The contraction principle for set valued mappings on a metric space with a graph

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].…”

A fixed point theorem for G-monotone multivalued mapping with application to nonlinear integral equations

“…and observe that each fixed point of f is a solution of integral equation (3). Of course, f is well-defined since and p are two closed bounded continuous functions.…”

“…Jachymski in [9], merged above theories to have a generalization of the Banach contraction principle for mappings of a metric space endowed with a graph. Then, Beg et al [3] extended some results of Jachymski to multivalued mappings; other generalizations of [9] are available in [1,4,5,10,15,17,18]. For completeness, we recall that Nadler [14] first extended the Banach contraction principle to multivalued mappings; then, Nadler's fixed point theorem has been generalized and extended in several directions, see for example [2, 6, 8, 11-13, 16, 19].…”

“…Let G = (V, E), where V = {1, 2, 3, 4, 6, 8} and E = {(1, 1), (1,3), (2,2), (2,4), (2,6), (2,8), (4,2), (4,4), (4,6), (4,8), (6,8)…”

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

“…Fixed point theorems for single valued and multivalued operators in such metric spaces have been studied by some authors since 2007 (see [5]- [10] and so on).…”