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Some results on set-valued contractions in abstract metric spaces. (English) Zbl 1228.54041
Summary: In this paper, the concept of contractive set-valued maps in the frame of abstract metric spaces is studied and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, several known fixed point results are either generalized or extended, including the corresponding recent fixed point results of D. Wardowski [Nonlinear Anal., Theory Methods Appl. 71, No. 1-2, A, 512–516 (2009; Zbl 1169.54023)] as well as D. Klim and D. Wardowski [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, A, 5170–5175 (2009; Zbl 1203.54042)]. Examples are given to show that our results are distinct from the existing ones.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
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