Let (X,d) be a complete metric space, T: X -» X, and a: [0, oo) -* [0, oo ) be nondecreasing with respect to each variable. Suppose that for the function y(t) = a(t,t,t,2t,2t), the sequence of iterates y" tends to 0 in [0, oo) and \imt^,x(t -y(t)) = oo. Furthermore, suppose that for each x G X there exists a positive integer n = n(x) such that for all y € X, d(T"x,Tny) < a(d(x,Tnx),d(x,Tny),d(x,y),d(T,'x,y),d(Tny,y)). Under these assumptions our main result states that T has a unique fixed

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... . This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.