Nonlinear set-valued contraction mappings in b-metric spaces
by
Stefan Czerwik
Silesian University of Technology, Gliwice, Poland

Definition. Let X be a set and s >= 1 a real number. A function d: X×X --> R₊ is said to be a b-metric if for all x, y, z in X

(1) d(x, y)=0 iff x=y,

(2) d(x, y)=d(y, x),

(3) d(x, z) <= s[d(x, y)+d(y, z)].

A pair (X, d) is called a b-metric space.

Let CL(X) denotes the space of all nonempty closed subsets of X.

Theorem 1. If (X, d) is a complete b-metric space, then (CL(X), H), where H means the Hausdorff b-metric incluced by d, is also a complete b-metric space.

Some generalization of classical fixed point theorems for set-valued mappings in b-metric spaces will be presented. Among others, we have the following

Theorem 2. Let (X, d) be a complete b-metric space and let d be a continuous function. Let F: X --> 2^X satisfy

H[F(x), F(y)] <= j[d(x, y)],     x, y in X,

where j: R₊ --> R₊ is any non-decreasing function such that lim_{n --> \infty} jⁿ(t)=0 for each fixed t > 0. Then F has a fixed point, that is there exist an u in X such that

u in F(u).

Date received: June 14, 2002