KKM Technique Research Articles

Existence and Approximation of Solution for Vector Mixed Variational-Like Inequality Under Densely (η, f)-C-Pseudomonotone Mappings

In this paper we study vector mixed variational inequality problem under (η, C)-pseudomonotonicity and densely (η, f)-C-pseudo-monotonocity in reflexive Banach space. The existence and uniqueness of solutions have been established with the help of KKM technique and further we have proposed iterative algorithm to find the approximate solution of vector mixed variational inequality by defining an auxiliary problem.

Existence results for mixed hemivariational-like inequalities involving set-valued maps

A hemivariational-like inequality involving a set-valued map and a (nonlinear) bifunction is considered in this paper. The presence of the set-valued map ensures that not one, but two types of solution can be defined, while the presence of the bifunction does not allow the inequality to be written equivalently as an inclusion, making nonsmooth critical point theory unavailable. Using topological methods we are able to prove that our inequality possesses at least one solution (strong solution) provided the set-valued map is upper semicontinuous (lower semicontinuous, respectively). One of the main points of interest in our approach is that we are able to prove the existence of at least one solution even if the KKM technique fails, therefore some of our results are new even in the particular case when the problem reduces to a classical hemivariational, or variational inequality. In the last section we provide a nontrivial example for which our theoretical results are valid. More precisely, we consider a differential inclusion involving the Φ-Laplacian and mixed boundary conditions whose variational formulation leads to a hemivariational-like inequality.

Second order differential variational inequalities involving anti-periodic boundary value conditions

This paper deals with a new class of second order nonlinear differential variational inequalities. Firstly, we study the properties of a solution set for a variational inequality by using the KKM technique and a monotonicity property. Then, we prove the existence of solutions to a second order nonlinear differential variational inequality with anti-periodic boundary value conditions not assuming the Lipschitz continuity of the nonlinear term. Our approaches, which are essentially different from previous research methods in differential variational inequalities, are based on the topological degree theory and the Filippov implicit function lemma.

New generalised mixed vector variational-like inequalities with semi-η-pseudomonotonicity

In this paper, we introduce the concept of semi-η-pseudomonotonicity for vector set-valued mappings. Using this concept, and utilising KKM technique and Nadler's Lemma, we establish some existence results for the new generalised mixed vector variational-like inequality problem in reflexive Banach spaces. Our results are the extension and improvement of some of the results given by Plubtieng and Thammathiwat (2014). Few examples are given to justify our new findings.

Existence Results and Optimal Control for a Class of Quasi Mixed Equilibrium Problems Involving the (f,\xa0g,\xa0h)-Quasimonotonicity

In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities.

Existence results for equilibrium problems under strong sign property

This paper concerns equilibrium problems in real metric linear spaces. Considering a modified notion of upper sign property for bifunctions, we obtain the relationship between the solution sets of the local Minty equilibrium problem and the equilibrium problem, where the technical conditions on $f$ used in the literature are relaxed. The KKM technique is used to generalize and unify some existence results for the relaxed $mu$-quasimonotone equilibrium problems in the literature.

Equilibrium and mixed equilibrium problems on Hadamard manifolds

In this paper, we study equilibrium problems and mixed equilibrium problems on Hadamard manifolds. Using the KKM technique, we establish the existence of solutions of the stated problems under the generalised monotonicity assumptions on the functions involved. We construct some examples to justify our work in Hadamard manifold. We also investigate the proximal point algorithm to solve mixed equilibrium problems on Hadamard manifolds. Our results and ideas may stimulate further research in this fascinating and interesting field of nonlinear space.

Equilibrium and mixed equilibrium problems on Hadamard manifolds

In this paper, we study equilibrium problems and mixed equilibrium problems on Hadamard manifolds. Using the KKM technique, we establish the existence of solutions of the stated problems under the generalised monotonicity assumptions on the functions involved. We construct some examples to justify our work in Hadamard manifold. We also investigate the proximal point algorithm to solve mixed equilibrium problems on Hadamard manifolds. Our results and ideas may stimulate further research in this fascinating and interesting field of nonlinear space.

Variational-Like Inequalities for Weakly Relaxed η-α Pseudomonotone Set-Valued Mappings in Banach Space

We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.

Existence results for a class of hemivariational inequalities involving the stable $(g,f,\alpha)$-quasimonotonicity

In this paper, by introducing a new concept of the stable (g; f; alpha)-quasimonotonicity and applying the properties of Clarke's generalized gradient and KKM technique, we show the existence results of solutions for hemivariational inequalities when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. In the last section, we also give an example to present the our main result.

Using KKM technique in set-valued optimization problems and variational-like inequalities

KKM technique has proved a very useful tool in many areas of analysis. This paper aims to use this technique to give some necessary and sufficient conditions for set-valued optimization problems. We also use the KKM technique to introduce sufficient conditions for solution existence of some kinds of variational-like inequalities and thereby setvalued optimization problems.

Generalized mixed equilibrium problems with generalized α -η -monotone bifunction in topological vector spaces

The purpose of this paper is to introduce a new class of the generalized mixed equilibrium problems with a new definition of the relaxed monotonicity for bi-functions in topological vector spaces. By employing the KKM technique and under some appropriate assumptions on the considering nonlinear mappings, we obtain the existence of a solution for the generalized mixed equilibrium problems with the new concept of the relaxed monotonicity and coercivity condition(in order to relax the compactness of the domains of the nonlinear mappings) in the setting of topological vector spaces. Moreover, the compactness and convexness of the solution set are investigated. The results in the paper extend and generalize the corresponding results, especially Sintunavarat (2013) [20] in this area by providing mild assumptions in order to guarantee the existence of a solution for the generalized mixed equilibrium problem.

Existence theorems for relaxed η-α pseudomonotone and strictly η-quasimonotone generalized variational-like inequalities

Abstract In this paper, we prove the existence of solutions for a variational-like inequality and a generalized variational-like inequality in the relaxed η-α pseudomonotone and strictly η-quasimonotone cases in Banach spaces by using the KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.

Generalised Mixed Vector Equilibrium Problem with New Relaxed Monotonicity

In this paper, we introduce a new notion of relaxed α—β— C monotonicity for bi-mappings. It has been shown that the α—β— C monotonicity is a proper generalisation of monotonicity. Using the KKM technique, we obtain the existence of solutions for generalised mixed vector equilibrium problem with relaxed α—β— C monotonicity in reflexive Banach spaces. Some examples are also given.

Equilibrium problems with generalized relaxed monotonicities in Banach spaces

In this paper, we introduce the concepts of generalized relaxed α-monotonicity and generalized relaxed α-pseudomonotonicity for bi-functions. By using the KKM technique, we obtain existence solution of the mixed equilibrium problems with the generalized relaxed α-monotonicity in reflexive Banach spaces. We also, obtain the existence results for classical equilibrium problem with the generalized relaxed α-pseudomonotonicity.

Vector equilibrium problems with new types of generalized monotonicity

In this paper, we introduce the concept of generalized relaxed ®- pseudomonotonicity for vector valued bi-functions. By using the KKM technique, we obtain some substantial results of the vector equilibrium problems with generalized relaxed ®-pseudomonotonicity assumptions in reflexive Banach spaces. Several examples are provided to illustrate our investigations.

Mixed Equilibrium Problems with Weakly Relaxedα-Monotone Bifunction in Banach Spaces

We introduce the class of mixed equilibrium problems with the weakly relaxedα-monotone bi-function in Banach spaces. Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxedα-monotone bi-function in Banach spaces. The results presented in this paper extend and improve the corresponding results in the existing literature.

On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxedη−αMonotone Mapping

We introduce two new concepts of weakly relaxedη-αmonotone mappings and weakly relaxedη-αsemimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxedη-αmonotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxedη-αsemimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

A New Fixed Point Theorem in FC-Metric Spaces with the Application to Saddle Points

In this paper,with the help of the KKM technique,a new fixed point theorem for transfer compactly open valued mappings is established in FC-metric spaces.As applications,a maximal element theorem,an intersection theorem,a minimax inequality and a saddle point theorem are obtained in FC-metric spaces.Our results unify,improve and generalize some known results in recent references.

Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces

In this paper, we establish some existence results for the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in reflexive Banach spaces. Using the concept of the stable $$f$$ -quasimonotonicity, the properties of Clarke's generalized directional derivative, Clarke's generalized gradient and KKM technique, some existence theorems of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Our main results extend various results existing in the current literatures.