Abstract
The aim of this article is to introduce the notion of a ϕ,ψ-metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology τϕ,ψ in such spaces and discuss their topological properties. We also establish the Banach contraction principle in the context of ϕ,ψ-metric spaces and we illustrate the significance of our main theorem by examples. Ultimately, as applications, the existence of a unique solution of Fredholm type integral equations in one and two dimensions is ensured and an example in support is given.
Highlights
Fixed-point technique offers a focal concept with many diverse applications in nonlinear analysis
In 1989, the class of of b-metric spaces has been introduced by Bakhtin [6], that is, the classical triangle inequality is relaxed in the right-hand term by a parameter s ≥ 1
We present a new generalization of the concept of metric spaces, namely, a (φ, ψ)−metric space
Summary
Fixed-point technique offers a focal concept with many diverse applications in nonlinear analysis. Let (=, d) be a (φ, ψ)-metric space and M be a subset of =. Let C be the closure of C with respect to the topology τ(φ,ψ) , that is, C is the intersection of all (φ, ψ)-closed subsets of = containing C. Note that the set of decreasing sequences of nonempty (φ, ψ)-closed subsets of C has a nonempty intersection. This implies that { Z n }n∈N is a decreasing sequence of nonempty (φ, ψ)-closed subsets of Z. Let (=, d) be a complete (φ, ψ)-metric space and T : = → = be a self-mapping. Assume that: Suppose that for each sequence {σn } ⊂ =, we have lim d (σn , σ ) = 0 ⇒ d (σ, ς) ≤ lim sup d (σn , ς) , ς ∈ =;.
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