Advertisement

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

  • Zeqing Liu
  • Haijiang Dong
  • Sun Young Cho
  • Shin Min KangEmail author
Article

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 

References

  1. 1.
    Bellman, R.: Some functional equations in the theory of dynamic programming, I. Functions of points and point transformations. Trans. Am. Math. Soc. 80, 55–71 (1955) CrossRefGoogle Scholar
  2. 2.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957) zbMATHGoogle Scholar
  3. 3.
    Bellman, R., Lee, E.S.: Functional equations arising in dynamic programming. Aequ. Math. 17, 1–18 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bellman, R., Roosta, M.: A technique for the reduction of dimensionality in dynamic programming. J. Math. Anal. Appl. 88, 543–546 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bhakta, P.C., Mitra, S.: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348–362 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Liu, Z.: Coincidence theorems for expansion mappings with applications to the solutions of functional equations arising in dynamic programming. Acta Sci. Math. 65, 359–369 (1999) zbMATHGoogle Scholar
  8. 8.
    Liu, Z.: Compatible mappings and fixed points. Acta Sci. Math. 65, 371–383 (1999) zbMATHGoogle Scholar
  9. 9.
    Liu, Z.: Existence theorems of solutions for certain classes of functional equations arising in dynamic programming. J. Math. Anal. Appl. 262, 529–553 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Liu, Z., Kang, S.M.: Properties of solutions for certain functional equations arising in dynamic programming. J. Glob. Optim. 34, 273–292 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Liu, Z., Kang, S.M.: Existence and uniqueness of solutions for two classes of functional equations arising in dynamic programming. Acta Math. Appl. Sin. 23, 195–208 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Liu, Z., Kang, S.M., Ume, J.S.: Solvability and convergence of iterative algorithms for certain functional equations arising in dynamic programming. Optimization 59(6), 887–916 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Liu, Z., Ume, J.S., Kang, S.M.: Some existence theorems for functional equations arising in dynamic programming. J. Korean Math. Soc. 43, 11–28 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Liu, Z., Ume, J.S., Kang, S.M.: Some existence theorems for functional equations and system of functional equations arising in dynamic programming. Taiwan. J. Math. 14(4), 1517–1536 (2010) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liu, Z., Xu, Y.G., Ume, J.S., Kang, S.M.: Solutions to two functional equations arising in dynamic programming. J. Comput. Appl. Math. 192, 251–269 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Liu, Z., Zhao, L.S., Kang, S.M., Ume, J.S.: On the solvability of a functional equation. Optimization 60(3), 365–375 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Erbe, L.H., Kong, Q.K., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York (1995) Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Zeqing Liu
    • 1
  • Haijiang Dong
    • 1
  • Sun Young Cho
    • 2
  • Shin Min Kang
    • 3
    Email author
  1. 1.Department of MathematicsLiaoning Normal UniversityDalianPeople’s Republic of China
  2. 2.Department of MathematicsGyeongsang National UniversityJinjuKorea
  3. 3.Department of Mathematics and RINSGyeongsang National UniversityJinjuKorea

Personalised recommendations