Related fixed points for set‐valued mappings on two uniform spaces

Duran Türkoğlu,Brian Fisher

doi:10.1155/s0161171204404268

Abstract

Some related fixed point theorems for set‐valued mappings on two complete and compact uniform spaces are proved.

Highlights

Let (X, ᐁ1) and (Y, ᐁ2) be uniform spaces
We may assume that β1, β2 themselves are a base by adjoining finite intersections of members of β1, β2, if necessary
An associated family for ᐁ1, ᐁ2 will be denoted by Ᏸ1, Ᏸ2, respectively

Summary

Introduction

Let (X, ᐁ1) and (Y , ᐁ2) be uniform spaces. Families {di[1 ]: i ∈ I being indexing set}, {di[2 ]: i ∈ I} of pseudometrics on X, Y , respectively, are called associated families for uniformities ᐁ1, ᐁ2, respectively, if families β1 = V1(i, r ) : i ∈ I, r > 0 , β2 = V2(i, r ) : i ∈ I, r > 0 , (1.1) whereV1(i, r ) = x, x : x, x ∈ X, di[1] x, x < r , V2(i, r ) = y, y : y, y ∈ Y , di[1] y, y < r , (1.2)are subbases for the uniformities ᐁ1, ᐁ2, respectively. If {An} and {Bn} are sequences of bounded, nonempty subsets of a complete uniform space (X, ᐁ) which converge to the bounded subsets A and B, respectively, sequence {δi(An, Bn)} converges to δi(A, B).

Results

Conclusion