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CALOGERO VETRO

MR3098564 Reviewed Al-Thagafi, M. A.; Shahzad, Naseer Krasnosel'skii-type fixed-point results. J. Nonlinear Convex Anal. 14 (2013), no. 3, 483–491. (Reviewer: Calogero Vetro) 47H10 (47H09)

Abstract

The Krasnosel'skii fixed-point theorem is a powerful tool in dealing with various types of integro-differential equations. The initial motivation of this theorem is the fact that the inversion of a perturbed differential operator may yield the sum of a continuous compact mapping and a contraction mapping. Following and improving this idea, many fixed-point results were proved.\\ The authors present significant and interesting contributions in this direction. In particular, they give the following main theorem: \begin{theorem} Let $M$ be a nonempty bounded closed convex subset of a Banach space $E$, $S:M \to E$ and $T:M \to E$. Suppose that \begin{itemize} \item[(a)] $S$ is 1-set-contractive; \item[(b)] $S(M) \subseteq (I-T)(M)$; \item[(c)] $(I-T)^{-1}$ is a single-valued $\alpha_K$-condensing map on $S(M)$. \end{itemize} Then there exists $x^* \in M$ such that $Tx^*+Sx^*=x^*$. Moreover $F(S+T)$ is compact. \end{theorem} We recall the following: \begin{definition} Let $M$ be a nonempty subset of a metric space $X:=(X,d)$ and $\mathcal{B}(M)$ the set of nonempty bounded subsets of $M$. Then, $T:M \to X$ is called: \begin{itemize} \item[(i)] 1-set-contractive if it is continuous and, for every $A \in \mathcal{B}(M)$, $T(A) \in \mathcal{B}(X)$ and $\alpha_K(T(A)) \leq \alpha_K(A)$, where $\alpha_K(A)=\inf \{\varepsilon>0 \, : \, A \subseteq \cup_{i=1}^nA_i \mbox{ and \rm diam}A_i \leq \varepsilon \}$; \item[(ii)] $\alpha_K$-condensing if it is continuous and, for every $A \in \mathcal{B}(M)$ with $\alpha_K(A)>0$, $T(A) \in \mathcal{B}(X)$ and $\alpha_K(T(A)) < \alpha_K(A)$. \end{itemize} \end{definition} Thus, in view of above theorem, the authors obtain existence results of $t_0$-periodic solutions, where $t_0>0$, for the following nonlinear integral equation with delay: $$x(t)=x^3(t- \tau)+ \sigma(t)x(t- \tau)+p(t)+(h \circ x)(t)+ \int_{- \infty}^t f(t-s)g(x(s))ds$$ for every $t \in \mathbb{R}$, where $\tau >0$ is a constant, $f$ and $g$ are continuous real functions, $\sigma$ and $p$ are $t_0$-periodic continuous real functions with supremum norm $\|\sigma \| = \max_{t \in [0,t_0]}|\sigma(t)|$ and $\|p \| = \max_{t \in [0,t_0]}|p(t)|$. Moreover, $h$ is a 1-set-contractive real function with $|(h \circ x)(t)| \leq \eta$ for some $\eta>0$ and all $t \in \mathbb{R}$.