The Study Of The Convexity Of Locally Convex Spaces And Fixed Point Theorem

The article mainly aims at the convexity and smoothness of locally convex spaces and the unique common fixed point of two types space. For one thing, we have results in locally convex space. At the base of midpoint locally k-uniformly convexity and midpoint locally k-uniformly smoothness, proved the relationship between midpoint locally k-uniformly convexity and midpoint locally (k+1)-uniformly convexity, then on the conditions of P-reflexivity, we obtain further the equivalont dual theorem between midpoint locally k-uniformly convexity and midpoint locally k-uniformly smoothness. Then in ordered gauge spaces, we present some fixed-point results for mappings satisfying cyclic (ψλ,A, B)-contraction for the pair (f, g). Lastly, we present some fixed-point results for three mappings under a condition related to altering distance function in G-Metric spaces.Chapter one, we introduce the concepts and properties of locally convex spaces. This chapter sort out variety of convexity (smoothness). (1) if (X,P) is midpoint locally k-uniformly convexity, then (X,P) is midpoint locally (k+1)-uniformly convexity. (2) if (X,P) is midpoint locally k-uniformly smoothness, then (X,P) is midpoint locally k+1)-uniformly smoothness. When (X, Tp) is P-reflexivity, then(1) if (x’,P*)is midpoint locally k-uniformly convexity, if and only if (X,P) is midpoint locally (k+1)-uniformly smoothness.(2) if (x’,P*)is midpoint locally k-uniformly smoothness, if and only if (X,P)is midpoint locally (k+1)-uniformly convexity.Chapter two, we introduce the concepts and properties of ordered gauge spaces, present some fixed-point results for mappings satisfying cyclic (ψλ,A, A,B)-contraction. Let (X,t,(?)) be an ordered gauge space and A,B be nonempty closed subsets of X f,g:Xâ†'Xbe two mappings such that the pair (f,g) is (A,B)-weakly increasing. Assume the following:(1)the pair (f,g) is a cyclic(ψλ,A,B)-contraction;(2)/org is continuous.Then f and g have a common fixed point.Chapter there, we introduce the concepts and properties of G-metric space and present a fixed-point result. Let (X, G)be a G-metric space and f be self mappings on X, for all x, y, z ∈ X satisfyingWhere 0< δ,α<1. Then f have a fixed point u ∈ X Then our obtained results can be applied, consider the integral equationWhere K:[0,T]×[0,T]×Râ†'R,h:Râ†'R...