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A generalization of Mizoguchi and Takahashi’s theorem for single-valued mappings in partially ordered metric spaces. (English) Zbl 1220.54023

Summary: We present a generalization of Mizoguchi and Takahashi’s fixed point theorem for single-valued mappings in partially ordered metric spaces. As an application of the main result, we give an existence and uniqueness theorem for the solution of a periodic boundary value problem.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Nadler, S. B., Multi-valued contraction mappings, Pacific J. Math., 30, 475-488 (1969) · Zbl 0187.45002
[2] Turinici, M., Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl., 117, 100-127 (1986) · Zbl 0613.47037
[3] Ran, A. C.M.; Reurings, M. C.B., A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132, 1435-1443 (2004) · Zbl 1060.47056
[4] Nieto, J. J.; Lopez, R. R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239 (2005) · Zbl 1095.47013
[5] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341, 1241-1252 (2008) · Zbl 1142.47033
[6] Bhaskar, T. G.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65, 1379-1393 (2006) · Zbl 1106.47047
[7] Lakshmikantham, V.; Ćirić, Lj. B., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70, 4341-4349 (2009) · Zbl 1176.54032
[8] Samet, B., Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72, 4508-4517 (2010) · Zbl 1264.54068
[9] Amini-Harandi, A.; O’Regan, D., Fixed point theorems for set-valued contraction type maps in metric spaces, J. Fixed Point Theory Appl., 1-7 (2010) · Zbl 1188.54014
[10] Reich, S., Fixed points of contractive functions, Boll. Unione Mat. Ital. (4), 5, 26-42 (1972) · Zbl 0249.54026
[11] Mizoguchi, N.; Takahashi, W., Fixed point theorems for multivalued mappings on complete metric space, J. Math. Anal. Appl., 141, 177-188 (1989) · Zbl 0688.54028
[12] Du, W.-S., Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasiordered metric spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 876372, 9 pages · Zbl 1194.54061
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