A fixed point theorem for product spaces

Abstract

We prove the following result in this paper: Let (X, d) be a complete metric space and 7 be a space having the fixed point property. Let f: X xY->X xY be a continuous map. If / is a contraction mapping in the first variable, then / has a fixed point. This result is a generalization to the result obtained in Nadler [5]. Other results are proved concerning the fixed point theorem for product spaces. The concept "continuous height-selection" is discussed and its relation to the existence of fixed points for a function is also discussed.

Fixed Point Theorems in Product Spaces

“…*-y||1,C2(H,t;)) for x,y G X and u,v g K2. It was shown in [7] (see also [2,5,6,8,9,10]) that if X has ATC-norm, 0 ^ Kx C X is weakly compact and convex, and Kx and K2 have the fixed point property for nonexpansive mappings, then every nonexpansive (with respect to the metric C^) T: Kx x K2 -► Kx x K2 has a fixed point. Here we generalize this result.…”

The Fixed Point Property of Non-Retractable Topological Spaces

“…In topology, it is well known that in general the product property of the FPP does not hold [16,17,21]. Comparing with the FPP in References [5,16,17,21], its Khalimsky topological version has its own feature. Since the term "Khalimsky" will be often used in this paper, hereafter we will use the terminology 'K-' instead of "Khalimsky" if there is no danger of confusion.…”

“…Furthermore, motivated by the Tarski-Davis theorem [8,9] on a lattice and Kuratowski's question [10,11] on the product property of the FPP on a peano continuum (or a compact, connected and locally connected metric space), many works dealt with the FPP for ordered sets and topological spaces. Some of these include References [3,10,[12][13][14][15][16][17][18]. Rival [19] considered irreducible points in arbitrary ordered sets, as follows: For a poset (P, ≤), consider two distinct points x, y ∈ P. If x < y and there is no z ∈ P such that x < z < y, then y is said to be an upper cover of x and x is called a lower cover of y.…”

“…Comparing a K-topological space with spaces dealt with in earlier papers [16,17,21], we can obtain a poset derived from the given K-topological space. Every two points of a poset (X, ≤) need not be retractable point and further, each element of (X, ≤) need not be irreducible either (see Property 1).…”

“…Hence we cannot use the Tarski-Davis [8,9], Rival [19] and Schrder [20] theorems to study the FPP for K-topological spaces. Henceforth, we need to study fixed point theory for K-topological spaces in some different approaches from those of References [16,17,21] (see Theorem 3).…”