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

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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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Authors and Affiliations

  1. Department of Mathematics, Liaoning Normal University, Dalian, Liaoning, 116029, People’s Republic of China

    Zeqing Liu & Haijiang Dong

  2. Department of Mathematics, Gyeongsang National University, Jinju, 660-701, Korea

    Sun Young Cho

  3. Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea

    Shin Min Kang

Authors
  1. Zeqing Liu
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  2. Haijiang Dong
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  3. Sun Young Cho
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  4. Shin Min Kang
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Corresponding author

Correspondence to Shin Min Kang.

Additional information

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

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