Abstract
In this manuscript, we introduce a new notion: a Berinde type ( α , ψ ) -contraction mapping. Thereafter, we investigate not only the existence, but also the uniqueness of a fixed point of such mappings in the setting of right-complete quasi-metric spaces. The result, presented here, not only generalizes a number of existing results, but also unifies several ones on the topic in the literature. An application of nonlinear fractional differential equations is given.
Highlights
Introduction and PreliminariesFixed point results have been studied in various directions since the introduction of Banach contraction theorem
Several new contractive conditions have been developed in an attempt to obtain more refined fixed point results
Some related fixed point results are known as α-ψ contraction type results
Summary
Fixed point results have been studied in various directions since the introduction of Banach contraction theorem. Berinde [1] introduced the concept of (θ, L)-weak contractions and studied some related fixed point results. In the present paper, inspired from the result of Berinde [1] and Popescu [3], we propose a new contraction, and we discuss fixed point existence problems for such mappings. A sequence {θn } in a (q.m.s.) (M, ρ) is called left-Cauchy if for each e > 0, there exists N = N (e) ∈ N, so that ρ(θn , θm ) < e for all n ≥ m > N. The notion of right-Cauchy is defined analogously A self-mapping φ, on the non-negative real numbers, is called a comparison function ([26]) if it is non-decreasing and satisfies lim φn (s) = 0, for each s ∈ [0, ∞).
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