Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations

Eskandar Ameer,Muhammad Arshad,Hassen Aydi,Manuel De La Sen

doi:10.3390/sym12030467

Abstract

In this paper, we initiate the notion of Ćirić type rational graphic Υ , Λ -contraction pair mappings and provide some new related common fixed point results on partial b-metric spaces endowed with a directed graph G. We also give examples to illustrate our main results. Moreover, we present some applications on electric circuit equations and fractional differential equations.

Highlights

Introduction and PreliminariesThe Banach principle [1] has been improved and generalized by several researchers for different kinds of contractions in various spaces
Let ( M, Pb ) be a partial b-metric space endowed with a directed graph G, s > 1 and φ, ψ be self-mappings of M
If φ = ψ, we say that φ is a Ćirić type rational graphic (Υ, Λ)-contraction

Summary

Introduction and Preliminaries

The Banach principle [1] has been improved and generalized by several researchers for different kinds of contractions in various spaces. A mapping T : M → M is said to be a θ-contraction, if there exist θ ∈ Θ and a real constant k ∈ (0, 1) such that ζ, η ∈ M, d( T (ζ ) , T (η )) 6= 0 =⇒ θ (d( T (ζ ) , T (η ))) ≤ [θ (d(ζ, η )]k , where Θ is the set of functions θ : (0, ∞) −→ (1, ∞) such that:. A mapping T : M → M is said to be a (Υ, Λ)-Suzuki contraction, if there exist comparison functions Υ and Λ ∈ Φ such that, for all ζ, η ∈ M with T (ζ ) 6= T (η ), d (ζ, T (ζ )) < d (ζ, η ) =⇒ Λ (d ( T (ζ ) , T (η ))) ≤ Υ [Λ (U (ζ, η ))] , where d (ζ, T (η )) + d (η, T (ζ )). Since ( M, Pb ) is a partial b-metric space, the weight assigned to each vertex ζ need not to be zero, and whenever a zero weight is assigned to some edge (ζ, η ), it reduces to a loop (ζ, ζ )

Main Results

Some Consequences

Application

Application to Fractional Differential Equations

Conclusions