α-β-ψ-φ Contraction in Digital Metric Spaces

Abstract

Samet et al. (Nonlinear Anal. 75, 2012, 2154-2165) introduced a new, simple and unified approach by using the concepts of α-ψ-contractive type mappings and α-admissible mappings in metric spaces and presented some nice fixed point results. Recently, Sridevi et.al (International Journal of Mathematics Trends and Technology, Volume 48, Number 3 August 2017) proposed the concept of α-ψ- contraction and generalized α-ψ- for self map in digital metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called α--ψ- contraction and generalized α--ψ- contractive pair of mappings and study various fixed point theorems for such mappings in digital metric spaces. For this, we introduce a new notion of α--admissible w.r.t T mapping which in turn generalizes the concept of g-monotone mapping recently given by “Ciric et al. (Fixed Point Theory Appl. 2008 (2008), Article ID 131294, 11 pages)”. Also, we give some fixed point theorems for cyclic contractive mapping in such spaces. The presented theorems hold without using completeness of the space and without the assumption of continuity of the given mappings. Our results extend, generalize and subsumes digital version of various known comparable results [[1-4,8,13,16,18-22], worth to mention here]. Some illustrative examples are quoted to demonstrate the main results.