No abstract

No abstract

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

In this paper, we apply the idea of k-local contraction of Rincón-Zapatero and Rodrigues-Palmero (2003, 2007 to study discounted stochastic dynamic programming models with unbounded returns. Our main results concern the existence of a unique solution to the Bellman equation and are applied to the theory of stochastic optimal growth. Also a discussion of some subtle issues concerning k-local and global contractions is included.