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Fixed point theorems for set-valued contractions in complete metric spaces. (English) Zbl 1133.54025

Let (M,d) be a metric space and let H(A,B) denote the Pompeiu-Hausdorff distance between the sets A,BM. The main results of this paper are fixed point theorems for set-valued contractions in complete metric spaces which are obtained by considering, instead of the classical contraction conditions of the form

H(Tx,Ty)φ(d(x,y))d(x,y),x,yM,

a more general condition: for each xM, there exists

yI b x :=yTx:bd(x,y)d(x,Tx),

for a certain b(0,1], such that

d(y,Ty)φ(d(x,y))d(x,y)·

Several related results in literature are thus extended or generalized.


MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces