On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. (English) Zbl 0593.47056

Lim, Teck-Cheong

J. Math. Anal. Appl. 110, 436-441 (1985).

The author studies the stability of fixed points of set-valued contractions. Firstly, he proves that if X is a complete metric space and \(T_ 1,T_ 2\) are \(\lambda\)-contractions from X into C(X) (family of nonempty closed subsets of X), then \(D(F(T_ 1),F(T_ 2))=(1- \lambda)^{-1}\sup_{x\in X}D(T_ 1(x),T_ 2(x))\). (Here \(F(T_ i)\) is the set of fixed points of T, D-Hausdorff distance.) From here follows that if for any sequence \(T_ i:X\to C(X)(i=0,1,...)\lim_{i\to \infty} D(T_ i(x),T_ 0(x))=0\) uniformly for all \(x\in X\), then \(\lim_{i\to \infty} D(F(T_ i),F(T_ 0))=0\). (Theorem 1). Further, the author proves two results concerning the limits of fixed points of a sequence of nonexpansive mappings. At the end of the paper Theorem 1 is applied to the problems on the stability of solution sets for generalized differential equations.

Reviewer: O.John

MSC:

47H10	Fixed-point theorems
47F05	General theory of partial differential operators
47H05	Monotone operators and generalizations

Keywords:

stability of fixed points of set-valued contractions; limits of fixed points of a sequence of nonexpansive mappings; stability of solution sets for generalized differential equations

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