Differential variational–hemivariational inequalities: existence, uniqueness, stability, and convergence

The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

Fixed point theorems for correspondences with properties weaker than lower semicontinuity

In this paper, we study existence of fixed points for correspondences having weak continuity properties. The obtained results extend or improve the corresponding results present in literature. We use continuous selection technique and also well-known KKM principle in order to establish our fixed point theorems.

Fixed point theorems in minimal generalized convex spaces

This paper deals with coincidence and fixed point theorems in minimal generalized convex spaces. By establishing a kind of KKM Principle in minimal generalized convex space, we obtain some results on coincidence point and fixed point theorems. Generalized versions of Ky Fan?s lemma, Fan-Browder fixed point theorem, Nash equilibrium theorem and some Urai?s type fixed point theorems in minimal generalized convex spaces are given.

Remarks on Weakly KKM Maps in Abstract Convex Spaces

A KKM space is an abstract convex space satisfying the KKM principle. We obtain variants of the KKM principle for KKM spaces related to weakly KKM maps and indicate some applications of them. These results properly generalize the corresponding ones inG-convex spaces andϕA-spaces(X,D;{ϕA}A∈〈D〉). Consequently, results by Balaj 2004, Liu 1991, and Tang et al. 2007 can be properly generalized and unified.

APPLICATIONS OF THE KKM-PRINCIPLE TO PROLLA TYPE THEOREMS

We prove some theroems of Prolla type [13] using a well known KKM-principle of Ky Fan [6], so generalizing several results known in the literature.