Abstract
In the recent paper (B. Samet, C. Vetro, and P. Vetro, Fixed point theorems forα-ψ-contractive type mappings, Nonlinear Analysis. Theory, Methods and Applications, 75 (2012), 2154-2165.), the authors introduced the concept ofα-admissible maps on metric spaces. Using this new concept, they presented some nice fixed point results. Also, they gave an existence theorem for integral equation to show the usability of their result. Then, many authors focused on this new concept and obtained a lot of fixed point results, which are used for existence theorems. In this paper, we not only extend some of the recent results about this direction but also generalize them. Then, we give some examples to show our results are proper extensions. Furthermore, we use our results to obtain the existence and uniqueness result for a solution of fourth order two-point boundary value problem.
Highlights
Introduction and PreliminariesFixed point theory contains many different fields of mathematics, such as nonlinear functional analysis, mathematical analysis, operator theory, and general topology
The study of fixed point theory has developed in two major branches: the first is fixed point theory for contraction or contraction type mappings on complete metric spaces and the second is fixed point theory for continuous operators on compact and convex subsets of a normed space
There has been a lot of activities in the first branch and several fundamental fixed point results have been extended and generalized by many authors in different directions
Summary
Fixed point theory contains many different fields of mathematics, such as nonlinear functional analysis, mathematical analysis, operator theory, and general topology. In their recent paper, Samet et al [2] introduced the notions of α-admissible and α-ψ-contractive mappings and gave some fixed point results for such mappings. Samet et al [2] introduced the notions of α-admissible and α-ψ-contractive mappings and gave some fixed point results for such mappings Their results are closely related to some ordered fixed point results. Let (X, d) be a complete metric space and let T : X → X be an α-admissible and α-ψ-contractive mapping. The aim of this paper is to extend and generalize the above results Note that, in these theorems, the function ψ belongs to the class Φ; that is, ψ is (c)-comparison function.
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