Mixed Variational-like Inequalities Research Articles

Solvability of the System of Extended Nonlinear Mixed Variational-Like Inequalities and Proximal Dynamical System

In this article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.

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

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.

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.

Second order evolutionary partial differential variational-like inequalities

<abstract><p>This work is dedicated to the study of a second order evolutionary partial differential variational-like inequality in Banach space. We obtain the mild solution of our problem by applying the concept of strongly continuous cosine family of bounded linear operators, fixed point theorem for condensing multi-valued operators and measure of non-compactness. It is proved that the solution set of mixed variational-like inequalities is non-empty, bounded, closed and convex.</p></abstract>

Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space

In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.

Fuzzy Mixed Variational-like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings.

A new shrinking projection algorithm for a generalized mixed variational-like inequality problem and asymptotically quasi-$$\phi $$-nonexpansive mapping in a Banach space

A new shrinking projection algorithm for a generalized mixed variational-like inequality problem and asymptotically quasi-$$\phi $$-nonexpansive mapping in a Banach space

On Strongly Generalized Preinvex Fuzzy Mappings

In this article, we introduce a new notion of generalized convex fuzzy mapping known as strongly generalized preinvex fuzzy mapping on the invex set. Firstly, we have investigated some properties of strongly generalized preinvex fuzzy mapping. In particular, we establish the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also prove that the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings can be characterized by strongly generalized fuzzy mixed variational-like inequalities, which can be viewed as a novel and innovative application. Several special cases are discussed. Results obtained in this paper can be viewed as improvement and refinement of previously known results.

Extended Perturbed Mixed Variational-Like Inequalities for Fuzzy Mappings

In this article, our aim is to consider a class of fuzzy mixed variational-like inequalities (FMVLIs) for fuzzy mapping known as extended perturbed fuzzy mixed variational-like inequalities (EPFMVLIs). As exceptional cases, some new and classically defined “FMVLIs” are also attained. We have also studied the auxiliary principle technique of auxiliary “EPFMVLI” for “EPFMVLI.” By using this technique and some new analytic results, some existence results and efficient numerical techniques of “EPFMVLI” are established. Some advanced and innovative iterative algorithms are also obtained, and the convergence criterion of iterative sequences generated by algorithms is also proven. In the end, some new and previously known existence results and algorithms are also studied. Results secured in this paper can be regarded as purification and development of previously familiar results.

Exponentially Preinvex Fuzzy Mappings and Fuzzy Exponentially Mixed Variational-Like Inequalities

In this article, our aim is to consider a class of nonconvex fuzzy mapping known as exponentially preinvex fuzzy mapping. With the support of some examples, the notions of exponentially preinvex fuzzy mappings are explored and discussed in some special cases. Some properties are also derived and relations among the exponentially preinvex fuzzy mappings (exponentially preinvex-FMs), exponentially invex fuzzy mappings (exponentially-IFMs), and exponentially monotonicity are established under some mild conditions. In the end, using the fact that fuzzy optimization and fuzzy variational inequalities have close relationships, we have proven that the optimality conditions of exponentially preinvex fuzzy mapping can be distinguished by exponentially fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality. These inequalities render the very interesting outcomes of our main results and appear to be the new ones. Presented results in this paper can be considered and the development of previously obtained results.

Perturbed mixed variational-like inequalities

Perturbed mixed variational-like inequalities is a newly developed class of variational-like inequalities. The existence of a unique solution of perturbed mixed variational-like inequalities is being conducted by using auxiliary principle technique. As special cases, many known and new results are obtained from our results. It is an interesting problem to develop some efficient numerical techniques for solving perturbed variational-like inequalities.

Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces

We study a new class of mixed equilibrium problem, in short MEP, under weakly relaxed $ \alpha $-monotonicity in Banach spaces. This class of problems extends and generalizes some related fundamental results such as mixed variational-like inequalities, variational inequalities, and classical equilibrium problems as special cases. Existence and uniqueness of the solution to the problem is established. Auxiliary principle technique is used to obtain an iterative algorithm. Solvability of the auxiliary problem is established in the paper and finally the convergence of the iterates to the exact solution is proved. As applications of the approach developed in this paper, we study the existence and algorithmic approach for a general class of nonlinear mixed variational-like inequalities. The results obtained in this paper are interesting and improve considerably many existing results in literature.

Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

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.

New generalised mixed vector variational-like inequalities with semi-<i>η</i>-pseudomonotonicity

New generalised mixed vector variational-like inequalities with semi-<i>η</i>-pseudomonotonicity

Hybrid projection method for a system of unrelated generalized mixed variational-like inequality problems

Abstract The aim of this paper is to consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems. Further, we prove some properties of a solution set of generalized mixed variational-like inequality problems. Furthermore, we use a hybrid projection method to find a common element of a solution set of a system of unrelated generalized mixed variational-like inequality problems for generalized relaxed α-monotone mappings and the set of fixed points of a common fixed point problem for a family of generalized asymptotically quasi-ϕ-nonexpansive mappings in a reflexive, uniformly smooth and strictly convex Banach space.

Solution of a class of equilibrium problems and variational inequalities in FC spaces

In this paper, we study a class of mixed variational-like inequalities with respect to generalized weakly relaxed \(\eta {-}\alpha \) monotone mappings, involving nonlinear bifunctions, in finitely continuous topological space, in short FC space. Existence of the solution to the problem is established relaxing convexity and linearity condition by using generalized RKKM theorem. We have proposed a proximal iterative scheme using auxiliary principle technique. Solvability of the auxiliary variational inequality problem is established. Finally convergence of the iterates to the exact solution is proved. Some results with application to equilibrium problem are also discussed.

Extended new generalized mixed vector variational-like inequalities in reflexive Banach spaces

In this paper, we introduce and study an extended new generalized mixed vector variational-like inequality (ENGMVVLI) problem in real reflexive Banach spaces. This problem is more general than many existing equilibrium problems and variational inequalities available in the literature. We prove the existence results of (ENGMVVLI) by utilizing the KKM lemma and Nadlar’s results with relaxed semi- α-monotone mappings. Also, by using the Brouwer’s fixed point theorem, we prove the existence of solutions for (ENGMVVLI) without using the monotonicity concept. Finally, we give an example to justify the validity of our results.

Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map

In this paper, we introduce and study a new class of generalized nonlinear vector mixed quasi-variational-like inequalities governed by a multi-valued map in Hausdorff topological vector spaces which includes generalized vector mixed general quasi-variational-like inequalities, generalized nonlinear mixed variational-like inequalities, and so on. By using the fixed point theorem, we prove some existence theorems for the proposed inequality.

Existence of Solutions of New Generalized Mixed Vector Variational-Like Inequalities in Reflexive Banach Spaces

In this paper, we extend the concept of monotonicity for a vector set-valued mapping to semimonotonicity for a vector set-valued mapping. Then, we prove solvability results for a class of new generalized mixed vector variational-like inequalities by applying the Fan-KKM theorem and Nadler's result. On the other hand, we introduce the concepts of complete semicontinuity and strong semicontinuity for vector multivalued mappings. Moreover, by using the Brouwer fixed point theorem, we prove the solvability for the class of generalized vector variational-like inequalities without monotonicity assumption. Using this result, we obtain a theorem and corollary that improve and extend some known results.