Common fixed points of a generalized ordered \(g\)-quasicontraction in partially ordered metric spaces. (English) Zbl 1286.54048

Summary: The concept of a generalized ordered \(g\)-quasicontraction is introduced, and some fixed and common fixed point theorems for a \(g\)-nondecreasing generalized ordered \(g\)-quasicontraction mapping in partially ordered complete metric spaces are proved. We also show the uniqueness of the common fixed point in the case of a generalized ordered \(g\)-quasicontraction mapping. Finally, we prove fixed point theorems for mappings satisfying the so-called weak contractive conditions in the setting of a partially ordered metric space. The presented theorems are generalizations of very recent fixed point theorems due to Z. Golubović et al. [Fixed Point Theory Appl. 2012, Article ID 20 (2012; Zbl 1273.54055)].


54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54E50 Complete metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces


Zbl 1273.54055
Full Text: DOI


[1] Agarwal, RP; El-Gebeily, MA; O’Regan, D, Generalized contractions in partially ordered metric spaces, Appl. Anal, 87, 109-116, (2008) · Zbl 1140.47042
[2] Agarwal, RP; O’Regan, D; Sambandham, M, Random and deterministic fixed point theory for generalized contractive maps, Appl. Anal, 83, 711-725, (2004) · Zbl 1088.47044
[3] Ahmad, B; Nieto, JJ, The monotone iterative technique for three-point second-order integrodifferential boundary value problems with \(p\)-Laplacian, No. 2007, (2007) · Zbl 1149.65098
[4] Boyd, DW; Wong, JSW, On nonlinear contractions, Proc. Am. Math. Soc, 20, 458-464, (1969) · Zbl 0175.44903
[5] Cabada, A; Nieto, JJ, Fixed points and approximate solutions for nonlinear operator equations, J. Comput. Appl. Math, 113, 17-25, (2000) · Zbl 0954.47038
[6] Ćirić, LB, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Belgr.), 12, 19-26, (1971) · Zbl 0234.54029
[7] Ćirić, LB, A generalization of banach’s contraction principle, Proc. Am. Math. Soc, 45, 267-273, (1974) · Zbl 0291.54056
[8] Ćirić, LB, Fixed points of weakly contraction mappings, Publ. Inst. Math. (Belgr.), 20, 79-84, (1976) · Zbl 0354.54031
[9] Ćirić, LB, Coincidence and fixed points for maps on topological spaces, Topol. Appl, 154, 3100-3106, (2007) · Zbl 1132.54024
[10] Ćirić, LB; Ume, JS, Nonlinear quasi-contractions on metric spaces, Prakt. Akad. Athēnōn, 76, 132-141, (2001)
[11] Ćirić, LB, Common fixed points of nonlinear contractions, Acta Math. Hung, 80, 31-38, (1998) · Zbl 0918.47045
[12] Drici, Z; McRae, FA; Vasundhara Devi, J, Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear Anal., Theory Methods Appl, 67, 641-647, (2007) · Zbl 1127.47049
[13] Bhaskar, TG; Lakshmikantham, V, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., Theory Methods Appl, 65, 1379-1393, (2006) · Zbl 1106.47047
[14] Gajić, L; Rakočević, V, Quasicontraction nonself-mappings on convex metric spaces and common fixed point theorems, Fixed Point Theory Appl, 3, 365-375, (2005) · Zbl 1104.54018
[15] Hussain, N, Common fixed points in best approximation for Banach operator pairs with ćrić type \(I\)-contractions, J. Math. Anal. Appl, 338, 1351-1363, (2008) · Zbl 1134.47039
[16] Golubović, Z; Kadelburg, Z; Radenović, S, Common fixed points of ordered \(g\)-quasicontractions and weak contractions in ordered metric spaces, No. 2012, (2012) · Zbl 1273.54055
[17] Jungck, G, Commuting mappings and fixed points, Am. Math. Mon, 83, 261-263, (1976) · Zbl 0321.54025
[18] Jungck, G, Compatible mappings and common fixed points, Int. J. Math. Math. Sci, 9, 771-779, (1986) · Zbl 0613.54029
[19] Ćirić, LB; Cakić, N; Rajović, M; Ume, JS, Monotone generalized nonlinear contractions in partially ordered metric spaces, No. 2008, (2008) · Zbl 1158.54019
[20] Das, KM; Naik, KV, Common fixed point theorems for commuting maps on a metric space, Proc. Am. Math. Soc, 77, 369-373, (1979) · Zbl 0418.54025
[21] Berinde, V, A common fixed point theorem for quasi contractive type mappings, Ann. Univ. Sci. Bp, 46, 81-90, (2003)
[22] Beg, I; Abbas, M, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, No. 2006, (2006) · Zbl 1133.54024
[23] Song, Y, Coincidence points for noncommuting \(f\)-weakly contractive mappings, Int. J. Comput. Appl. Math, 2, 51-57, (2007) · Zbl 1256.54083
[24] Jungck, G; Hussain, N, Compatible maps and invariant approximations, J. Math. Anal. Appl, 325, 1003-1012, (2007) · Zbl 1110.54024
[25] Al-Thagafi, MA; Shahzad, N, Banach operator pairs, common fixed points, invariant approximations and ∗-nonexpansive multimaps, Nonlinear Anal, 69, 2733-2739, (2008) · Zbl 1170.47034
[26] Das, KM; Naik, KV, Common fixed point theorems for commuting maps on a metric space, Proc. Am. Math. Soc, 77, 369-373, (1979) · Zbl 0418.54025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.