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Multi-valued nonlinear contraction mappings. (English) Zbl 1179.54053

Among three fixed point theorems established in the paper there is the following: Let (X,d) be a complete metric space and let φ:[0,)[a,1), 0<a<1, be such that lim ¯ rt+ (r)<1 for all t[0,). If T:XCl(X) (= all nonempty closed sets of X) is such that xd(x,Tx) is lower-semicontinuous and for any xX there is yTx with φ(d(x,y))d(x,y)d(x,Tx) and d(y,Ty)φ(d(x,y))d(x,y), then zTz for some z.

This theorem generalizes results of D. Klim and D. Wiatrowski [J. Math. Anal. Appl. 334, No. 1, 132–139 (2007; Zbl 1133.54025)], Y. Feng and S. Liu [ibid., 317, No. 1, 103–112 (2006; Zbl 1094.47049)], N. Mizoguchi and W. Takahashi [ibid., 141, No. 1, 177–188 (1989; Zbl 0688.54028)].

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)