Two fixed point existence theorems for a more general contractive type mapping than the one given by B. E. Rhoades [Trans. Am. Math. Soc. 226, 257-290 (1977; Zbl 0365.54023)] have been derived. A comparison of the results obtained in this paper with the earlier known result due to B. Fisher [Proc. Am. Math. Soc. 75, 321-325 (1979; Zbl 0411.54049)] and due to the author [J. Sichuan Univ., Nat. Sci. Ed. No.1, 45-67 (1981; Zbl 0473.54038)] has also been given.