A note on a fixed point theorem and the Hyers-Ulam stability. (English) Zbl 1254.39012

Badora, Roman; Brzdȩk, Janusz

J. Difference Equ. Appl. 18, No. 6, 1115-1119 (2012).

Let \(S\) and \(Z\) be two nonempty sets and let \(Z^{S}\) stand for the set of all functions mapping \(S\) into \(Z\) . We say that \(\mathcal{T}: Z^{S} \to Z^{S}\) is an operator of \(substitution\) if \[ \mathcal{T}(\alpha)(t) = G(t, \alpha(g(t))) \] for \(\alpha \in Z^{S}\), \(t \in S\) with some \(g:S \to S\) and \(G: S \times Z = Z\).
Let \(\mathbb{R}_{+} := [0,\infty)\), \(S\) be a nonempty set, \((X,d)\) be a complete metric space. Let functions \(f: S \to S\), \(\varepsilon: S \to \mathbb{R}_{+}\), and \(\Lambda: S \times \mathbb{R}_{+} \to \mathbb{R}_{+}\) be given and operator \(\mathcal{L}_{f}^{\Lambda}: \mathbb{R}_{+}^{S} \to \mathbb{R}_{+}^{S}\) be defined by \(\mathcal{L}_{f}^{\Lambda}(\alpha)(t) := \Lambda(t,\alpha(f(t)))\) for \(\alpha \in \mathbb{R}_{+}^{S}\) and \(t \in S\). Assume that function \(\Lambda_{t} = \Lambda(t, \cdot)\) is nonincreasing for \(t \in S\) and \[ h(t) := \sum_{n=0}^{\infty}(\mathcal{L}_{f}^{\Lambda})^{n}(\varepsilon)(t) < \infty \] for all \(t \in S\) .
If \(\mathcal{T}: \mathbb{R}_{+}^{S} \to \mathbb{R}_{+}^{S}\) is such that \[ d(\mathcal{T}(\alpha)(t), \mathcal{T}(\beta)(t)) \leq \Lambda(t, d(\alpha(f(t)), \beta(f(t)))) \] for \(\alpha, \beta \in X^{S}\), \(t \in S\) and \[ d(\mathcal{T}(\varphi)(t), \varphi(t)) \leq \varepsilon(t) \] for \(t \in S\) and \(\varphi: S \to X\) , then the limit \[ \Phi(t) := \lim_{n \to \infty}\mathcal{T}^{n}(\varphi)(t) \] exists for \(t \in S\) and \[ d(\varphi(t), \Phi(t)) \leq h(t) \] for \(t \in S\).
Moreover, the following two statements are true.
(i) If \(\mathcal{T}\) is a continuous operator of substitution or \(\Lambda_{t}\) is continuous at zero for each \(t \in S\), then \(\Phi\) is a fixed point of \(\mathcal{T}\).
(ii) If \(\Lambda_{t}\) is subadditive for each \(t \in S\), then \(\mathcal{T}\) has at most one fixed point \(\Phi \in X^{S}\) such that \(d(\phi(t),\Phi(t)) \leq M h(t)\) for \(t \in S\) and for some \(M \in \mathbb{N}\).

Reviewer: Andrzej Smajdor (Kraków)

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

Hyers-Ulam stability; fixed points; complete metric space