Weak contractions, common fixed points, and invariant approximations. (English) Zbl 1183.47049
The concept of weak contraction was extended to the notion of \((\theta, L)\)-weak contraction by V.Berinde [“On the approximation of fixed points of weak contractive mappings”, Carpathian J. Math.19, No.1, 7–22 (2003; Zbl 1114.47045)]. In this paper, the authors extend the concept of \((\theta, L)\)-weak contraction and introduce the notion of \((f,\theta, L)\)-weak contraction. For the underlying concept, the authors first establish a common fixed point result and then apply it to get a result in the field of approximation.
Reviewer: Hemant Kumar Nashine (Raipur)
MSC:
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
weak contraction; \((\theta, L)\)-weak contraction; fixed point; best approximation; Opial’s conditionCitations:
Zbl 1114.47045References:
| [1] | Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York, NY, USA; 2001:x+302. · Zbl 1318.47001 · doi:10.1002/9781118033074 |
| [2] | Berinde V: Approximating fixed points of weak contractions using the Picard iteration.Nonlinear Analysis Forum 2004,9(1):43-53. · Zbl 1078.47042 |
| [3] | Berinde V, Păcurar M: Fixed points and continuity of almost contractions.Fixed Point Theory 2008,9(1):23-34. · Zbl 1152.54031 |
| [4] | Singh SP, Watson B, Srivastava P: Fixed Point Theory and Best Approximation: The KKM-Map Principle, Mathematics and Its Applications. Volume 424. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997. · Zbl 0901.47039 · doi:10.1007/978-94-015-8822-5 |
| [5] | Ćirić LB: A generalization of Banach’s contraction principle.Proceedings of the American Mathematical Society 1974,45(2):267-273. · Zbl 0291.54056 |
| [6] | Ćirić LB: Contractive type non-self mappings on metric spaces of hyperbolic type.Journal of Mathematical Analysis and Applications 2006,317(1):28-42. 10.1016/j.jmaa.2005.11.025 · Zbl 1089.54019 · doi:10.1016/j.jmaa.2005.11.025 |
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