New results on coupled fixed point theory in metric spaces endowed with a directed graph

Abstract

The purpose of this paper is to present some existence results for coupled fixed points of contraction type operators in metric spaces endowed with a directed graph. Our results generalize the results obtained by Gnana Bhaskar and Lakshmikantham in (Nonlinear Anal. 65:1379-1393, 2006. As an application, the existence of a continuous solution for a system of Fredholm and Volterra integral equations is obtained. MSC: 47H10; 54H25Keywords: fixed point; coupled fixed point; metric space; connected graph PreliminariesIn fixed point theory, the importance of study of coupled fixed points is due to their applications to a wide variety of problems. Bhaskar and Lakshmikantham [] gave some existence results for coupled fixed point for a mixed monotone type mapping in a metric space endowed with partial order, using a weak contractivity type of assumption.The purpose of this paper is to generalize these results using the context of metric spaces endowed with a graph. This new research direction in the theory of fixed points was initiated by Jachymski Let (X, d) be a metric space and be the diagonal of X × X. Let G be a directed graph, such that the set V (G) of its vertices coincides with X and ⊆

Coincidence point theorems for $\varphi$-$\psi$-contraction mappings in metric spaces involving a graph

“…Our results extend and improve the results obtained by Eshi et al in [12], Işık and Türkoglu in [11], Chifu and Petrusel in [9] so on. Moreover, we have an application to some integral system to support the results.…”

“…In the case where (X, ) is partially ordered complete metric space, taking E (G) = {(x, y) ∈ X × X : x y}, the functions ϕ (t) = t and ψ (t) = kt, for t ∈ [0, ∞) and k ∈ [0, 1), Theorem 2 generalize and improve the results obtained by Bhaskar and Lakshmikantham ( [1], Theorem 2.1) and Chifu and Petrusel ( [9], Theorem 2.1). If we take the function ψ (t) = ϕ (t) − ψ 1 (t), for t ∈ [0, ∞), where ψ 1 ∈ Ψ, Theorem 2 generalize the results given by Luong and Thuan ( [3], Theorem 2.1).…”

Coupled fixed point results for (varphi,G)-contractions of type (b) in b-metric spaces endowed with a graph

“…Our results also generalize and extend some fixed point and coupled fixed point theorems in partially ordered complete metric spaces and b-metric spaces given by Harjani and Sadarangani [10], Nieto and Rodríguez-López [14,16], Nieto et al [15], Jleli et al [13], O'Regan and Petruşel [17], Ran and Reurings [18], Gnana Bhaskar and Lakshmikantham [8], and Chifu and Petrusel in [6].…”

Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application

“…Other results for single valued and multivalued operators in such metric spaces were given by Beg et al [20], Bajor [21], Alfuraid [22,23], Chifu and Petrusel [24] and Suantai et al [25].…”

“…In 2014, Chifu and Petrusel [24] introduced the notion of G continuity for a mapping F W X 2 ! X and the property A as follows.…”