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Jachymski [ Proc. Amer. Math. Soc., 136 (2008), 1359-1373] gave modified version of a Banach fixed point theorem on a metric space endowed with a graph. In the present paper, (G, Φ)-graphic contractions have been de ned by using a comparison function and studied the existence of fixed points. Also, Hardy-Rogers G-contraction have been introduced and some fixed point theorems have been proved. Some examples are presented to support the results proved herein. Our results generalized and extend various comparable results in the existing literature.                                            Also,     Also, Hardy- Rogers G-contractions have been introduced and some xed point theorems have been proved.

کلیدواژه ها:

Connected graph ، Fixed point ، Φ-contraction ، Hardy-Rogers contraction