Fixed point theorems for partial α−ψ contractive mappings in generalized metric spaces

Abstract

In this paper, we introduce the concept of partial α-ψ contractive mappings along with generalized metric distance. We also establish the existence of fixed point theorems for such mappings in generalized metric spaces. Our results extend and unify main results of Karapinar [E. Karapinar, Abstr. Appl. Anal., 2014 (2014), 7 pages] and several well-known results in literature. We give some examples to illustrate the usability of our results. Moreover, we prove the fixed point results in generalized metric space endowed with an arbitrary binary relation and the fixed point results in generalized metric space endowed with graph.

Various contractions in generalized metric space

“…This is different from the classical concept of Cauchy sequence. In 2003, Akram et al [30] introduced the notion of A-contraction mapping as a generalization of Kannan's maps, where A is collection of all functions α : R 3 + → R + satisfying 1. α is continuous on the set R 3 + of all triplets of non-negative reals; 2. a ≤ Kb for some k ∈ [0, 1) whenever a ≤ α(a, b, b) or a ≤ α(b, b, a) for all a, b. Definition 1.10.…”