Variational-like Inequalities Research Articles

SOME PROPERTIES OF EXPONENTIALLY PREINVEX FUNCTIONS

In this paper, we introduce some new concepts of the exponentially preinvex functions. We investigate several properties of the exponentially preinvex functions and discuss their relations with convex functions. Optimality conditions are characterized by a class of variational-like inequalities. Several interesting results characterizing the exponentially preinvex functions are obtained. Results obtained in this paper can be viewed as significant improvement of previously known results.

Stability analysis of a class of non-newtonian fluids based on generalized vector variational-like inequalities on riemannian manifold

Fluid mechanics is a branch of mechanics. It is the science of studying fluid phenomena and related mechanical behaviors. So far, the mutual soaking and fusion between fluid mechanics and other disciplines have formed many branches, many physicists and Mathematicians are working on this aspect of research. The classical Newtonian fluid mechanics believe that in parallel flow, the shear force is proportional to the shear rate, and the proportional coefficient is called the viscosity coefficient. In recent years, with the increasing importance of non-Newtonian fluids, people have found that research on non-Newtonian flow is necessary. The variational inequalities based on Riemannian manifolds were first proposed by S.Z.Németh. S.Z.Németh studied the existence of solutions to variational inequalities on Hadamard manifolds. For the equivalence of non-differentiable vector optimization problems on Riemannian manifolds and generalized weak vector variational-like inequalities, we give some new conclusions.

Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization

Abstract The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.

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.

Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem

Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem

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.

Existence results for generalized mixed vector variational-like inequality problems with exponential type invexities

In this paper, we study a new kind of existence of solution for set valued exponential type mixed vector variational-like inequality problem in Euclidean space and proposed αg-relaxed exponentially (γ,η)-monotone mapping. Moreover, we established an example in order to illustrate the main problem. We proved the existence results by KKM-technique with αg-relaxed exponentially (γ,η)-monotone mapping. Further, we give some consequences of the main result. The results presented in this paper unifies and extends some known results in this area.

New classes of strongly exponentially preinvex functions

In this paper, some new classes of the strongly exponentially generalized preinvex functions involving an auxiliary non-negative function and a bifunction are introduced. New relationships among various concepts of strongly exponentially generalized preinvex functions are established. It is shown that the optimality conditions of differentiable strongly exponentially generalized preinvex functions can be characterized by exponentially variational-like inequalities. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Sensitivity analysis for a parametric generalized mixed multi-valued quasi-variational inclusions

In this paper, using P-η-proximal mapping, we study the existence and sensitivity analysis of solution of a parametric generalized variational-like inequality problem in uniformly smooth Banach space. The approach used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis of solution for various important classes of variational inequalities (inclusions) given by many authors, see for example [2-7,9-11,13-16,18,19].

A vector auxiliary principle technique approach to systems of mixed vector equilibrium problems

ABSTRACTIn this paper, we introduce and study a system of generalized mixed vector equilibrium problems involving set-valued variational-like inequalities (SGMVEP, in short) in the general setting of Banach spaces. First, we construct an associated vector auxiliary problem and establish existence results for it. This leads to the generation of an iterative algorithm for solving SGMVEP. Then, under weaker assumptions, the strong convergence of the proposed algorithm is established. The method adopted here is based on the vector auxiliary principle and the obtained results improve some existing results in the literature.

Pascoletti-Serafini scalarization and vector optimization with a variable ordering structure

In this paper, optimality concepts for vector optimization problems with a variable ordering structure are considered. The notion of Pascoletti-Serafini scalarization for the vector optimization problems is extended. Then, some relations between optimal elements of the vector optimization problem and efficient solutions of scalarized problem are obtained. Also, by using the generalized Pascoletti-Serafini scalarization method, nonsmooth variational-like inequalities are defined and their relations with the scalarized optimization problems are studied.

The continuous-time problem with interval-valued functions: applications to economic equilibrium

ABSTRACTThe aim of this paper is to define the Continuous-Time Problem in an interval context and to obtain optimality conditions for this problem. In addition, we will find relationships between solutions of Interval Continuous-Time Problem and Interval Variational-like Inequality Problems, both Stampacchia and Minty type. Pseudo invex monotonicity condition ensures the existence of solutions of the problem. These results generalize similar conclusions obtained in Euclidean or Banach spaces inside classical mathematical programming problems or Continuous-Time Problems. We will finish generalizing the existence of Walrasarian equilibrium price model and the Wardrop's principle for traffic equilibrium problem to an environment of interval-valued functions.

On vector variational-like inequalities involving right upper-Dini-derivative functions

In this paper, we introduce (weak) Stampacchia and Minty arcwise connected vector variational-like inequalities in the term of right upper-Dini-derivative and establish not only the relations of introduced inequalities with vector optimization problems but also the existence results, by using KKM-Fan theorem and Brouwer fixed point theorem. Examples are provided to illustrate the derived 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

On vector variational-like inequalities and vector optimization problems with (G, α)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.

Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming

In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.

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.