In this paper we introduced the definitions of pre-semi-convex sets and locally pre-semi-convex space.

n this paper, solutions of vector-valued integral equations of a class on an interval (a,b) to a complete Hausdorff locally semi--convex space are presented by the fixed point method.

Existence of solutions to Cauchy initial problem in locally convex spaces;

f-(mutually) fartest points on locally convex spaces;

Differential operator in locally convex spaces and it′s fixed point theorem;

Generalization of the Arrow-Barankin-Blackwell theorem in a locally convex space;

A study on absolutely convergent series in locally convex spaces;

In the locally convex space X ,we constructureed bounded sets B ∪ and a bounded set family B ∪.

The compactness and covergence of vector-valued regular functions in local convex space;

The solution of the doubly periodic Riemannian boundary value of a class with range in a separeted local convex space C ∞ is presented via local uniform differentiability and the representation of parameter.

Moreover, in this paper, by Fan-kakutani fixed point theorem, we gen-eralized the extreme minimum fixed point theorem for set-valued mappings in thelocal convex space in Xu’s paper.