Existence and Iterative Approximations of Solutions for Certain Functional Equation and Inequality

Zeqing Liu (), Haijiang Dong (), Sun Young Cho () and Shin Min Kang ()
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Zeqing Liu: Liaoning Normal University
Haijiang Dong: Liaoning Normal University
Sun Young Cho: Gyeongsang National University
Shin Min Kang: Gyeongsang National University

Journal of Optimization Theory and Applications, 2013, vol. 157, issue 3, No 8, 716-736

Abstract: Abstract This paper deals with a functional equation and inequality arising in dynamic programming of multistage decision processes. Using several fixed-point theorems due to Krasnoselskii, Boyd–Wong and Liu, we prove the existence and/or uniqueness and iterative approximations of solutions, bounded solutions and bounded continuous solutions for the functional equation in two Banach spaces and a complete metric space, respectively. Utilizing the monotone iterative method, we establish the existence and iterative approximations of solutions and nonpositive solutions for the functional inequality in a complete metric space. Six examples which dwell upon the importance of our results are also included.

Keywords: Functional equation; Functional inequality; Dynamic programming; Solution; Bounded solution; Bounded continuous solution; Nonpositive solution; Krasnoselskii’s fixed-point theorem; Boyd–Wong’s fixed-point theorem; Liu’s fixed-point theorem; Monotone iterative method (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10957-012-0185-4

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