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In this paper, we introduce the (G,ψ)-contraction and the (G,ψ)-graphic contraction in a metric space by using a graph. We explain some conditions for a mapping which is a (G,ψ)-contraction to have a unique fixed point and also we give conditions as regards the existence of a fixed point for (G,ψ)-graphic contraction by applying the connectivity of the graph in both cases. Moreover, we give examples to show that our results are a substantial improvement of some known results in the literature.

On some fixed-point theorems for ψ-contraction on metric space involving a graph