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

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2024年4月22日

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Janusz Matkowski

摘要

A mapping T of a metric space $\left( X,d\right) $ $(X, d)$ into a metric space $ \left( Y,\rho \right) $ $(Y, ρ)$ is called restrictive Lipschitz if there exist: a positive decreasing to zero sequence $\left( t_{n}:n\in \mathbb {N} \right) $ $(t_{n} : n \in N)$ and a nonnegative sequence $\left( L_{n}:n\in \mathbb {N}\right) ,$ $(L_{n} : n \in N),$ with $L:=\liminf _{n\rightarrow \infty }L_{n}<\infty ,$ $L : = {lim inf}_{n \to \infty} L_{n} < \infty,$ such that for all $ x,y\in X,$ $x, y \in X,$ $n\in \mathbb {N}$ $n \in N$ $\begin{aligned} d\left( x,y\right) =t_{n}\Longrightarrow \rho \left( Tx,Ty\right) \le L_{n}t_{n}\text {.} \end{aligned}$ $\begin{matrix} d (x, y) = t_{n} ⟹ ρ (T x, T y) \leq L_{n} t_{n} . \end{matrix}$ Using a basis property of the sequence $\left( t_{n}:n\in \mathbb {N}\right) $ $(t_{n} : n \in N)$ (Lemma 1), we prove that if T is a continuous and restrictive Lipschitz mapping of a complete metrically convex space $\left( X,d\right) $ $(X, d)$ into a metric space $\left( Y,\rho \right) ,$ $(Y, ρ),$ then T is Lipschitz continuous with the constant L, that is $\begin{aligned} \rho \left( Tx,Ty\right) \le Ld\left( x,y\right) , \quad x,y\in X, \end{aligned}$ $\begin{matrix} ρ (T x, T y) \leq L d (x, y), x, y \in X, \end{matrix}$ and, in the case when the set $\left\{ n\in \mathbb {N}:L_{n}<L\right\} $ $\{n \in N : L_{n} < L\}$ is infinite, even essentially more, namely $\begin{aligned} \rho \left( Tx,Ty\right) \le L\alpha \left( d\left( x,y\right) \right) , \quad x,y\in X, \end{aligned}$ $\begin{matrix} ρ (T x, T y) \leq L α (d, (x, y)), x, y \in X, \end{matrix}$ where the function $\alpha :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $ $α : [0, \infty) \to [0, \infty)$ is continuous, increasing, concave (so subadditive) and such that $\alpha \left( t\right) <t$ $α (t) < t$ for all $t>0$ $t > 0$ . This result leads to the following fixed-point principle: Every continuous selfmapping T of a nonempty metrically convex complete metric space $ \left( X,d\right) $ $(X, d)$ that is restrictive Lipschitz with a sequence $\left( L_{n}:n\in \mathbb {N}\right) ,$ $(L_{n} : n \in N),$ such that $\ 0\,\le L_{n}<1\ (n\in \mathbb {N) }$ $0 \leq L_{n} < 1 (n \in N)$ and $\liminf _{n\rightarrow \infty }L_{n}\le 1,$ ${lim inf}_{n \to \infty} L_{n} \leq 1,$ has a unique fixed point, and either it is a Banach contraction, or there is an increasing concave function $\alpha :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $ $α : [0, \infty) \to [0, \infty)$ , such that $\alpha \left( t\right) <t$ $α (t) < t$ for $t>0$ $t > 0$ and $\begin{aligned} d\left( Tx,Ty\right) \le \alpha \left( d\left( x,y\right) \right) , \quad x,y\in X. \end{aligned}$ $\begin{matrix} d (T x, T y) \leq α (d, (x, y)), x, y \in X . \end{matrix}$ Some applications of these results to the theory of iterative functional equations are proposed.