Abstract
Generalized Meir-Keeler α-contractive functions and pairs are introduced and their fixed and common fixed point theorems are obtained. Also, the so-called generalized Meir-Keeler α-f-contractive maps commuting with f are introduced and their coincidence and common fixed point theorems are investigated. New sufficient conditions different from those in (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012) are used. An application to the coupled fixed point is established as well. An example is given to show that the α-Meir-Keeler generalization is real. AMS Subject Classification: 47H10, 54H25.
Highlights
Fixed point theory is of wide and endless applications in many fields of engineering and science
The Banach contraction principle, has attracted many researchers who tried to generalize it in different aspects
Some dealt with the contractive condition itself, of worth mentioning Meir-Keeler contractive type [ – ], some extended it to more generalized metric-type spaces [ – ] and others applied to common [ ], coupled and tripled versions
Summary
Fixed point theory is of wide and endless applications in many fields of engineering and science. We assume that the following condition (H) holds: If {xn} is a sequence in X such that α(xn, xn+ ) ≥ for all n and xn → x implies α(xn, x) ≥ for all n, uniqueness of the fixed point is obtained. Corollary Let (X, d) be an f -orbitally complete metric space, where f is a self-mapping of X. We assume that the following condition (H) holds: If {xn} is a sequence in X such that α(xn, xn+ ) ≥ for all n and xn → x, α(xn, x) ≥ for all n, uniqueness of the fixed point is obtained. The existence and uniqueness of the fixed point cannot be proved in the category of Meir-Keeler contractions, but can be proved by means of Corollary. Such an admissibility condition was used in obtaining the main result in Theorem . of [ ]
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