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

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 $$M \in \mathbb{N}$$.

### 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