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Best proximity point theorems via fixed point theorems for multivalued mappings. (English) Zbl 1505.54063

Summary: It is well known that the concept of a best proximity point includes that of a fixed point as a special case. In this paper, we show that the best proximity point theorems of S. S. Basha and N. Shahzad [Fixed Point Theory Appl. 2012, Paper No. 42, 9 p. (2012; Zbl 1282.41012)] and of A. Fernández-León [J. Nonlinear Convex Anal. 15, No. 2, 313–324 (2014; Zbl 1301.54043)] can be regarded as a fixed point theorem for multivalued mappings which is modified as regards the results of N. Mizoguchi and W. Takahashi [J. Math. Anal. Appl. 141, No. 1, 177–188 (1989; Zbl 0688.54028)] and of O. Kada et al. [Math. Japon. 44, No. 2, 381–391 (1996; Zbl 0897.54029)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

[1] Basha, SS, Shahzad, N: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl. 2012, 42 (2012) · Zbl 1282.41012 · doi:10.1186/1687-1812-2012-42
[2] Fernández-León, A: Best proximity points for proximal contractions. J. Nonlinear Convex Anal. 15(2), 313-324 (2014) · Zbl 1301.54043
[3] Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1), 177-188 (1989) · Zbl 0688.54028 · doi:10.1016/0022-247X(89)90214-X
[4] Kada, O, Suzuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 44(2), 381-391 (1996) · Zbl 0897.54029
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[6] Basha, SS: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151(1), 210-216 (2011) · Zbl 1226.90135 · doi:10.1007/s10957-011-9869-4
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[8] Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976) · Zbl 0305.47029 · doi:10.1090/S0002-9947-1976-0394329-4
[9] Hu, S, Papageorgiou, NS: Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and Its Applications, vol. 419, xvi+964 pp. Kluwer Academic, Dordrecht (1997). ISBN 0-7923-4682-3 · doi:10.1007/978-1-4615-6359-4
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