Fixed point results for generalized quasicontraction mappings in abstract metric spaces. (English) Zbl 1189.54036

Summary: We introduce the concept of generalized quasicontraction mappings in an abstract metric space. By using this concept, we construct an iterative process which converges to a unique fixed point of these mappings. The result presented in this paper generalizes the Banach contraction principle in the setting of metric space and a recent result of L.-G. Huang and X. Zhang [ibid. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)] for contractions. We also validate our main result by an example.


54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
47H10 Fixed-point theorems
54E35 Metric spaces, metrizability


Zbl 1118.54022
Full Text: DOI


[1] Berinde, V., A fixed point theorem of maia type in \(K\)-metric spaces, Semin. fixed point theory (cluj-napoca), 3, 7-14, (1991) · Zbl 0782.54036
[2] Collatz, L., Functional analysis and numerical mathematics, (1966), Academic Press New York · Zbl 0221.65088
[3] De Pascale, E.; Marino, G.; Pietromala, P., The use of the \(E\)-metric spaces in the search for fixed points, Le math., 48, 367-376, (1993) · Zbl 0833.47049
[4] Rus, I.A.; Petrusel, A.; Petrusel, G., Fixed point theory, (2008), Cluj University Press Cluj-Napoca · Zbl 1171.54034
[5] Schröder, J., Anwendung funktionalanalytischer methoden zur numerischen behandlung von gleichungen, Z. angew. math. mech., 36, 260-261, (1956) · Zbl 0073.33602
[6] Schröder, J., Das iterationverfahren bei allgemeinerem abstandsbegriff, Math. Z., 66, 111-116, (1956) · Zbl 0073.33503
[7] Schröder, J., Nichtlineare majoranten beim verfahren der schrittweisen, Arch. math., 7, 471-484, (1956) · Zbl 0080.10605
[8] Schröder, J., Anwendung von fixpunktsätzen bei der numerischen behandlung nichtlinearer gleichungen in halbgeordneten Räumen, Arch. ration. mech. anal., 4, 177-192, (1959) · Zbl 0090.10101
[9] Zabreiko, P.P., \(K\)-metric and \(K\)-normed linear spaces: survey, Collect. math., 48, 825-859, (1997) · Zbl 0892.46002
[10] Zabreiko, P.P., The fixed point theory and the Cauchy problem for partial differential equations, Nonlinear anal. appl., 93-106, (1998), (in Russian). Tr. Inst. Mat. (Minsk) 1, Natl. Akad. Nauk Belarusi, Inst. Mat., Minsk · Zbl 0924.34057
[11] Huang, L.G.; Zhang, X., Cone metric space and fixed point theorems of contractive type mappings, J. math. anal. appl., 332, 1467-1475, (2007)
[12] D. Ilic, V. Rakocevic, Quasi-contraction on cone metric space, Appl. Math. Lett. (2008) (doi:10.1016/j.aml.2008.08.011) · Zbl 1179.54060
[13] Ćirić, Lj.B., A generalization of banach’s contraction principle, Proc. amer. math. soc., 45, 267-273, (1974) · Zbl 0291.54056
[14] Ćirić, Lj.B., Fixed point theory: contraction mapping principle, (2003), FME Press · Zbl 1093.47517
[15] Banach, S., Sur LES operations dans LES ensembles abstraits et leur applications aux equations integrales, Fund. math., 3, 133-181, (1922) · JFM 48.0201.01
[16] Pathak, H.K.; Shahzad, N., Fixed points for generalized contractions and applications to control theory, Nonlinear anal., 68, 2181-2193, (2008) · Zbl 1144.54031
[17] Denkowski, Z.; Migorski, S.; Papageorgiou, N.S., An introduction to nonlinear analysis: applications, (2003), Kluwer Academic Publishers · Zbl 1030.35106
[18] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press Boston · Zbl 0661.47045
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.