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.

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle iterative sequences generated by the algorithm technique. The convergence of three-step is also proved.

ALGORITHM OF SOLUTIONS FOR MIXEDNONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACE

In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.

Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces

A new bilevel generalized mixed equilibrium problem （BGMEP） involving generalized mixed variational-like inequality problems （GMVLIPs） is introduced and studied in the reflexive Banach spaces. First, an auxiliary generalized mixed equilibrium problem （AGMEP） is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs. By using a minimax inequality, the existence and the unique- ness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions. By using an auxiliary principle technique, the new iterative algorithms are proposed and analyzed, with which the approximate solutions of the BGMEP are computed. The strong convergence of the iterative sequence generated by the algorithms is shown under mild conditions without any coercive assumptions. These new results can generalize some recent results in this field.

g-η-Monotone mapping and resolvent operator technique for solving generalized implicit variational-like inclusions

A new class of g-η-monotone mappings and a class of generalized implicit variational-like inclusions involving g-η-monotone mappings are introduced. The resolvent operator of g-η-monotone mappings is defined and its Lipschitz continuity is presented, An iterative algorithm for approximating the solutions of generalized implicit wriational- like inclusions is suggested and analyzed. The convergence of iterative sequences generated by the algorithm is also proved,

<i>H</i>(.,.)<i>- φ - η -</i>Accretive Operators and Generalized Variational-Like Inclusions

In this paper, we generalize H(.,.) accretive operator introduced by Zou and Huang [1] and we call it H(.,.)- φ - η - accretive operator. We define the resolvent operator associated with H(.,.)- φ - η - accretive operator and prove its Lipschitz continuity. By using these concepts an iterative algorithm is suggested to solve a generalized variational-like inclusion problem. Some examples are given to justify the definition of H(.,.)- φ - η - accretive operator.

Iterative Algorithm for Finding Approximate Solutions of a Class of Mixed Variational-like Inequalities

The purpose of this paper is to investigate the iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities in a real Hilbert space, where the iterative algorithm is presented by virtue of the auxiliary principle technique. On one hand, the existence of approximate solutions of this class of mixed variational-like inequalities is proven. On the other hand, it is shown that the approximate solutions converge strongly to the exact solution of this class of mixed variational-like inequalities.

ROLE OF α-PSEUDO-UNIVEX FUNCTIONS IN VECTOR VARIATIONAL-LIKE INEQUALITY PROBLEMS

In this paper, we introduce a new class of generalized convex function, namely, a-pseudounivex function, by combining the concepts of pseudo-univex and α-invex functions. Further, we establish some relationships between vector variational-like inequality problems and vector optimization problems under the assumptions of α-pseudo-univex functions. Results obtained in this paper present a refinement and improvement of previously known results.

Iterative Algorithm with Mixed Errors for Solving a New System of Generalized Nonlinear Variational-Like Inclusions and Fixed Point Problems in Banach Spaces

A new system of generalized nonlinear variational-like inclusions involving A- maximal m-relaxed η-accretive （so-called, （A, η）-accretive in [36]） mappings in q-uniformly smooth Banach spaces is introduced, and then, by using the resolvent operator technique associated with A-maximal m-relaxed ~/-accretive mappings due to Lan et al., the exis- tence and uniqueness of a solution to the aforementioned system is established. Applying two nearly uniformly Lipschitzian mappings 81 and 82 and using the resolvent operator technique associated with A-maximal m-relaxed ~？-accretive mappings, we shall construct a new perturbed N-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = （S1, S2） which is the unique solution of the aforesaid system. We also prove the convergence and stability of the iterative sequence generated by the suggested perturbed iterative algorithm under some suitable conditions, The results presented in this paper extend and improve some known results in the literature.

Generalized Nonlinear Implicit Variational-Like Inequalities

A new class of generalized nonlinear implicit variational-like inequality problems （for short, GNIVLIP） in the setting of locally convex topological vector spaces is introduced and studied in this paper. Under suitable conditions, some existence theorems of solutions for （GNIVLIP） are presented by using some fixed point theorems.

Perturbed Proximal-Projection Methods for Nonlinear Mixed Variational-Like Inequalities

In this paper we introduce a new perturbed proximal-projection algorithm for finding the common element of the set of fixed points of non-expansive mappings and the set of solutions of nonlinear mixed variational-like inequalities. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. Our approach and results generalize many known results in this field.