Abstract
The main purpose of this paper is to give some common fixed point theorems of mappings and set-valued mappings of a symmetric space with some applications to probabilistic spaces. In order to get these results, we define the concept of E-weak compatibility between set-valued and single-valued mappings of a symmetric space.
Highlights
The main purpose of this paper is to give some common fixed point theorems of mappings and set-valued mappings of a symmetric space with some applications to probabilistic spaces
In order to get these results, we define the concept of E-weak compatibility between set-valued and single-valued mappings of a symmetric space
Let d be symmetric on a set X, and for r > 0 and any x ∈ X, let B(x, r) = {y ∈ X : d(x, y) < r}
Summary
We recall some basic definitions from the theory of symmetric spaces. A symmetric function on a set X is a nonnegative real-valued function d on X × X such that (1) d(x, y) = 0 if and only if x = y, (2) d(x, y) = d(y,x). Let d be symmetric on a set X, and for r > 0 and any x ∈ X, let B(x, r) = {y ∈ X : d(x, y) < r}. Note that lim n→∞d(xn, x) = 0 if and only if xn→x in the topology t(d). There are several concepts of completeness in this setting (see [1]). (i) X is S-complete if for every d-Cauchy sequence (xn), there exists x in X with lim n→∞d(x, xn) = 0. (ii) X is d-Cauchy complete if for every d-Cauchy sequence {xn}, there exists x in X with xn→x in the topology t(d)
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