Abstract
In this paper, we improve the Proinov theorem by adding certain rational expressions to the definition of the corresponding contractions. After that, we prove fixed point theorems for these modified Proinov contractions in the framework of dislocated b-metric spaces. We show some illustrative examples to indicate the validity of the main results.
Highlights
Introduction and preliminariesIn the nature of mathematics, there is the purpose of generalizing, expanding, and obtaining the most general forms of existing concepts and results
The concept of metric, which is the most fundamental and solid basis of the analysis study, has been constantly expanded and generalized with this motivation
Two of the new and original generalizations of metric notions are b-metrics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and dislocated metrics [17,18,19,20,21]. These two notions have emerged under the name of dislocated b-metric [22, 23]
Summary
Theorem 11 Let (S, db, s) be a complete db-ms, , ∈ , a number α ∈ [1, ∞), and two continuous mappings Z, U : S → S such that, for every distinct v, w ∈ S with db(Zv, Uw) > 0, the following inequality sαdb(Zv, Uw) ≤ R1(v, w). Corollary 12 Let (S, db, s) be a complete db-ms, , ∈ , a number α ∈ [1, ∞), and a continuous mapping Z : S → S such that, for every distinct v, w ∈ S with db(Zv, Zw) > 0, the following inequality sαdb(Zv, Zw) ≤ R∗1(v, w). Theorem 13 Let (S, db, s) be a complete db-ms, , ∈ , a number α ∈ [1, ∞), and two mappings Z, U : S → S such that, for every distinct v, w ∈ S with db(Zv, Uw) > 0, the following inequality sαdb(Zv, Uw) ≤ R2(v, w).
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