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Remarks on contractive conditions of integral type. (English) Zbl 1215.54021
In this excellent paper, the author shows that a number of contractive conditions of integral type are equivalent to corresponding contractive conditions of non-integral type. I shall illustrate this by stating Theorem 1 of the paper. Let $\mathfrak{F} := \{ F : \mathbb{R_+} \to \mathbb{R_+} : F$ is increasing, continuous and satisfies $ F^{-1}(0) = \{0\}\}$ and $\Phi := \{\varphi : \mathbb{R_+} \to \mathbb{R_+} : \varphi$ is nondecreasing, right upper continuous, and satisfies $\varphi(t) < t \text{ for each } t > 0\}$. Let $A, B, S, $ and $T$ be selfmaps of a metric space $(X, d)$. For $x,y \in X$, set $M(x,y) :=\max\{d(Ax,by), d(Ax,Sx), d(By,Ty), [d(Ax,By) + d(By,Sx)]/2\}.$ Then the following are equivalent: {\parindent=8mm \item{(i)} there exist $\varphi \in \Phi$ and $f \in \mathfrak{F}$ such that $$ \int_0^{d(Sx,Ty)}f(s)\,ds \leq \varphi\Big(\int_0^{M(x,y)}f(s)\,ds\Big)\text{ for all }x,y \in X; $$ \item{(ii)} there exist $F \in \mathfrak{F}$ and $\varphi \in \Phi$ such that $F(d(Sx,Ty)) \leq \varphi(F(M(x,y))$ for all $x,y \in X$; \item{(iii)} for every $\alpha \in (0,1)$, there exists $F \in\mathfrak{F}$ such that $F(d(Sx,Ty)) \leq \alpha(F(M(x,y))$ for all $x,y \in X$; \item{(iv)} there exists $\varphi \in\Phi$ such that $d(Sx, Ty) \leq \varphi(M(x,y))$ for all $x,y \in X$. \par} The author also proves the following fixed point theorem. Theorem 8. Let $T$ be a selfmap of a complete metric space $(X,d)$ such that, for some $\varphi \in \Phi$ and $F \in \mathfrak{F}$, $F(d(Tx,Ty)) \leq \varphi(F(d(x,y)))$ for all $x, y \in X$. Then $T$ has a unique fixed point $x_{\epsilon}$ and, for any $x \in X, \lim_nT^nx = x_{\epsilon}$.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
WorldCat.org
Full Text: DOI
References:
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