Journal Article

Restrictive Lipschitz continuity, basis property of a real sequence, and fixed-point principle in metrically convex spaces

2024

Overview

A mapping T of a metric space X,d into a metric space Y,ρ is called restrictive Lipschitz if there exist: a positive decreasing to zero sequence tn:n∈N and a nonnegative sequence Ln:n∈N, with L:=lim infn→∞Ln<∞, such that for all x,y∈X,n∈Ndx,y=tn⟹ρTx,Ty≤Lntn.Using a basis property of the sequence tn:n∈N (Lemma 1), we prove that if T is a continuous and restrictive Lipschitz mapping of a complete metrically convex space X,d into a metric space Y,ρ, then T is Lipschitz continuous with the constant L, that is ρTx,Ty≤Ldx,y,x,y∈X,and, in the case when the set n∈N:Ln0. This result leads to the following fixed-point principle: Every continuous selfmapping T of a nonempty metrically convex complete metric space X,d that is restrictive Lipschitz with a sequence Ln:n∈N, such that0≤Ln<1(n∈N) and lim infn→∞Ln≤1, has a unique fixed point, and either it is a Banach contraction, or there is an increasing concave function α:0,∞→0,∞, such that αt0 and dTx,Ty≤αdx,y,x,y∈X.Some applications of these results to the theory of iterative functional equations are proposed.

• Publisher: Springer Nature B.V
• Subject: Functional equations / Lipschitz condition / Mapping / Metric space / Numbers