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Fixed point theorems on generalized metric spaces for mappings in a class of almost \(\varphi\)-contractions. (English) Zbl 1343.54014

In this paper, the authors obtained some sufficient conditions for the existence and uniqueness of fixed point for a contraction mapping satisfying an implicit relation in the framework of generalized metric space in the sense of A. Branciari [Publ. Math. 57, No. 1–2, 31–37 (2000; Zbl 0963.54031)]. The results obtained generalize and extend the corresponding results of V. Berinde [Nonlinear Anal. Forum 9, No. 1, 43–53 (2004; Zbl 1078.47042)], R. M. Tiberio Bianchini [Boll. Unione Mat. Ital., IV. Ser. 5, 103–108 (1972; Zbl 0249.54023)], R. Kannan [Am. Math. Mon. 76, 405–408 (1969; Zbl 0179.28203)] and of S. Reich [Can. Math. Bull. 14, 121–124 (1971; Zbl 0211.26002)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
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[1] A. Azam, M. Arshad, Kannan fixed point theorem on generalized metric spaces, J. Nonlinear Sci. Appl. 1(1) (2008), 45-48.; · Zbl 1161.54022
[2] V. Berinde, Approximating fixed point of weak contractions using the Picard iteration, Nonlinear Anal. Forum 9(1) (2004), 43-53.; · Zbl 1078.47042
[3] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, Heidelberg, New York, 2007.; · Zbl 1165.47047
[4] V. Berinde, M. Pacurar, Fixed points and continuity of almost contractions, Fixed Point Theory 9(1) (2008), 23-34.; · Zbl 1152.54031
[5] V. Berinde, Stability of Picard iteration for contractive mappings satisfying an implicit relation, Carpathian J. Math. 27(1) (2011), 13-23.; · Zbl 1265.54152
[6] R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. Un. Mat. Ital. 5 (1972), 103-108.; · Zbl 0249.54023
[7] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31-37.; · Zbl 0963.54031
[8] P. Das, A fixed point theorem on a class of generalized metric spaces, Korean J. Math. Sci. 1 (2002), 29-33.;
[9] P. Das, L. K. Dey, A fixed point theorem in a generalized metric space, Soochow J. Math. 33 (2007), 33-9.; · Zbl 1137.54024
[10] P. Das, L. K. Dey, Fixed point of contractive mappings in a generalized metric space, Math. Slovaca 59(4) (2009), 499-504.; · Zbl 1240.54119
[11] P. Das, L. K. Dey, Porosity of certain classes of operators in generalized metric spaces, Demonstratio Math. 42(1) (2009), 163-174.; · Zbl 1173.54018
[12] R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405-408.; · Zbl 0179.28203
[13] L. Kikina, K. Kikina, A fixed point theorem in generalized metric spaces, Demonstratio Math. 46(1) (2013), 181-190.; · Zbl 1272.54037
[14] L. Kikina, K. Kikina, Fixed points on two generalized metric spaces, Int. J. Math. Anal. 5(30) (2011), 1459-1467.; · Zbl 1259.54021
[15] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 256-290.; · Zbl 0365.54023
[16] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121-124.; · Zbl 0211.26002
[17] B. Samet, Discussion on a fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces by Branciari, Publ. Math. Debrecen 76(3-4) (2010), 493-494.; · Zbl 1224.54106
[18] I. R. Sarma, J. M. Rao, S. S. Rao, Contractions over generalized metric spaces, J. Nonlinear Sci. Appl. 2(3) (2009), 180-182.; · Zbl 1173.54311
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