Abstract
In this paper, we prove some fixed point theorems in a b-metric-like space setting using a new class of admissible mappings and types of α − k and ( ψ , ϕ )-contractive conditions. Our results are supported by the application of finding solutions of integral equations and generalizing some well-known results of the literature.
Highlights
Introduction and PreliminariesRecently, several authors investigated fixed point theorems in generalized metric spaces, such as b-metric spaces, metric-like spaces, b-metric-like spaces, and so on, where “metric” d takes its values in more generalized conditions
Many generalizations under (ψ − φ), α − ψ, and (α − ψ − φ)-contractive conditions was provided in many works
In our work, following this direction, using the notion of α-admissible mapping, in the first part of the paper, we proved some fixed point theorems for contractions of rational types, by means of a function γ : R+ × R+ → R+
Summary
Several authors investigated fixed point theorems in generalized metric spaces, such as b-metric spaces, metric-like spaces, b-metric-like spaces, and so on, where “metric” d takes its values in more generalized conditions. Let ( M, σb ) be a b-metric-like space with parameter s, and let {tn } be any sequence in M with t ∈ M such that lim σb (tn , t) = 0. Let ( M, σb ) be complete b-metric-like space with parameter s ≥ 1, and let {tn } be a sequence such that lim σb (tn , tn+1 ) = 0. Let {tn } be a sequence in a b-metric-like space ( M, σb ) with parameter s ≥ 1, such that σb (tn , tn+1 ) ≤ λσb (tn−1 , tn ) for all n ∈ N, for some λ, where 0 ≤ λ < 1/s. For the proof of the previous lemma, one can use the following clear inequalities: σb (tn+1 , tn+2 ) ≤ λσb (tn , tn+1 ) ≤ λ2 σb (tn−1 , tn ) ≤ . . . ≤ λn+1 σb (t0 , t1 ), and σb (tm , tn ) ≤ sσb (tm , tm+1 ) + s2 σb (tm+1 , tm+2 ) + . . . + sn−m−1 σb (tn−2 , tn−1 ) + sn−m σb (tn−1 , tn ), where m, n ∈ N and n > m
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