Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

A survey of ergodicity on the set-valued mappings

In this paper, we give a survey on the PhD thesis of the first author. There theexistence and ergodicity on invariant measures of set-valued mappings are discused.

Upper and Lower Semicontinuity of Solution Sets for Parametric Generalized Vector Quasi-equilibrium Problems

In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generalizes one given in Han and Gong(Optimization 65:1337–1347,2016).Then,we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping.Finally,in terms of the semicontinuity of the level mapping,we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

An Equilibrium Version of Set-Valued Ekeland Variational Principle and its Applications to Set-Valued Vector Equilibrium Problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland varia- tional principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of X complete metric spaces （Z, d）, we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X C Z is countably compact in any Hausdorff topology weaker than that induced by d. When （Z, d） is a Fechet space （i.e., a complete metrizable locally convex space）, our existence result only requires that the domain C Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.

EQUIVALENT CONDITIONS FOR THE UNIFORM CONTINUITY OF DUALITY MAPPING

For a duality mapping F(x), this paper gives equivalent conditions for F(x)being single-valued and uniformly continuous, or uniformly lower semi-continuous, or forexistence a uniformly continuous support functions.