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Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application

Journal of Inequalities and Applications volume 2007, Article number: 056161 (2007) Cite this article

Abstract

By using the iterative technique and Nadler's theorem, we construct a new iterative algorithm for solving a system of nonlinear inclusions in Banach spaces. We prove some new existence results of solutions for the system of nonlinear inclusions and discuss the convergence of the sequences generated by the algorithm. As an application, we show the existence of solution for a system of functional equations arising in dynamic programming of multistage decision processes.

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References

  1. Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Analysis. Theory, Methods & Applications 1987,11(5):623–632. 10.1016/0362-546X(87)90077-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.

    MATH  Google Scholar 

  3. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics. Volume 27. Pitman, Boston, Mass, USA; 1985:x+236.

    Google Scholar 

  4. Zhang Z: New fixed point theorems of mixed monotone operators and applications. Journal of Mathematical Analysis and Applications 1996,204(1):307–319. 10.1006/jmaa.1996.0439

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang S-S, Guo WP: On the existence and uniqueness theorems of solutions for the systems of mixed monotone operator equations with applications. Applied Mathematics. A Journal of Chinese Universities. Series B 1993,8(1):1–14. 10.1007/BF02661986

    Article  MathSciNet  MATH  Google Scholar 

  6. Huang N-J, Tang Y-Y, Liu Y-P: Some new existence theorems for nonlinear inclusion with an application. Nonlinear Functional Analysis and Applications 2001,6(3):341–350.

    MathSciNet  MATH  Google Scholar 

  7. Huang N-J, Fang Y-P: Fixed points for multi-valued mixed increasing operators in ordered Banach spaces with applications to integral inclusions. Journal for Analysis and Its Applications 2003,22(2):399–410.

    MathSciNet  MATH  Google Scholar 

  8. Chadli O, Konnov IV, Yao J-C: Descent methods for equilibrium problems in a Banach space. Computers & Mathematics with Applications 2004,48(3–4):609–616. 10.1016/j.camwa.2003.05.011

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding XP, Yao J-C: Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces. Computers & Mathematics with Applications 2005,49(5–6):857–869. 10.1016/j.camwa.2004.05.013

    Article  MathSciNet  MATH  Google Scholar 

  10. Fang Y-P, Huang N-J: Iterative algorithm for a system of variational inclusions involving-accretive operators in Banach spaces. Acta Mathematica Hungarica 2005,108(3):183–195. 10.1007/s10474-005-0219-6

    Article  MathSciNet  MATH  Google Scholar 

  11. Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.

    Article  MathSciNet  MATH  Google Scholar 

  12. Rudin W: Functional Analysis, International Series in Pure and Applied Mathematics. 2nd edition. McGraw-Hill, New York, NY, USA; 1991:xviii+424.

    MATH  Google Scholar 

  13. Baskaran R, Subrahmanyam PV: A note on the solution of a class of functional equations. Applicable Analysis 1986,22(3–4):235–241. 10.1080/00036818608839621

    Article  MathSciNet  MATH  Google Scholar 

  14. Bellman R, Lee ES: Functional equations in dynamic programming. Aequationes Mathematicae 1978,17(1):1–18. 10.1007/BF01818535

    Article  MathSciNet  MATH  Google Scholar 

  15. Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 1984,98(2):348–362. 10.1016/0022-247X(84)90254-3

    Article  MathSciNet  MATH  Google Scholar 

  16. Chang S-S, Ma YH: Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 1991,160(2):468–479. 10.1016/0022-247X(91)90319-U

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang N-J, Lee BS, Kang MK: Fixed point theorems for compatible mappings with applications to the solutions of functional equations arising in dynamic programmings. International Journal of Mathematics and Mathematical Sciences 1997,20(4):673–680. 10.1155/S0161171297000926

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China

    Ke-Qing Wu & Nan-Jing Huang

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Jen-Chih Yao

Authors
  1. Ke-Qing Wu
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  2. Nan-Jing Huang
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  3. Jen-Chih Yao
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Correspondence to Ke-Qing Wu.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Wu, KQ., Huang, NJ. & Yao, JC. Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application. J Inequal Appl 2007, 056161 (2007). https://doi.org/10.1155/2007/56161

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