Remarks on monotone multivalued mappings on a metric space with a graph

Let (X,d)$(X,d)$ be a metric space and J:X→^2X$J: X\rightarrow2^{X}$ be a multivalued mapping. In this work, we discuss the definition of G-contraction mappings introduced by Beg et al. (Comp. Math. Appl. 60:1214-1219, 2010) and show that it is restrictive and fails to give the main result of (Beg et al. in Comp. Math. Appl. 60:1214-1219, 2010). In this work, we give a new definition of the G-contraction and obtain sufficient conditions for the existence of fixed points for such mappings.