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A best proximity point theorem for weakly contractive non-self-mappings. (English) Zbl 1228.54046

Let $A,B$ be nonempty subsets of a metric space $\left(X,d\right)$. A mapping $T:A\to B$ is called weakly contractive if there exists a continuous nondecreasing function $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ with $\psi \left(0\right)=0,\phantom{\rule{4pt}{0ex}}\psi \left(t\right)>0,\phantom{\rule{4pt}{0ex}}t>0,$ and ${lim}_{t\to \infty }\psi \left(t\right)=0$, such that $d\left(x,y\right)\le d\left(x,y\right)-\psi \left(d\left(x,y\right)\right)$ for all $\left(x,y\right)\in A×B$. If $B=A$ and $\psi \left(t\right)=\left(1-k\right)t$ for some $0, then $T$ is a $k$-contraction on $A$. It was proved by Y. Alber and S. Guerre-Delabriere [Oper. Theory, Adv. Appl. 98, 7–22 (1997; Zbl 0897.47044)] that, if $Y$ is a closed convex set of a Hilbert space, then every weakly contractive mapping $T:Y\to Y$ has a unique fixed point. The result was extended by B. E. Rhoades [Nonlinear Anal., Theory Methods Appl. 47, No. 4, 2683–2693 (2001; Zbl 1042.47521)] to weakly contractive mappings on an arbitrary complete metric space $\left(X,d\right)$.

The author extends the above result of Rhoades to weakly contractive mappings $T:A\to B$. A best proximity point for $T$ is a point ${x}^{*}\in A$ such that $d\left({x}^{*},T{x}^{*}\right)=d\left(A,B\right)\phantom{\rule{0.277778em}{0ex}}\left(=inf\left\{d\left(x,y\right):x\in A,\phantom{\rule{4pt}{0ex}}y\in B\right\}\right)$. Recall that Ky Fan [Math. Z. 112, 234–240 (1969; Zbl 0185.39503)] proved that, if $A$ is a nonempty compact convex subset of a normed space $X$, then for every continuous mapping $T:A\to X$ there exists $x\in A$ such that $\parallel x-Tx\parallel =d\left(Tx,A\right)\phantom{\rule{0.277778em}{0ex}}\left(=inf\left\{\parallel Tx-a\parallel :a\in A\right\}\right)$. One says that the pair $A,B$ of sets has the property $P$ if $d\left({x}_{1},{y}_{1}\right)=d\left({x}_{2},{y}_{2}\right)=d\left(A,B\right)$ implies $d\left({x}_{1},{x}_{2}\right)=d\left({y}_{1},{y}_{2}\right)$ for all ${x}_{1},{x}_{2}\in A$ and ${y}_{1},{y}_{2}\in B$. Denote ${A}_{0}=\left\{x\in A:\exists y\in B,\phantom{\rule{4pt}{0ex}}d\left(x,y\right)=d\left(A,B\right)\right\}$ and ${B}_{0}=\left\{y\in B:\exists x\in A,\phantom{\rule{4pt}{0ex}}d\left(x,y\right)=d\left(A,B\right)\right\}$.

The main result of the paper (Theorem 3.1) asserts that, if $A,B$ are subsets of a complete metric space $\left(X,d\right)$ satisfying the property $P$ and such that ${A}_{0}\ne \varnothing$, then for every weakly contractive mapping $T:A\to B$ satisfying $T\left({A}_{0}\right)\subset {B}_{0}$ there exists a unique point ${x}^{*}\in A$ such that $d\left({x}^{*},T{x}^{*}\right)=d\left(A,B\right)$.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E40 Special maps on metric spaces 54E50 Complete metric spaces 41A65 Abstract approximation theory