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Zbl 0904.54034
Jungck, G.; Rhoades, B.E.
Fixed points for set valued functions without continuity. (English)
[J] Indian J. Pure Appl. Math. 29, No.3, 227-238 (1998). ISSN 0019-5588; ISSN 0975-7465/e

Let $(X,d)$ be a metric space and ${\cal B} (X)$ the set of all nonempty bounded subsets of $X$. If $A,B\in {\cal B} (X)$ then $\delta (A,B)= \{\sup d(a,b)$: $a\in A$ and $b\in B\}$ and $d(x,B) =\inf\{d(a,b): b\subset B\}$. The authors extend the concepts of compatible (weak) maps and generalized Meir-Keeler contractions.\par Definition 1. Let $(X,d)$ be a metric space and let $A,B: X\to {\cal B}(X)$. Then $A$ and $B$ are $(\varepsilon, \gamma) (p)$ contractions relative to maps $S,T: X\to X$ if $\bigcup A(X) \subset T(X), \bigcup B(X)\subset S(X) $, and there exist functions $p:X\times X\to [0,\infty)$, $\gamma: (0,\infty) \to(0,\infty)$ such that $\gamma (\varepsilon)> \varepsilon$ for all $\varepsilon$ and for $x,y\in X$: $$0< \varepsilon <p (x,y) <\gamma (\varepsilon) \Rightarrow \delta(Ax,By) <\varepsilon. $$ Definition 2. Let $(X,d)$ be a metric space and let $S:X\to X$ and $A:X\to {\cal B} (X)$. The pair $\{A,S\}$ is a weakly compatible pair if $Ax=\{Sx\}$ implies $SAx=ASx$. Every compatible pair $\{A,S\}$ [{\it G. Jungck} and {\it B. E. Rhoades}, Int. J. Math. Math. Sci. 16, No. 3, 417-428 (1993; Zbl 0783.54038)] is weakly compatible. Examples of weakly compatible pairs which are not compatible are given in the paper. In this paper the authors prove some fixed point theorems for set valued functions without appeal to continuity. These theorems extend results of {\it T.-H. Chang} [Math. Jap. 38, No. 4, 675-690 (1993; Zbl 0805.47049)], generalize results by {\it J. Jachymski} [ibid. 42, No. 1, 131-136 (1995; Zbl 0845.47044)] and by {\it S. M. Kang} and {\it B. E. Rhoades} [ibid. 37, No. 6, 1053-1059 (1992; Zbl 0767.54037)] and produce as byproducts generalizations of theorems for point valued functions. \par Theorem 4.1: Let $S$ and $T$ be self maps of a metric space $(X,d)$ and let $A, B: X\to {\cal B} (X)$. Suppose $\bigcup A(X) \subset T(X)$, $\bigcup B(X) \subset S(X)$, and one of $S (X)$, $T(X)$ is complete. Let $p:X\times X\to [0,\infty)$ and $\varphi$: $[0,\infty) \to(0,\infty)$ be maps, and suppose that $\varphi(t) <t$ for $t>0$. If $\delta (Ax,By)\le\varphi(p(x,y))$ for $x,y\in X$, then there exists a unique point $z\in X$ such that $\{z\}= \{Sz\}= \{Tz\} =Az=Bz$ provided that both $\{A,S\}$ and $\{B,T\}$ are weakly compatible pairs, and one of (a), (b) below is true:\par (a) $p= \max \{d(Sx,Ty), {1\over 2} (d(Ax,Ty) +d(Sx,By))\}$, $\delta (Ax,By)=0$, whenever $m=0$, and $\varphi$ is u.s.c. from the right; and\par (b) $p=\max \{d(Sx,Ty)$, $\delta (Ax,Sx)$, $\delta (By,Ty)$, ${1\over 2} (d(Ax,Ty) +d(Sx,By))\}$ and $\varphi$ is u.s.c.
[V.Popa (Bacau)]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54C60 Set-valued maps

Citations: Zbl 0783.54038; Zbl 0805.47049; Zbl 0845.47044; Zbl 0767.54037

Cited in: Zbl 1237.54049 Zbl 1253.54032 Zbl 1246.54048 Zbl 1194.54054 Zbl 1144.47054 Zbl 1150.54351 Zbl 1029.54010 Zbl 0958.54044

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