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Partially ordered cone metric spaces and coupled fixed point results. (English) Zbl 1205.54044
Summary: {\it T. G. Bhaskar} and {\it V. Lakshmikantham} [Nonlinear Anal., Theory Methods Appl. 65, No. 7 (A), 1379--1393 (2006; Zbl 1106.47047)] studied the coupled coincidence point of a mapping $F$ from $X\times X$ into $X$ and a mapping $g$ from $X$ into $X$. {\it E. Karapinar} [Comput. Math. Appl. 59, No. 12, 3656--3668 (2010; Zbl 1198.65097)] proved some results of the coupled coincidence point of a mapping $F$ from $X\times X$ into $X$ and a mapping $g$ from $X$ into $X$ over normal cones without regularity. In the present paper, we prove that coupled coincidence fixed point theorems over cone metric spaces are not necessarily normal. Our results generalize several well known comparable results in the literature.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Haung, L. G.; Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings, J. math. Anal. appl. 332, 1468-1476 (2007) · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087
[2] Abbas, M.; Jungck, G.: Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. math. Anal. appl. 341, 416-420 (2008) · Zbl 1147.54022 · doi:10.1016/j.jmaa.2007.09.070
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[4] Abdeljawad, T.; Karapinar, E.: Quasicone metric spaces and generalizations of caristi kirk’s theorem, Fixed point theory appl. (2009) · Zbl 1197.54051 · doi:10.1155/2009/574387
[5] Amini-Harandi, A.; Fakhar, M.: Fixed point theory in cone metric spaces obtained via the scalarization, Comput. math. Appl. (2010) · Zbl 1197.54055
[6] Kadelburg, Z.; Pavlovic, M.; Radenović, S.: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. math. Appl. 59, 3148-3159 (2010) · Zbl 1193.54035 · doi:10.1016/j.camwa.2010.02.039
[7] Karapinar, E.: Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. math. Appl. (2010) · Zbl 1285.47065
[8] Karapinar, E.: Fixed point theorems in cone Banach spaces, Fixed point theory appl. 2009 (2009) · Zbl 1204.47066 · doi:10.1155/2009/609281
[9] Llić, D.; Rakočević, V.: Common fixed points for maps on cone metric space, J. math. Anal. appl. 341, 876-882 (2008) · Zbl 1156.54023 · doi:10.1016/j.jmaa.2007.10.065
[10] Rezapour, Sh.; Hamlbarani, R.: Some notes on the paper ”cone metric spaces and fixed point theorems of contractive mappings”, J. math. Anal. appl. 347, 719-724 (2008) · Zbl 1145.54045 · doi:10.1016/j.jmaa.2008.04.049
[11] Turkoglu, D.; Abuloha, M.: Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta math. Sin. (Engl. Ser.) 26, No. 3, 489-496 (2010) · Zbl 1203.54049 · doi:10.1007/s10114-010-8019-5
[12] Turkoglu, D.; Abuloha, M.; Abdeljawad, T.: KKM mappings in cone metric spaces and some fixed point theorems, Nonlinear anal. TMA 72, No. 1, 348-353 (2010) · Zbl 1197.54076 · doi:10.1016/j.na.2009.06.058
[13] Bhaskar, T. G.; Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal. 65, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017
[14] 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 · doi:10.1016/j.na.2008.09.020