Best proximity point results for p -proximal contractions
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BEST PROXIMITY POINT RESULTS FOR p-PROXIMAL CONTRACTIONS I. ALTUN1,2,∗ , M. ASLANTAS3 and H. SAHIN4 1 2
Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam e-mail: [email protected] 3
4
Department of Mathematics, Faculty of Science, C ¸ ankırı Karatekin University, 18100 C ¸ ankırı, Turkey e-mail: [email protected]
Department of Mathematics, Faculty of Science and Arts, Amasya University, Amasya, Turkey e-mail: [email protected] (Received October 31, 2019; revised February 8, 2020; accepted February 11, 2020)
Abstract. We introduce the concepts of p-proximal contraction and p-proximal contractive mappings on metric spaces. Then we give some best proximity point results for such mappings. Also we provide some illustrative examples to compare our results with some earliers.
1. Introduction and preliminaries Since fixed point theory has a great number of applications in fields such as mathematics, computer science and economics, there are a lot of results on this topic. Suzuki [20] categorized metrical fixed point results into four types of classes which are Leader type, Unnamed type, Subrahmanyam type and Caristi type. Suzuki mentioned that fixe d point theorems for the Unnamed type class can be constructed from a Leader type class. However, this gives meaningless results. Besides, he presented a fixed point theorem which belongs to Unnamed type class and cannot be obtained from Leader type class. In this sense, very recently inspired by the definition of p-contraction mapping, Altun et. al. [2] gave the following notion of p-contractive mapping and obtained a fixed point theorem which also belongs to Unnamed type class: ∗ Corresponding
author. Key words and phrases: best proximity point, p-proximal contraction, fixed point, complete metric space. Mathematics Subject Classification: primary 54H25, secondary 47H10. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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I. I. ALTUN, ALTUN, M. M. ASLANTAS ASLANTAS and and H. H. SAHIN SAHIN
Let (Ω, d) be a metric space and T : Ω → Ω be a self mapping. If there exists a k in [0, 1) such that d(T ζ, T η) ≤ k d(ζ, η) + |d(ζ, T ζ) − d(η, T η)| for all ζ, η ∈ Ω, then T is said to be a p-contraction mapping. If the mapping T satisfies d(T ζ, T η) < d(ζ, η) + |d(ζ, T ζ) − d(η, T η)| for all ζ, η ∈ Ω with ζ �= η, then it is called a p-contractive mapping. On the other hand, very recently, well known fixed point theorems in the literature have been extended taking into account best proximity point theorems for nonself mappings. In this paper, we obtain some new best proximity point theorems for nonself p-contraction and p-contractive mappings. Before giving our results, we recall some fundamental concepts about best proximity points. Let (Ω, d) be a metric space and Φ, Ψ be nonempty subsets of Ω. If Φ ∩ Ψ = ∅, then the nonself mapping T : Φ → Ψ h
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