Stability Of Solution Set Research Articles

Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application

This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong (2008), He (2007), and Zhong and Huang (2010).

Stability of Solutions in Parametric Variational Relation Problems

The purpose of this paper is to investigate topological properties and stability of solution sets in parametric variational relation problems. The results of the paper give a unifying way to treat these questions in the theory of variational inequalities, variational inclusions and equilibrium problems.

Stability of solution sets for generalized differential equations

where R(t, x) is a compact subset of Euclidean n-space for real t and x E Rfl, has been proved by Castaing [l, Theorem 21 for convex-valued R and by Filippov [2, Th eorem I] for R satisfying a Lipschitz condition. In [1] it was also shown that the set of solutions to (1) has upper semicontinuous dependence on the initial point. Theorem 2 below will show that under the joint assumption of convexity and a Lipschitz condition on R, the set of solutions to (1) has continuous dependence on the initial point. This result is based on the following theorem on the stability of the fixed point set of a set-valued contraction.