Abstract
Extending the Presic type operators to modular spaces, we introduce generalised Presic typew-contractive mappings and stronglyw-contractive mappings in a modular metric space and establish fixed-point theorems for such contractions in modular spaces. Ulam–Hyers stability of the fixed-point equation involving Presic type operators is also discussed. Our results extend and generalise some known results in the literature. The results are supported by appropriate example and an application to Caratheodory type integral equation.
Highlights
Maurice Ren Frechet [1] introduced the general and axiomatic form of distance as “L—space.” Felix Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalised in numerous ways
Bakhtin [3], Branciari [4], George et al [5], and Mitrovicand Radenovic [6] introduced the notions of a b-metric, a rectangular metric, a rectangular b-metric, and bsv-metric, respectively. e basic concepts and theory of modular space was formulated in [7]
We see that Chaipunya et al [12] introduced Geraghty type theorems and Turkoglu and Kilinc [13] introduced Caristi type theorems in a modular metric space and gave its applications in integral equations
Summary
Maurice Ren Frechet [1] introduced the general and axiomatic form of distance as “L—space.” Felix Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalised in numerous ways. In [8], the authors established fixed-point theorems in a modular function space. Chistyakov, in [10], established the existence of fixed point for contractive maps and strongly contractive maps in modular metric spaces. Umit et al [11] introduced Bogin type w-contractions and proved fixed-point theorems for such contractions in a w-complete modular metric space and provided application to antiperiodic boundary value problems. E aim of this work is to introduce generalised Presic type contractions (which includes Ciric–Presic type contraction and Presic type contractions) in modular metric spaces and establish fixed-point theorems for such contraction mappings in a w-complete modular metric space. We have introduced Ulam–Hyers stability of fixed-point equations involving Presic type operators in a modular metric space. If w is (not necessarily convex) modular on X, vλ(x, y) (wλ(x, y)/λ) is always convex modular on X
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