is unbounded, as follows:

Theorem 2. 

Let K be a non-empty closed and convex subset of $\mathbb{B}$, and let $\eta :K\times K\to \mathbb{B}$ and $\alpha :\mathbb{B}\to \mathbb{R}$ be the mappings. Suppose $N:K\times K\to {\mathbb{B}}^{\ast }$ is $\eta -$hemicontinuous and a weakly relaxed $\eta -\alpha$ quasimonotone operator, and suppose $b:\mathbb{B}\times \mathbb{B}\to \mathbb{R}$ is convex and a lower semicontinuous mapping in the second argument. Furthermore, suppose the following conditions hold:

(i) 

(Coercivity) For each $w_0\in K$, we have the following:

$$\underset{\parallel v\parallel \to \infty }{lim}{\displaystyle \frac{\langle N(w,y)-N(w_0,y),\eta (v,w_0)\rangle +b(w,v)-b(w_0,v)}{\parallel \eta (v,w_0)\parallel }}=\infty ;$$

(ii) 

η is antisymmetric;

(iii) 

N is η−convex in the first argument and lower semicontinuous for both the arguments;

(iv) 

$\alpha \left(v_{\beta }\right)$ is weakly lower semicontinuous.

Then, the NMVLIP (1) has at least one solution.

Proof. 

Considering the problem, we find $w_r\in K\cap B_r$, such that the following holds:

$$\langle N(w_r,y),\eta (v,w_r)\rangle +b(w_r,v)-b(w_r,w_r)\ge 0,\forall v\in K\cap B_r,$$

(2)

where $B_r=\{v\in E:\parallel v\parallel \le r\}.$

As shown in Theorem 1, Problem (2) has a solution, namely $w_r\in K\cap B_r.$ Choosing $\parallel w_0\parallel <r,$ we can put $w_0$ in the place of v in (2), thereby leading to the following:

$$\langle N(w_r,y),\eta (w_0,w_r)\rangle +b(w_r,w_0)-b(w_r,w_r)\ge 0.$$

Now, the following is true:

$$\begin{array}{cc}\hfill & \langle N(w_r,y),\eta (w_0,w_r)\rangle +b(w_r,w_0)-b(w_r,w_r)\hfill \\ \hfill & =-\langle N(w_r,y),\eta (w_r,w_0)\rangle +\langle N(w_0,y),\eta (w_0,w_r)\rangle +\langle N(w_0,y),\eta (w_r,w_0)\rangle \hfill \\ & +b(w_r,w_0)-b(w_r,w_r)\hfill \\ \hfill & =-\langle N(w_r,y)-N(w_0,y),\eta (w_r,w_0)\rangle +b(w_r,w_0)-b(w_r,w_r)+\langle N(w_0,y),\eta (w_0,w_r)\rangle \hfill \\ \hfill & \le \parallel \eta (w_r,w_0)\parallel \left({\displaystyle \frac{-\langle N(w_r,y)-N(w_0,y),\eta (w_r,w_0)\rangle +b(w_r,w_r)-b(w_r,w_0)}{\parallel \eta (w_r,w_0)\parallel }}+\parallel N(w_0,y)\parallel \right).\hfill \end{array}$$

If $\parallel w_r\parallel =r$ and $r\to \infty ,$ then, using the $\eta -$coercivity of N with respect to b in the second variable, the above inequality reduces to the following:

$$\langle N(w_r,y),\eta (w_0,w_r)\rangle +b(w_r,w_0)-b(w_r,w_r)<0.$$

This is a contradiction, as $\langle N(w_r,y),\eta (w_0,w_r)\rangle +b(w_r,w_0)-b(w_r,w_r)\ge 0.$ Therefore, $\parallel w_r\parallel <r.$ Now, for any $v\in K,$ we choose $0<ϵ<1,$ such that the following is true:

$$w_r+ϵ(v-w_r)\in K\cap B_r.$$

Using Assumption $\left(3\right)$ and the convexity of $b,$ we obtain the following from Equation (2):

$$\begin{array}{cc}\hfill 0& \le \langle N(w_r,y),\eta (w_r+ϵ(v-w_r),w_r)\rangle +b(w_r+ϵ(v-w_r))-b(w_r,w_r)\hfill \\ \hfill & \le (1-ϵ)\langle N(w_r,y),\eta (w_r,w_r)\rangle +ϵ\langle N(w_r,y),\eta (v,w_r)\rangle +(1-ϵ)b(w_r,w_r)+ϵb(v,w_r)\hfill \\ & -b(w_r,w_r)\hfill \\ \hfill & =ϵ\langle N(w_r,y),\eta (v,w_r)\rangle +ϵb(w_r,v)-ϵb(w_r,w_r).\hfill \end{array}$$

Therefore, the following is true:

$$\langle N(w_r,y),\eta (v,w_r)\rangle +b(w_r,v)-b(w_r,w_r)\ge 0,\forall y\in K,$$

and $w_r\in K$ is a solution of Problem $\left(1\right).$ Hence, Problem (1) is solvable. □

Remark 3. 

Considering Remark 2, it is clear that Theorem 2 is a proper generalization of Theorem 2.3 from Fang and Huang [12] and Theorem 3 from Pany and Pani [10].

4. Convergence Criteria

There are numerous methods used to find solutions to variational inequality problems. However, the related convergence criteria measure the effectiveness of a corresponding solution procedure. Thus, along with iterative methods, it is equally important to study the convergence criteria. This section first introduces an iterative algorithm for finding approximate solutions to (1). We used the auxiliary principle technique, which was initially proposed by [1]. We first constructed an auxiliary problem and then formulated the associated auxiliary variational inequality problem.

The auxiliary minimizing problem is defined as follows:

$${\min }_{w\in K}\{\alpha \left(w\right)+\langle \rho N(v,y),\eta (w,v)\rangle -\langle {\alpha }^{\prime }\left(v\right),w\rangle +\rho b(v,w)\},$$

(3)

where $w\in E,v\in K$, and $\rho$ is a positive constant. If N is $\eta$−convex in the first argument, then (3) is equivalent to following auxiliary variational inequality, as follows:

$$\langle {\alpha }^{\prime }\left(w\right)-{\alpha }^{\prime }\left(v\right),u-w\rangle \ge -\rho \langle N(v,y),\eta (u,w)\rangle +\rho b(v,w)-\rho b(v,u),\mathrm{for}\mathrm{all}u\in K.$$

(4)

Problem (4) is referred to as the auxiliary variational inequality problem.

Remark 4. 

If w equals to $v,$ then v satisfies (1).

Now, we are in a position to propose an iterative algorithm, as follows:

1.

Let $v_0$ be the initial approximation for $n=0.$

2.

Let $v^{\ast }=v_n$ be the solution of the auxiliary variational inequality at the nth step, and let $v_{n+1}$ be the solution.

3.

If $\parallel v_{n+1}-v_n\parallel \le ϵ,\mathrm{where}ϵ>0,$ stop; otherwise, go to step $\left(2\right).$

The proof for the existence of a solution for the auxiliary problem and the strong convergence of the iterates to the exact solution follow a similar pattern to that of the proofs in [10]. For the sake of completeness, we state and prove the result as follows:

Theorem 3. 

Let K be a non-empty closed and convex subset of $\mathbb{B}$, and let $\eta :K\times K\to \mathbb{B}$ and $\alpha :\mathbb{B}\to \mathbb{R}$ be the mappings, such that α is differentiable and convex on $\mathbb{B}$. Let $N:K\times K\to {\mathbb{B}}^{\ast }$ and $b:\mathbb{B}\times \mathbb{B}\to \mathbb{R}$ be the mappings, such that N and b aew $\eta -$hemicontinuous, convex lower semicontinuous functional in the second variable, linear in the first argument, bounded, and $b(u,v)-b(u,w)\le b(u,v-w)$. In addition, we assume the following conditions:

(i) 

N is η−convex in the first argument and lower semicontinuous in both the arguments;

(ii) 

N satisfies weakly lower semicontinuity, weakly relaxed $\eta -\alpha$ monotonicity, and $\eta -$convexity, with respect to the first argument, for any $w,y$ in K;

(iii) 

N satisfies strong $\eta -$monotonicity, $\eta -$antimonotonicity, and weakly relaxed monotonicity in first and second arguments, respectively;

(iv) 

N is Lipschitz continuous in both the arguments with Lipschitz constants ${\sigma }_1$ and ${\sigma }_2$, respectively;

(v) 

For all $u,v,w\in K$, $\eta (v,w)=\eta (v,u)+\eta (u,w)$;

(vi) 

For all $u,v,w\in K$, $\eta (v,u)+\eta (u,w)=0$;

(vii) 

For a real constant $\delta >0$, η is Lipschitz continuous;

(viii) 

The mappings $x\mapsto N(x,y)$ and $x\mapsto \eta (x,y)$ are $0-$diagonally convex;

(ix) 

The mapping $w\mapsto {\alpha }^{\prime }\left(w\right)$ is continuous from a weak to a strong topology, and ${\alpha }^{\prime }$ is strongly monotone;

(x) 

$\alpha \ne 2\mu ,({\sigma }_1+{\sigma }_2)\delta +\mu >0$ and $0<\rho <{\displaystyle \frac{\rho ({\sigma }_1+{\sigma }_2)\delta }{\alpha -2\mu }}$.

Then, both the nonlinear mixed variational inequality problem and the corresponding auxiliary Problem (3) are solvable, and the related auxiliary variational inequality and the approximates exhibit strong convergence to the exact solution.

Proof. 

Since N is $\eta -$convex with respect to the first argument for any $w,y\in K,$ Assumption $\left(ii\right)$ of Theorem 1 holds, through the condition that $\left(vi\right),$$\eta (v,v)=0.$ Hence, condition $\left(i\right)$ of Theorem 1 also holds. As a result, all the conditions of Theorem 1 are satisfied, and, hence, a solution to Problem (1) exists. In order to prove the second part, we need to prove that all the conditions of Lemma 3 are satisfied. To this aim, let us define a mapping $\varphi :K\times K\to R$ as follows:

$$\varphi (u,w)=\langle {\alpha }^{\prime }\left(v_n\right)-{\alpha }^{\prime }\left(w\right),u-w\rangle -\rho \langle N(v_n,y_n),\eta (u,w)\rangle +\rho b(v_n,w)-\rho b(v_n,u).$$

(5)

Since the mapping $w\mapsto {\alpha }^{\prime }\left(w\right)$ is continuous from a weak topology to a strong topology, the mapping $w\mapsto \langle {\alpha }^{\prime }\left(w\right),w\rangle$ is weak continuous on $K.$ Therefore, $w\mapsto \varphi (u,w)$ is weakly lower semicontinuous, and, thus, Assumption $\left(i\right)$ of Lemma 3 holds good. For the second condition, we assume the contrary. Therefore, there exists $\{u_1,\cdots ,u_n\}\subset K$ and w, which is a convex combination of $u_i,$ such that $\varphi (u_i,w)>0$. Thus, from Equation (5), we obtain the following:

$$\sum _{i=1}^n{\lambda }_i\langle {\alpha }^{\prime }\left(v_n\right)-{\alpha }^{\prime }\left(w\right),u_i-w\rangle -\rho \langle N(v_n,y_n),\eta (u_i,w)\rangle +\rho b(v_n,w)-\rho \sum _{i=1}^n{\lambda }_{i}b(v_n,u_i)>0.$$

As b is convex in the second argument, we obtain the following:

$$\sum _{i=1}^n{\lambda }_i\langle {\alpha }^{\prime }\left(v_n\right)-{\alpha }^{\prime }\left(w\right),u_i-w\rangle -\rho \langle N(v_n,y_n),\eta (u_i,w)\rangle >0.$$

This contradicts Assumption $\left(viii\right)$, so Condition $\left(ii\right)$ of Lemma 3 holds. Now, we consider a set $D=\{v\in K: \parallel v-u^{\ast }\parallel \le \theta \},$ where $\theta ={\displaystyle \frac{1}{\alpha }}[\mu \parallel u^{\ast }\parallel ]+\delta \parallel N(u^{\ast },y)\parallel$; using Conditions $\left(ii\right),\left(iii\right),\left(vi\right),and\hspace{0.17em}\left(viii\right)$ and the conditions on b, we conclude that Condition $\left(iii\right)$ of Lemma 3 holds. Thus, in Lemma 3, there exists $w_0\in K$, such that, for all $u\in K$, we obtain the following:

$$\langle {\alpha }^{\prime }\left(w_0\right)-{\alpha }^{\prime }\left(v_n\right),u-w_0\rangle \ge -\rho \langle N(v_n,y_n),\eta (u,w_0)\rangle +\rho b(v_n,w_0)-\rho b(v_n,u).$$

(6)

Hence, a solution to the Problem (3) exists.

For the convergence analysis, we now consider the following functional $\mathrm{\Gamma }:K\to (-\infty ,+\infty ],$ defined as follows:

$$\mathrm{\Gamma }\left(v\right)=\alpha \left(v_0\right)-\alpha \left(v\right)-\langle {\alpha }^{\prime }\left(v\right),v_0-v\rangle ,$$

where $v_0$ is assumed to be the unique solution of Problem (1). Considering the strong monotonicity of ${\alpha }^{\prime },$ we obtain the following:

$$\mathrm{\Gamma }\left(v\right)=\alpha \left(v_0\right)-\alpha \left(v\right)-\langle {\alpha }^{\prime }\left(v\right),v_0-v\rangle \ge {\displaystyle \frac{\sigma }{2}}{\parallel v-v_0\parallel }^2.$$

Putting $w_0=v_{n+1},u=v_0$ in $\left(vi\right)$ and considering the antisymmetricity of $\eta$ and the strong monotonicity of ${\alpha }^{\prime },$ we obtain the following:

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left(v_n\right)-\mathrm{\Gamma }\left(v_{n+1}\right)& \ge {\displaystyle \frac{\sigma }{2}}{\parallel v_n-v_{n+1}\parallel }^2+\rho \langle N(v_n,y_n),\eta (v_{n+1},v_0)\rangle +\rho b(v_n,v_{n+1})-\rho b(v_n,v_0)\hfill \\ \hfill & ={\displaystyle \frac{\sigma }{2}}{\parallel v_n-v_{n+1}\parallel }^2+\rho \langle N(v_n,y_n)-N(v_0,y_0),\eta (v_{n+1},v_0)\rangle \hfill \\ \hfill & +\rho \langle N(v_0,y_0),\eta (v_{n+1},v_0)\rangle +\rho b(v_n,v_{n+1})-\rho b(v_n,v_0).\hfill \end{array}$$

As $v_0$ is a solution of (1), it follows that the below holds:

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left(v_n\right)-\mathrm{\Gamma }\left(v_{n+1}\right)& \ge {\displaystyle \frac{\sigma }{2}}{\parallel v_n-v_{n+1}\parallel }^2+\rho \langle N(v_n,y_n)-N(v_0,y_0),\eta (v_{n+1},v_0)\rangle \hfill \\ & +\rho [b(v_0,v_0)-b(v_0,v_{n+1})+b(v_n,v_{n+1})-b(v_n,v_0)]\hfill \\ & ={\displaystyle \frac{\sigma }{2}}{\parallel v_n-v_{n+1}\parallel }^2+M.\hfill \end{array}$$

Now, using the conditions on b and the Assumptions $\left(iii\right),\left(iv\right),\left(v\right)$, and $\left(vii\right)$, we obtain the following:

$$\begin{array}{cc}\hfill M=& \rho \langle N(v_n,y_n)-N(v_0,y_0),\eta (v_{n+1},v_0)\rangle \hfill \\ & -\rho [b(v_n-v_0,v_0)-b(v_n-v_0,v_{n+1})+b(v_n-v_0,v_n)-b(v_n-v_0,v_n)]\hfill \\ & \ge \rho [\langle N(v_n,y_n)-N(v_0,y_0),\eta (v_{n+1},v_n)\rangle +\langle N(v_n,y_n)-N(v_0,y_0),\eta (v_n,v_0)\rangle ]\hfill \\ & -\rho [b(v_n-v_0,v_0-v_n)+b(v_n-v_0,v_n-v_{n+1})]\hfill \\ & \ge \rho [\langle N(v_n,y_n)-N(v_0,y_n),\eta (v_n,v_0)\rangle \hfill \\ & +\langle N(v_0,y_n)-N(v_0,y_0),\eta (v_n,v_0)\rangle \hfill \\ & +\langle N(v_n,y_n)-N(v_0,y_n),\eta (v_{n+1},v_n)\rangle \hfill \\ & +\langle N(v_0,y_n)-N(v_0,y_0),\eta (v_{n+1},v_n)\rangle ]\hfill \\ & -\rho \mu [\parallel v_n-v_0{\parallel }^2+\parallel v_n-v_0\parallel \parallel v_n-v_{n+1}\parallel ]\hfill \\ & \ge \rho \alpha \parallel v_n-v_0{\parallel }^2-\rho {\sigma }_1\delta \parallel v_n-v_0\parallel \parallel v_{n+1}-v_n\parallel \hfill \\ & -\rho {\sigma }_2\delta \parallel v_n-v_0\parallel \parallel v_{n+1}-v_n\parallel -\rho \mu [\parallel v_n-v_0{\parallel }^2+\parallel v_n-v_0\parallel \parallel v_n-v_{n+1}\parallel ].\hfill \end{array}$$

Thus, we conclude the following:

$$\mathrm{\Gamma }\left(v_n\right)-\mathrm{\Gamma }\left(v_{n+1}\right)\ge \rho [\alpha -(2\mu +({\sigma }_1+{\sigma }_2)\delta )]{\parallel v_n-v_0\parallel }^2.$$

From Assumption $\left(x\right)$, we deduce that $\left\{\mathrm{\Gamma }\left(v_n\right)\right\}$ is a strictly decreasing sequence and it is non-negative, due to its strong monotonicity property; hence, it converges. So, the sequence $\left\{v_n\right\}$ converges strongly to $v_0$ when $n\to \infty .$ This completes the proof. □

Remark 5.

1. 

By fixing $v\in K$ and defining a mapping $\psi :K\to {\mathbb{B}}^{\ast }$ by $\psi \left(u\right):=N(u,v)$, and by fixing $u\in \mathbb{B}$ and defining a mapping $\phi :\mathbb{B}\to \mathbb{R}$ by $\phi \left(v\right):=b(u,v)$, and, finally, taking $\eta (u,v)=u-v$, we see that our Problem 1 reduces to the following problem: find $w\in K$, such that, for all $v\in K$, we have the following:

$$\langle \psi \left(w\right),v-w\rangle +\phi \left(v\right)-\phi \left(w\right)\ge 0.$$

(7)

Problem 7 has been considered and solved by Cohen [18] under the assumption that $\psi$ is monotone. In Theorem 3, we weakened the very strong assumption of $\psi$ being monotone to the assumption that N is a weakly relaxed $\eta -\alpha$ monotone. Therefore, Theorem 3 is a proper generalization and extension of Theorem 2.2 from Cohen [18].

2. 

In Theorem 3, we give an alternative and simple proof of Theorem 4 from Pany and Pani [10] by replacing the common source of the KKM technique with the relaxed KKM condition, with the help of Lemma 2.

5. Application to the Mathematical Programming Problem

This section deals with the formulation of the gap function corresponding to the nonlinear mixed variational-like Inequality (1). The variational-like inequality problem is then studied as an optimization problem, which involves the gap function.

Definition 11 

([25]). For a mapping $T:K\to {\mathbb{B}}^{\ast }$, we consider the following problem: find $x\in K$, such that the following holds:

$$\langle T\left(x\right),y-x\rangle \ge 0,\forall y\in K.$$

(8)

A gap function for the variational inequality Problem (8) is defined as $\varphi :K\to \mathbb{R}\cup \{+\infty \}$, such that the following hold:

• $\varphi \left(x\right)\le 0,\mathrm{for}\mathrm{all}x\in K,$
• $\varphi \left(x^{\ast }\right)=0$, if and only if $x^{\ast }$ is a solution of the variational inequality.

It has been established by Yang [25] that solving the following variational-like inequality,

$$x\in K:\langle T\left(x\right),\eta (y,x)\rangle \ge 0,\mathrm{for}\mathrm{all}y\in K,$$

is equivalent to solving the following maximization problem:

$$\underset{x\in K}{max}\varphi \left(x\right),$$

where the objective function is the gap function corresponding to the variational-like inequality problem.

The following problem, reported by Chen and Luo [9], is known as a relaxed Minty variational-like inequality problem, as follows:

$$x\in K,\langle T\left(y\right),\eta (x,y)\rangle \ge \alpha (y-x),\mathrm{for}\mathrm{all}y\in K,\mathrm{where}T:K\to X^{\ast }.$$

(9)

The gap function corresponding to the Problem 9 is as follows:

$${\varphi }_1\left(x\right)=\underset{y\in K}{min}\langle T\left(y\right),\eta (y,x)\rangle -\alpha (y-x).$$

Subsequently, Khan and Chen [26] considered generalized mixed quasivariational inequality problems using regularized gap function and D-gap function, where the global error bounds for the solution of the problem were also analyzed. However, the framework of study was a Hilbert space, while, in this work, we attempted to carry out our study in the framework of Banach spaces. With this aim, we consider the following function for the NMVLIP (1):

$${\varphi }_0\left(x\right)\equiv \underset{w\in K}{min}\langle N(w,y),\eta (v,w)\rangle +b(w,w)-b(w,v).$$

(10)

Theorem 4. 

The function ${\varphi }_0\left(x\right)$ defined in (10) is the gap function for the NMVLIP (1), if $\eta (x,x)=0$.

Proof. 

For any $w\in K$, the following holds:

$$\begin{array}{cc}\hfill {\varphi }_0\left(w\right)& =\underset{w\in K}{min}\langle N(w,y),\eta (v,w)\rangle +b(w,w)-b(w,v)\hfill \\ \hfill & \le \langle N(w,w),\eta (w,w)\rangle +b(w,w)-b(w,w)\hfill \\ \hfill & =0.\hfill \end{array}$$

Thus, ${\varphi }_0\left(x\right)$ satisfies the first condition of gap function. Now, if ${\varphi }_0\left(x\right)=0,$

$$\hspace{0.17em}\underset{w\in K}{min}\langle N(w,y),\eta (v,w)\rangle =b(w,w)-b(w,v)$$

$$\mathrm{or}\langle N(w,y),\eta (v,w)\rangle \ge b(w,w)-b(w,v).$$

This completes the proof. □

Recalling the concept in [25], it may be stated that solving NMVLIP (1) is the same as solving the following optimization Problem (11):

$$\underset{x\in K}{max}{\varphi }_0\left(x\right).$$

(11)

6. Discussion

In this paper, we explored the existence of the solutions for the NMVLIP (1) under a kind of generalized monotonicity, called weakly relaxed $\eta -\alpha$ quasimonotonicity, in the case of both bounded and unbounded sets. We attempted to relax the KKM condition while deriving its existence criteria. Further, using the auxiliary principle technique, we constructed an algorithm and established that the iterates scheme converges strongly with the exact solution. However, the important aspect of finding the rate of convergence still needs to be achieved. In future, we aim to obtain the corresponding error bounds for the type of gap function proposed here, as well as study some other types of gap functions, like regularized gap functions, in this connection. Studies in this regard can be carried out involving the generalized convex function. The research in this direction may lead to some interesting applications in elliptic boundary value problems, as studied in [30], for variational and hemivariational inequalities.