zbMATH — the first resource for mathematics

Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition. (English) Zbl 1133.54024
Let \(X\) be a metric space. A mapping \(T: X \mapsto X\) is called weakly contractive with respect to \(f:X \mapsto X\) if for each \(x, y\in X,\) \[ d(Tx,Ty)\leq d(fx,fy)-\varphi d(fx, fy) \] where \(\varphi :[0,+\infty )\rightarrow [0,+\infty )\) is continuous, nondecreasing, positive on \((0,+\infty )\), \(\varphi(0)=0\) and \(\lim_{t\rightarrow \infty}\varphi(t)=\infty\).
The authors prove the existence of coincidence points and common fixed points for a weakly contractive mapping \(T\) with respect to \(f\). Related results on invariant approximation are also derived.

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
Full Text: DOI EuDML
[1] Ahmed, MA, Common fixed point theorems for weakly compatible mappings, The Rocky Mountain Journal of Mathematics, 33, 1189-1203, (2003) · Zbl 1065.47051
[2] Alber, YaI; Guerre-Delabriere, S; Gohberg, I (ed.); Lyubich, Yu (ed.), Principle of weakly contractive maps in Hilbert spaces, No. 98, 7-22, (1997), Basel · Zbl 0897.47044
[3] Beg, I; Abbas, M, Fixed points and best approximation in manger convex metric spaces, Archivum Mathematicum, 41, 389-397, (2005) · Zbl 1109.47047
[4] Beg, I; Abbas, M, Fixed-point theorems for weakly inward multivalued maps on a convex metric space, Demonstratio Mathematica, 39, 149-160, (2006) · Zbl 1099.54030
[5] Beg, I; Azam, A, Common fixed points for commuting and compatible maps, Discussiones Mathematicae, Differential Inclusions, 16, 121-135, (1996) · Zbl 0912.47033
[6] Brosowski, B, Fixpunktsätze in der approximationstheorie, Mathematica—Revue d’Analyse Numérique et de Théorie de l’Approximation, 11 (34), 195-220, (1969) · Zbl 0207.45502
[7] Hussain, N; Khan, AR, Common fixed-point results in best approximation theory, Applied Mathematics Letters, 16, 575-580, (2003) · Zbl 1063.47055
[8] Jungck, G, Common fixed points for commuting and compatible maps on compacta, Proceedings of the American Mathematical Society, 103, 977-983, (1988) · Zbl 0661.54043
[9] Jungck, G; Rhoades, BE, Fixed points for set valued functions without continuity, Indian Journal of Pure and Applied Mathematics, 29, 227-238, (1998) · Zbl 0904.54034
[10] Kamran, T, Coincidence and fixed points for hybrid strict contractions, Journal of Mathematical Analysis and Applications, 299, 235-241, (2004) · Zbl 1064.54055
[11] Kaneko, H; Sessa, S, Fixed point theorems for compatible multi-valued and single-valued mappings, International Journal of Mathematics and Mathematical Sciences, 12, 257-262, (1989) · Zbl 0671.54023
[12] Meinardus, G, Invarianz bei linearen approximationen, Archive for Rational Mechanics and Analysis, 14, 301-303, (1963) · Zbl 0122.30801
[13] Pant, RP, Common fixed points of noncommuting mappings, Journal of Mathematical Analysis and Applications, 188, 436-440, (1994) · Zbl 0830.54031
[14] Rhoades, BE, Some theorems on weakly contractive maps, Nonlinear Analysis, 47, 2683-2693, (2001) · Zbl 1042.47521
[15] Sessa, S, On a weak commutativity condition of mappings in fixed point considerations, Institut Mathématique. Publications. Nouvelle Série, 32(46), 149-153, (1982) · Zbl 0523.54030
[16] Shahzad, N, Invariant approximations and [inlineequation not available: see fulltext.]-subweakly commuting maps, Journal of Mathematical Analysis and Applications, 257, 39-45, (2001) · Zbl 0989.47047
[17] Shahzad, N, Generalized [inlineequation not available: see fulltext.]- nonexpansive maps and best approximations in Banach spaces, Demonstratio Mathematica, 37, 597-600, (2004) · Zbl 1095.41017
[18] Singh, SP, An application of a fixed-point theorem to approximation theory, Journal of Approximation Theory, 25, 89-90, (1979) · Zbl 0399.41032
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.