A fixed point theorem for mappings on the l∞-sum of a metric space and its application

Abstract

The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings f : ℓ∞(X) → X, where X is a metric space and ℓ∞(X) is the space of all bounded sequences of elements from X. Our result generalizes the theorem obtained by Miculescu and Mihail in 2008, who proved a counterpart of the Banach principle for mappings f : X m → X, where X m is the Cartesian product of m copies of X. We also compare our result with a recent one due to Secelean, who obtained a weaker assertion under less restrictive assumptions. We illustrate our result with several examples and give an application.2010 Mathematics Subject Classification. Primary: 54E40, 54H25, 47H10; Secondary: 54B10, 54E35. Key words and phrases. generalized fixed points, contractive fixed points, ℓ∞-sum of a metric space, Tychonoff product topology.

On generalized iterated function systems defined on ℓ∞-sum of a metric space

“…The condition (12) looks more natural than (13). However, as was observed in [JMS,Remark 3.8], they are equivalent. We will need an extended version of this observations, so we give here short explanation:…”

“…The following result ( [JMS,Theorem 3.7]) is a counterpart of Theorem 2.2; in fact, as was proved in the last section of [JMS], it implies Theorem 2.2.…”

“…Hence if the function f satisfies (12) (or, equivalently, (13)) then it also satisfies assumptions of Theorem 2.4 (i.e., L s,1 (f ) < 1), but the converse need not be true. Indeed, the function f : ℓ ∞ ([0, 1]) → [0, 1] defined by f ((x k )) := 1 2 sup{x k : k ∈ N * } satisfies L s,1 (f ) = 1 2 < 1, but the sequence of generalized iterates (x k ) does not converge for any sequence (x k ) so that x i > 0 for some i ∈ N * (see [JMS,Example 3.11] for details).…”

“…It turns out that, under assumptions of Theorem 2.10, the fixed point x * of the map f is a limit of fixed points of certain restrictions of f (see [JMS,Theorem 3.13]):…”

“…Remark 2.14. The map f from Remark 2.12 shows that the thesis of Theorem 2.13 does not hold if we just assume that L s,1 (f ) < 1, i.e., under the assumptions of Theorem 2.4 -see [JMS,Example 3.14].…”