Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 905858 19 pages doi 2010 905858 Research Article On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming Zeqing Liu 1 Jeong Sheok Ume 2 and Shin Min Kang3 1 Department of Mathematics Liaoning Normal University Dalian Liaoning 116029 China 2 Department of Applied Mathematics Changwon National University Changwon 641-773 Republic of Korea 3 Department of Mathematics and Research Institute of Natural Science Gyeongsang National University Chinju 660-701 Republic of Korea Correspondence should be addressed to Jeong Sheok Ume jsume@ 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. 1. Introduction and Preliminaries The existence uniqueness and successive approximations of solutions for the following functional equations arising in dynamic programming f x maxWx y q x y f a x y l Vx e S y -ye Jff - 1 M 2 y V Z J f x max n x y f fl x y l Vx e S y
