# zbMATH — the first resource for mathematics

Meir-Keeler-type conditions in abstract metric spaces. (English) Zbl 1220.54027
Summary: Various Meir-Keeler-type conditions for mappings acting in abstract metric spaces are presented and their connections are discussed. Results about associated symmetric spaces, obtained in [S. Radenović and Z. Kadelburg, Banach J. Math. Anal. 5, No. 1, 38–50, electronic only (2011; Zbl 1297.54058)] are used to show that the regularity condition for the underlying cone can be dropped in some fixed point results that have appeared recently.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E35 Metric spaces, metrizability
Zbl 1297.54058
Full Text:
##### References:
 [1] Meir, A.; Keeler, E., A theorem on contraction mappings, J. math. anal. appl., 28, 326-329, (1969) · Zbl 0194.44904 [2] Park, S.; Rhoades, B.E., Meir – keeler type contractive conditions, Math. japon., 26, 13-20, (1981) · Zbl 0454.54030 [3] Jachymski, J., Equivalent conditions and the meir – keeler type theorems, J. math. anal. appl., 194, 293-303, (1995) · Zbl 0834.54025 [4] Suzuki, T., Fixed-point theorem for asymptotic contractions of meir – keeler type in complete metric spaces, Nonlinear anal., 64, 971-978, (2006) · Zbl 1101.54047 [5] Haghi, R.H.; Rezapour, Sh., Fixed points of multifunctions on regular cone metric spaces, Expo. math., 28, 71-77, (2010) · Zbl 1193.47058 [6] Huang, L.G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. math. anal. appl., 332, 2, 1468-1476, (2007) · Zbl 1118.54022 [7] Chen, Chi-Ming; Chang, Toni-Huei, Common fixed point theorems for a weaker meir – keeler type function in cone metric spaces, Appl. math. lett., 23, 11, 1336-1341, (2010) · Zbl 1196.54067 [8] Khoajasteh, F.; Goodarzi, Z.; Razani, A., Some fixed point theorems of integral type contraction in cone metric spaces, Fixed point theory appl., (2010), Article ID 189684, 13 pages [9] Eisenfeld, J.; Lakshmikantham, V., Comparison principle and nonlinear contractions in abstract spaces, J. math. anal. appl., 49, 504-511, (1975) · Zbl 0296.47037 [10] Eisenfeld, J.; Lakshmikantham, V., Remarks on nonlinear contraction and comparison principle in abstract cones, J. math. anal. appl., 61, 116-121, (1977) · Zbl 0435.47059 [11] Radenović, S.; Kadelburg, Z., Quasi-contractions on symmetric and cone symmetric spaces, Banach J. math. anal., 5, 1, 38-50, (2011) · Zbl 1297.54058 [12] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag · Zbl 0559.47040 [13] Vandergraft, J.S., Newton method for convex operators in partially ordered spaces, SIAM J. numer. anal., 4, 3, 406-432, (1967) · Zbl 0161.35302 [14] Zabreĭko, P.P., K-metric and K-normed spaces: survey, Collect. math., 48, 4-6, 825-859, (1997) · Zbl 0892.46002 [15] Wilson, W.A., On semimetric spaces, Amer. J. math., 53, 361-373, (1931) · JFM 57.0735.01 [16] I.A. Bahtin, The contraction mapping principle in quasimetric spaces (in Russian), Funktsional’niĭanaliz 30, pp. 26-37, Ul’yanovsk Gos. Univ., Ul’yanovsk, 1989. [17] V. Berinde, Generalized contractions in quasimetric spaces, in: Seminar on Fixed Point Theory, 3-9, Preprint 93-3, Babes-Bolyai Univ., Cluj-Napoca, 1993. · Zbl 0878.54035 [18] Berinde, V., Sequences of operators and fixed points in quasimetric spaces, Studia univ. babes-bolyai, 41, 4, 23-27, (1996) · Zbl 1005.54501
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.