Abstract
In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.
Highlights
Introduction and preliminariesIt is well known that the Banach contraction principle [5] is one of the most essential and attractive results in nonlinear analysis and mathematical analysis in general
The whole fixed point theory is a significant subject in different fields as geometry, differential equations, informatics, physics, economics, engineering, and many others
We dropped the property of Hausdorffness of the rectangular metric space (M, d) and the continuity of the mapping d
Summary
It is well known that the Banach contraction principle [5] is one of the most essential and attractive results in nonlinear analysis and mathematical analysis in general. Karapınar [13] extended the concepts given in [24] to obtain the existence and uniqueness of a fixed point of α−ψ-contraction mappings in the setting of rectangular metric spaces. Let (M, d) be a Hausdorff and complete rectangular metric space, and let T : M → M be an α-admissible mapping with respect to η. Rectangular metric space (M, d) is not Hausdorff, and the mapping T has no fixed point. Many known proofs of fixed point results in rectangular metric spaces become much simpler and shorter using both these lemmas. Let {pn+1}n∈N∪{0} = {T pn}n∈N∪{0} = {T np0}n∈N∪{0}, T 0p0 = p0 be a Picard sequence in rectangular metric space (M, d) inducing by the mapping T : M → M and initial point p0 ∈ X.
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