Coincidence point theorems for weak graph preserving multi-valued mapping

Abstract

In this paper, we prove some coincidence and fixed point theorems for a new type of multi-valued weak G-contraction mapping with compact values. The results of this paper extend and generalize several known results from a complete metric space endowed with a graph. Some examples are given to illustrate the usability of our results. MSC: 47H04; 47H10

Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph

“…The first work in this direction was initiated by Jachymski [8] in which the author introduced the concept of a graph preserving mapping and G-contraction for a single valued mapping defined on a metric space endowed with a graph. Later Phonon et al [9] generalised the concept of a graph preserving mapping by introducing weak graph preserving mapping and proved fixed point theorems for multivalued mappings in a metric space endowed with a graph. More results in this direction were considered wherein Bojor [10] considered Reich type contraction, Mohanta and Patra [11] discussed common fixed point of a hybrid pair of mappings in a b-metric space endowed with a graph, Cholamjiak et al [12] discussed viscosity approximation method for fixed point problems in a Hilbert space endowed with graph, Sauntai et al [13] proved the existence of coupled fixed point for θ-ψ contraction mapping whereas Sultan and Vetrivel [14] considered Mizoguci-Takahashi contractions in a metric space endowed with a graph.…”

“…Corollary 1 and hence Theorems 1, 2 and 4 are proper extension and generalization of the results of[4][5][6][7]. Corollary 2 and hence Theorem 2 is a proper extension and generalization of Theorem 3.1 of[9] and Theorem 1.13 of[15] and some of the references therein. For s = 1 Theorem 3 reduces to Theorem 2.2 of [3] wherein we do not require the metric space (X G , d G ) to be complete and also we do not require the condition∆ = {(x G , x G ) : x G ∈ X G } ⊂ E(G).Hence Theorem 3 is a substantial improvement and generalisation of Theorem 2.2 of [3].…”

(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

“…He defined the Banach Gcontraction for single-valued mapping, which was later extended by Beg et al [4] for the multivalued mappings. After these, there was a lot of work done in the direction of fixed points in metric spaces endowed with graphs, see [5][6][7][8][9][10][11][12][13][14].…”