Fixed point theorems for multivalued mappings in Banach spaces. (English) Zbl 0765.47016

Canetti, A.; Marino, G.; Pietramala, P.

Nonlinear Anal., Theory Methods Appl. 17, No. 1, 11-20 (1991).

The authors have extend some results of Kirk and Ray; and Marino and Pietramala for single-valued mappings to multi-valued mappings. The notations and definitions of closed-bounded subsets of a Banach space, closure, convex-hull, Hausdorff distance, and Lipschitzian pseudo contractive mapping are standard ones. The following geometric sets used in the results are defined as:
For \(x,y\in X\), and \(\varepsilon>0\) \[ \begin{aligned} G(x,y;K) &=\{z\in K:\;\| z-y\| \leq \| z-x\|\},\\ G(x,y;\varepsilon;K) &=\{z\in K:\;\| z-y\| \leq \| z-x\|+\varepsilon\},\\ G(x,A;K) &=\{z\in K:\;\exists a\in A:\;\| z-a\| \leq \| z-x\|\}\\ &=\bigcup_{a\in A} G(x,a;K), \qquad\text{and}\\ G(x,A;K;\varepsilon) &=\{z\in K:\;\exists a\in A:\;\| z-a\| \leq \| z-x\|+\varepsilon\}\\ &= \bigcup_{a\in A} G(x,a;\varepsilon;K).\end{aligned} \] The main results of the paper are stated as follows:
Theorem 1: Let \(X\) be a Banach space whose bounded closed convex subsets have the fixed point property for multivalued nonexpansive point-compact self mappings. Let \(K\) be a closed convex subset of \(X\) and \(T: K\to {\mathcal K}(K)\) nonexpansive. Suppose that for some \(x_ 0\in K\) the set \(G(x_ 0,\bar c_ 0(Tx_ 0);K)\) is bounded. Then \(T\) has a fixed point in \(K\).
Theorem 2: Let \(K\) be a closed convex subset of a Banach space \(X\). Let \(T:K\to{\mathcal K}(X)\) be a nonexpansive mapping satisfying the inwardness condition: \(Tx\subseteq\overline{I_ k(x)}\), where \[ I_ K(x):=\{x+c(u- x):\;u\in K,\;c\geq 1\}. \] Suppose that for some bounded \(A\subseteq K\) the set \(G(A,TA;K):=\bigcap_{a\in A} G(a,Ta;K)\) is bounded. Then there exists a bounded sequence \(\{x_ n\}\subseteq K\) such that \(d(x_ n,Tx_ n)\to 0\) as \(n\to\infty\).
Theorem 3: Let \(X\) be a Banach space whose bounded closed convex subsets have the fixed point property for multi-valued nonexpansive point-compact self mappings. Let \(K\) be a closed convex subset of \(X\) and let \(T: K\to{\mathcal K}(K)\) be a Lipschitzian pseudo contractive mapping. Suppose that:
(i) there exist \(x_ 0\in K\), \(\varepsilon>0\) such that \(G(x_ 0,\bar c_ 0(Tx_ 0);\varepsilon;K)\) is bounded;
(ii) there exists \(0<\alpha<\min\{1,1/L,\varepsilon(2LH(\{x_ 0\},Tx_ 0)+L\varepsilon)^{-1}\}\)
such that the set \(\{x\in K:\;z\in(1-\alpha)y+\alpha Tx\}\) is compact for any \(y\in K\).
Then \(T\) has a fixed point.
Various other interesting results in the form of corollaries are also proved.

Reviewer: L.S.Dube (Quebec)

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H04	Set-valued operators

Keywords:

multi-valued mappings; Lipschitzian pseudo contractive mapping; fixed point property; multivalued nonexpansive point-compact self mappings; inwardness

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