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Existence and Uniqueness of Solutions for two Classes of Functional Equations Arising in Dynamic Programming

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Abstract

In this paper we establish the existence, uniqueness and iterative approximation of solutions for two classes of functional equations arising in dynamic programming of multistage decision processes. The results presented here extend, and unify the corresponding results due to Bellman, Bhakta and Choudhury, Bhakta and Mitra, Liu and others.

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References

  1. Belbas, S.A. Dynamic programming and maximum principle for discrete goursat systems. J. Math. Anal. Appl., 161: 57–77 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellman, R. Dynamic programming. Princeton University Press, Princeton, New Jersey, 1957

  3. Bellman, R. Methods of nonlinear analysis Vol. 2. Academic Press, New York, 1973

  4. Bellman, B., Lee, E.S. Functional Equations Arising in Dynamic Programming. Aequationes Math., 17: 1–18 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bellman, R., Roosta, M. A technique for the reduction of dimensionality in dynamic programming. J. Math. Anal. Appl., 88: 543–546 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bhakta, P.C., Choudhury, S.R. Some existence theorems for functional equations arising in dynamic programming II. J. Math. Anal. Appl., 131: 217–231 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bhakta, P.C. Mitra, S. Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl., 98: 348–362 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boyd, D.W., Wong, J.S.W. On nonlinear contractions. Proc. Amer. Math. Soc. 20: 458–464 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chang, S.S., Ma, Y.H. Coupled fixed points for mixed monotone condensing operators and an existence theorem of solutions for a class of functional equations arising in dynamic programming. J. Math. Anal. Appl., 160: 468–479 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, N.J., Lee, B.S., Kang, M.K. Fixed point theorems for compatible mappings with applications to the solutions of functional equations arising in dynamic programmings. Int. J. Math. & Math. Sci., 20: 673–680 (1997)

    Article  MATH  Google Scholar 

  11. Liu, Z. Coincidence theorems for expansion mappings with applications to the solutions of functional equations arising in dynamic programming. Acta Sci. Math. (Szeged), 65: 359–369 (1999)

    MATH  Google Scholar 

  12. Liu, Z. Compatible mappings and fixed points. Acta Sci. Math. (Szeged), 65: 371–383 (1999)

    MATH  Google Scholar 

  13. Liu, Z. Existence theorems of solutions for certain classes of functional equations arising in dynamic programming. J. Math. Anal. Appl., 262: 529–553 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, Z., Agarwal, R.P., Kang, S.M. On solvability of functional equations and system of functional equations arising in dynamic programming. J. Math. Anal. Appl., 297: 111–130 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, Z., Kang, S.M. Properties of solutions for certain functional equations arising in dynamic programming. J. Global Optim., 34: 273–292 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, Z., Kim, J.K. A common fixed point theorem with applications in dynamic programming. Nonlinear Funct. Anal. Appl., 11: 11–19 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Liu, Z., Ume, J.S. On properties of solutions for a class of functional equations arising in dynamic programming. J. Optim. Theory Appl., 117: 533–551 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu, Z., Ume, J.S., Kang, S.M. Some existence ttheorems for functional equations arising in dynamic programming. J. Korean Math. Soc., 43: 11–28 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, Z., Xu, Y., Ume, J.S., Kang, S.M. Solutions to two functional equations arising in dynamic programming. J. Comput. Appl. Math., 192: 251–269 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang, S.S. Some existence theorems of common and coincidence solutions for a class of functional equations arising in dynamic programming. Appl. Math. Mech., 12: 31–37 (1991) (in Chinese)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, 116029, China

    Ze-qing Liu

  2. Department of Mathematics, and The Research Institute of Natural Science, Gyeongsang National University, Chinju, 660-701, Korea

    Shin Min Kang

Authors
  1. Ze-qing Liu
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  2. Shin Min Kang
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Correspondence to Shin Min Kang.

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Liu, Zq., Kang, S.M. Existence and Uniqueness of Solutions for two Classes of Functional Equations Arising in Dynamic Programming. Acta Mathematicae Applicatae Sinica, English Series 23, 195–208 (2007). https://doi.org/10.1007/s10255-007-0363-6

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  • DOI: https://doi.org/10.1007/s10255-007-0363-6

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