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A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces. (English) Zbl 1246.54035

Let (X,d) be a b-metric space with parameter s, in the sense of S. Czerwik [Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993; Zbl 0849.54036)]. Let S and T be two weakly compatible self-mappings of X such that: (1) S and T satisfy property (E.A) of M. Aamri and D. El Moutawakil [J. Math. Anal. Appl. 270, No. 1, 181–188 (2002; Zbl 1008.54030)]; (2) T(X)S(X); and (3) 

G(d(Tx,Ty),d(Sx,Sy),d(Sx,Tx),d(Sy,Ty),d(Sx,Ty),d(Sy,Tx))>0

for all x,yX such that xy, where G: + 6 is continuous and satisfies: (a) G is nondecreasing in variable t 1 and nonincreasing in variable t 2 ; (b) G(st,0,0,t,1 st,0)<0 for all t>0; and (c) G(t,t,0,0,t,t)0 for all t>0. The author proves that if S(X) or T(X) is a closed subspace of X, then T and S have a unique common fixed point. If the b-metric d is weakly continuous (i.e., if lim n d(x n ,x)=0 implies lim n d(x n ,y)=d(x,y) for every sequence {x n } in X and all x,yX), then the same conclusion holds with weaker assumptions for the function G.

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