A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

Gupta, Ankit; Kumar, Satish; Sarma, Ratna Dev; Garg, Pankaj Kumar; George, Reny

doi:https://doi.org/10.1155/2021/4488217

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Nonlinear and Variational Analysis and their Applications 2021

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Research Article | Open Access

Volume 2021 | Article ID 4488217 | https://doi.org/10.1155/2021/4488217

A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

Ankit Gupta,¹Satish Kumar,²Ratna Dev Sarma,³Pankaj Kumar Garg,³and Reny George⁴

Academic Editor: Yuanfang Ru

Received30 Apr 2021

Revised01 Jul 2021

Accepted07 Jul 2021

Published28 Jul 2021

Abstract

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in the proof. The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.

1. Introduction

Variational inequalities have appeared as a working and important tool to investigate various fields of mathematics as well as of sciences including elasticity, vector equilibrium problems, and optimization problems [1–4]. In mid-sixties, Browder [5] formulated and proved the basic existence results for the solutions to a class of nonlinear variational inequality problems. He used a reflexive Banach space and a monotone nonlinear map from the space to its dual space , to set up the nonlinear variational inequality problem. Browder used the property of hemicontinuity and monotonicity of mapping along with the lower semicontinuity of , for providing the existence of the solution of nonlinear variational inequality problem. After that, this problem has been generalized and extended in various directions under different set-ups using different techniques. Liu et al. [6], Zhao et al. [7], and Ahmad and Irfan [8] are a few, who extended Browder’s results to more generalized nonlinear variational inequalities. In 2009, Farajzadeh et al. [9] considered new kinds of generalized variational-like inequality problems under the frame work of topological vector spaces.

In the subsequent period, generalized quasi-variational inequalities were studied by Hung and others [10–12]. In 2017, Irfan et al. introduced a new generalized variational-like inclusion problem involving relaxed monotone operators [13]. A class of -generalized operator variational-like inequalities were introduced by Kim et al. in 2018 [14]. In the same year, Tavakoli et al. studied the -pseudomonotone property for the set-valued mappings in order to solve a generalized variational inequality problems [15]. On the other hand, vector equilibrium problems for the set-valued mappings were studied by Farajzadeh et al. and Chen et al. during this period [16, 17]. This wide range of literature is a clear indication of the importance that variational inequality problems have gained in the recent years. In this paper, we further add to this literature by providing solutions to a generalized nonlinear vector variational-like inequality problem, using topological methods.

Variational-like inequalities have number of applications which make it an interesting discipline for research. Vector variational inequality on flow equilibrium problem on a network has been discussed in [18]. Application of variational-like inequality in fuzzy optimization problem is discussed in [19]. More such studies are available in the literature [20–22].

Motivated by these studies, here, we investigate a generalized nonlinear variational like inequality problem, which was proposed by Farajzadeh et al. [9] as follows:

Generalized nonlinear variational-like inequality problem: let be a dual system of Hausdorff topological vector spaces and be a nonempty convex subset of . Given the mappings and , set-valued map , and a map , consider the following generalized nonlinear variational-like inequality problem (GNVLIP)

In this paper, we consider two variants of nonlinear vector variational-like inequality problems in a more general set-up as follows:

Let and be two topological vector spaces, and let be a nonempty, closed, and convex subset of and be the space of all continuous linear mappings from the space to the space . Clearly, is nonempty as the zero mapping, that is, defined as for all is always linear and continuous, hence belongs to .

Further, let be set-valued mappings and be single-valued mappings. Suppose the maps and are two bifunctions.

Problem 1. Suppose is a closed, convex, pointed cone with . Then, the generalized nonlinear vector variational-like inequality problem (I) (GNVVLIP (I)) is to find , such that for each , there exist , , and such that

Problem 2. Suppose is a set-valued map such that for every , is a proper, closed, convex, pointed cone with nonempty interior. Then, the generalized nonlinear vector variational-like inequality problem (II) (GNVVLI (II)) is to find such that for each , , , and such that

Here, we are trying to provide solutions to the above stated generalized nonlinear vector variational-like inequality problems (I and II) from a topological point of view. We consider and to be any topological vector spaces and use the concept of admissibility of the function space topology along with net theory to prove the existence of solutions of these generalized nonlinear vector variational-like inequality problems.

2. Preliminaries

Below, we provide some definitions and results related mainly to set-valued maps between topological spaces.

Definition 3 [23]. Suppose and are two topological spaces and is a set-valued map.(i) is called upper semicontinuous (in short, u.s.c.) at a point , if for every open set in such that , there exists an open set in with such that ;(ii) is called lower semicontinuous (in short, l.s.c.) at a point , if for every open set in such that , there exists an open set in with such that for each , ;(iii) is said to be continuous at if it is both upper semicontinuous and lower semicontinuous at ;(iv) is said to be continuous (resp. u.s.c. and l.s.c.) if it is so at each point of .

Lemma 4 [24]. Suppose and are topological spaces and is a set-valued map. Then,(i)if is upper semicontinuous at and is compact then for every net and with and , we have ;(ii) is lower semicontinuous at if and only if for every and every net with , there exists a subnet , where is a directed subset of and a net such that with .

Theorem 5 [23]. Let and be two topological spaces. Let be a set-valued map. Then, is lower semicontinuous at if for any net in converging to , the image net converges to .

Lemma 6 [25]. Suppose and are topological spaces and is a set-valued upper semicontinuous function. If is compact for each , then image of every compact subset of under is compact.

Definition 7 [23, 26]. Let and be two topological spaces. Let be the space of all continuous mappings from to . A topology on is called admissible, if the evaluation map , defined by , is continuous.

Definition 8 [26]. Let be a net in . Then, is said to continuously converge to if for each net in converging to , converges to in .

Theorem 9 [23]. Let and be two topological spaces. A topology on , the family of continuous mappings from to , is admissible if and only if for any net in , converges to in implies continuous convergence of to .

Definition 10 [27]. Suppose is a set-valued map from to . The graph of , denoted by , is

Definition 11 [28]. Suppose is a nonempty subset of some topological vector space . A set-valued map is called a KKM-mapping if for every nonempty finite set of , we have

The following result is taken from [28].

Lemma 12 (KKM-Theorem). Suppose is a nonempty subset of some topological vector space and is a KKM-mapping such that for every , is a closed subset of . If there exists a point such that is compact, then .

3. Main Results

Theorem 13. Suppose and are two topological vector spaces and is the space of all continuous linear mappings from the space to the space equipped with an admissible topology. Let be a nonempty, compact, closed, and convex subset of . Suppose is a closed, convex, pointed cone with . Further, let be set-valued lower semicontinuous mappings and be continuous mappings. Suppose the maps and are affine mappings such that is continuous in the second argument and is continuous in the first argument, respectively, with for all . Then, the generalized nonlinear vector variational-like inequality problem (I) has a solution. That is, there exists such that for each , there exist , , and such that

Proof. We define a set-valued map byClearly, is nonempty as . As , we have . Thus, for each , , and , we have . Since is a closed convex and pointed cone, thus .
The proof of the theorem is divided into two parts: (i) is a KKM-mapping on :Let be any finite subset of .
We show that . Let, if possible, for some . Then, we have for some and . Also, as , for all , , and , we have , for each . Since is convex and with , therefore . As , , and belong to they are linear. Therefore, we have Again, and are affine; hence, as by the given hypothesis. Therefore, . Thus, we have , where is the zero vector in . Thus, , which is a contradiction. Therefore, we have . Hence, is a KKM-mapping on .(ii) is closed for each :Let be a net in , converging to some in . As is closed, . We have to show that , that is, there exist , , and such that . Since , therefore there exist some , , and such that . Now, the maps , , and are set-valued lower semicontinuous functions; therefore, for each and converging to , there exists a subnet of with such that converges to , in view of Lemma 4. Now, is a net in itself converging to and ; therefore, there exists a subnet of with . Similarly, we have a subnet of with such that the subnets , , and converge to , , and , respectively, in view of Lemma 4. As these nets are subnets of , , and , respectively, thus without loss of generality, we denote the subnets , , and by , , and , respectively, which converge to , , and , respectively.
Since the single-valued map , , and are continuous, therefore we have , , and converge to , , and , respectively.
By the given hypothesis, that is, is continuous in the second argument, we have converges to . Since the space is given to be admissible, thus we have converges to , by Theorem 9. As the map is continuous in the first component, therefore converges to . Hence, converges to , in view of the fact that is admissible.
Now, we will show that .
Let, if possible, . Then, by the convergence of net, we have eventually, which leads to contradiction. Hence, . Thus, we have .
Now is closed, and is compact. This implies that is a compact subset of . Therefore, by KKM-Theorem, . Hence, there exists some such that . That is, for each , there exist , , and such that , hence the result.☐

In the above theorem, we have proved that, along with other conditions, lower semicontinuity of ensures existence of solutions for GNVVLIP (I).

In the next theorem, we are providing another set of conditions for the existence of solutions for these class of problems.

Theorem 14. Suppose and are two topological vector spaces and is the space of all continuous linear mappings from the space to the space equipped with an admissible topology. Let be a nonempty, closed, compact, and convex subset of . Suppose is a closed, convex, pointed cone with . Further, let be set-valued upper semicontinuous functions with nonempty compact values, that is, , , and are compact for every . Let be continuous mappings. Suppose the maps and are affine mappings such that is continuous in the second argument and is continuous in the first argument, respectively, with for all . Then, there exists a solution to the generalized nonlinear vector variational inequality problem. That is, there exists such that for each , there exist , , and such that

Proof. Consider a set-valued map defined asThe proof of the theorem is divided into two parts:(i) is a KKM-mapping on ;(ii) is closed for each .Proof of part (i) is similar to that of Theorem 13. Therefore, we are providing the proof of part (ii) only.
Let be a net in , converging to some . As is closed, . We have to show that , that is, there exist , , and such that . Since , therefore there exist some , , and such that . Now, is a compact subset of , and , , and are set-valued upper semicontinuous functions such that , , and are compact. Therefore, , , and are also compact by Lemma 6. As , , and are nets in , , and , respectively, there exist subnets , , and such that , , and converge to some , , and , respectively.
Now, we construct a directed set , defined in the following way:
By the order property of the directed set , for each triplet , there exists some such that , , and . We denote the collection of such s by . It can be easily verified that is a directed set under the induced ordering of .
Thus, we have subnets , , and of , , and , respectively, such that , , and converge to , , and , respectively.
Then, proceeding as in Theorem 13, we have . Thus, is closed, and is compact. This implies that is a compact subset of . Therefore, by KKM-Theorem, there exists some for each . That is, for each , there exist , , and such that , hence the result.☐

In the next theorem, we investigate the properties of solution sets of GNVVLIP (I).

Theorem 15. Let be the set of all solutions of a generalized nonlinear vector variational-like inequality problem as obtained in Theorem 13 (respectively, Theorem 14). Then, is closed and compact in .

Proof. Suppose is the solution set of the generalized nonlinear vector variational-like inequality problem. Then, by Theorem 13, we have , whereIt has been proved in Theorem 13 that each is closed. Therefore, is a closed set and hence a closed subset of . Since is compact, is compact as well.☐

In the next set of theorems, we consider the other variant of the generalized nonlinear vector variational-like inequality problem and provide the conditions for the solution.

Theorem 16. Suppose and are two topological vector spaces and is the space of all continuous linear mappings from the space to the space equipped with an admissible topology. Let be a nonempty, closed, compact, and convex subset of . Suppose is a set-valued map such that for every , is a proper closed, convex, pointed cone with . Further, suppose that is also a set-valued map defined by such that the graph of , , is a closed set in . Further, let be set-valued lower semicontinuous mappings and be continuous mappings. Suppose the maps and are affine mappings such that is continuous in the second argument and is continuous in the first argument, respectively, with for all . Then, the generalized nonlinear vector variational-like inequality problem (II) has a solution. That is, there exists such that for each , , , and such that

Proof. Consider a set-valued map defined asLikewise in Theorem 13, the proof of this theorem is also divided into two parts:(i) is a KKM-mapping on ;(ii) is closed for each .Proof of part (i) is similar to that of Theorem 13. Therefore, we are providing the proof of part (ii) only.

As in Theorem 13, it follows that the net converges to , in view of admissibility of , lower semicontinuity of , and the other given conditions of the hypothesis.

Now, we will show that . As in Theorem 13, we have the net is convergent and converges to . Also, the net is contained in , and the graph is closed; therefore, . Hence, we have . Thus, we have . Therefore, is closed.

Now is closed, and is compact. This implies that is a compact subset of . Therefore, by KKM-Theorem, . Hence, there exists some such that . That is, for each , there exist , , and such that , hence the result.☐

Another solution of the generalized nonlinear vector variational-like inequality problem (II) is provided below.

Theorem 17. Suppose and are two topological vector spaces and is the space of all continuous linear mappings from the space to the space equipped with an admissible topology. Let be a nonempty, closed, compact, and convex subset of . Suppose is a set-valued map such that for every , is a proper, closed, convex, pointed cone with . Further, suppose that is also a set-valued map defined by such that graph of , , is a closed set in . Further, let be set-valued upper semicontinuous functions with nonempty compact values, that is, , , and are compact for every and let be continuous mappings. Suppose the maps and are affine mappings such that is continuous in the second argument and is continuous in the first argument with , respectively, for all . Then, the generalized nonlinear vector variational-like inequality problem (II) has a solution. That is, there exists such that for each , , , and such that

Proof. The result can be proved on the similar lines as that of Theorem 14 and Theorem 16.☐

We can draw the following conclusions from the results obtained so far:(i)Suppose is defined as the constant map with for all and is defined as , and and are taken as zero functions. If are also defined as zero functions, that if for all and is taken as an identity map, that is, for all . Then, Theorem 16 ensures solution to the generalized vector variational inequality problem discussed in [29]. Also, we have(a)Theorem 16 reduces to Theorem 3.6 of [29];(b)Theorem 17 reduces to Theorem 3.1 of [29].

Further, if is single-valued, then Theorem 13 reduces to Theorem 3.1 and Theorem 17 reduces to Theorem 3.2 of [30], respectively.

4. Conclusion

In this paper, we have provided solutions to two variants of the generalized vector variational-like inequality problem. Our approach and the result obtained here differ significantly from those of the existing literature. To be precise,(i)The spaces and considered in this paper are topological vector spaces. In [15], the space is taken to be , whereas [13] deals with real Hilbert spaces;(ii)We have used conditions of upper semicontinuity as well as lower semicontinuity on the set-valued mappings to obtain our results. In [15], the concepts of transfer closed and intersectionally closed are used along with an assumption milder than -pseudomonotonicity on the set-valued mappings. On the other hand, relaxed monotonicity and relaxed Lipschitz’s continuity are used in [13] and generalized -quasi-convexity is used in [17] for the set-valued mappings.

The approach adopted in our paper is topological and varies significantly from the rest literature. Net theory is extensively used in all the main results. We have used admissibility of the function space topology to obtain our results. The authors are not aware of any such results in the literature of variational inequality which are proved adopting similar techniques.

Data Availability

No data is used.

Conflicts of Interest

The authors declare there is no conflict of interest.

Acknowledgments

The authors thank the referees for their valuable suggestions. The authors also thank Professor C. S. Lalitha for her valuable suggestions and remarks which helped improve the paper.

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