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We have been inspired by results of the papers "Important tools and possible applications in metric fixed point theory, Nonlinear Analysis, 47(2001), 3479-3490" by P. P. Murthy, "A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226(1977), 257-290" and "Contractive definitions revisited, Topological methods in non-linear functional analysis, Contemporary Math. A. M. S. 21(1983), 189-205" both by Billy E. Rhoades, "On nonlinear contractions, Proc. Amer. Math. Soc., 20(1969), 458-468" by D. W. Boyd and J. S. W. Wong and "A note on contractive mappings, Proc....