Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 701519, 30 pages
doi:10.1155/2011/701519
Research Article

Solvability and Algorithms for Functional Equations Originating from Dynamic Programming

Guojing Jiang,1 Shin Min Kang,2 and Young Chel Kwun3

1Organization Department, Dalian Vocational Technical College, Dalian, Liaoning 116035, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 5 January 2011; Accepted 11 February 2011

Academic Editor: Yeol J. Cho

Copyright © 2011 Guojing Jiang 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.

Abstract

The main purpose of this paper is to study the functional equation arising in dynamic programming of multistage decision processes 𝑓 ( 𝑥 ) = o p t 𝑦 ∈ 𝐷 o p t { 𝑝 ( 𝑥 , 𝑦 ) , 𝑞 ( 𝑥 , 𝑦 ) 𝑓 ( 𝑎 ( 𝑥 , 𝑦 ) ) , 𝑟 ( 𝑥 , 𝑦 ) 𝑓 ( 𝑏 ( 𝑥 , 𝑦 ) ) , 𝑠 ( 𝑥 , 𝑦 ) 𝑓 ( 𝑐 ( 𝑥 , 𝑦 ) ) } f o r a l l 𝑥 ∈ 𝑆 . A few iterative algorithms for solving the functional equation are suggested. The existence, uniqueness and iterative approximations of solutions for the functional equation are discussed in the Banach spaces 𝐵 𝐶 ( 𝑆 ) and 𝐵 ( 𝑆 ) and the complete metric space 𝐵 𝐵 ( 𝑆 ) , respectively. The properties of solutions, nonnegative solutions, and nonpositive solutions and the convergence of iterative algorithms for the functional equation and other functional equations, which are special cases of the above functional equation, are investigated in the complete metric space 𝐵 𝐵 ( 𝑆 ) , respectively. Eight nontrivial examples which dwell upon the importance of the results in this paper are also given.