Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 905858, 19 pages
Research Article

On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Republic of Korea

Received 12 July 2010; Accepted 26 October 2010

Academic Editor: Manuel De la Sen

Copyright © 2010 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We introduce and study two new functional equations, which contain a lot of known functional equations as special cases, arising in dynamic programming of multistage decision processes. By applying a new fixed point theorem, we obtain the existence, uniqueness, iterative approximation, and error estimate of solutions for these functional equations. Under certain conditions, we also study properties of solutions for one of the functional equations. The results presented in this paper extend, improve, and unify the results according to Bellman, Bellman and Roosta, Bhakta and Choudhury, Bhakta and Mitra, Liu, Liu and Ume, and others. Two examples are given to demonstrate the advantage of our results over existing results in the literature.