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.
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This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380). The authors are grateful to the editor and the referees for their valuable comments and suggestions.
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Liu, Z., Dong, H., Cho, S.Y. et al. Existence and Iterative Approximations of Solutions for Certain Functional Equation and Inequality. J Optim Theory Appl 157, 716–736 (2013). https://doi.org/10.1007/s10957-012-0185-4
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DOI: https://doi.org/10.1007/s10957-012-0185-4