Taiwanese Journal of Mathematics

SOME EXISTENCE THEOREMS FOR FUNCTIONAL EQUATIONS AND SYSTEM OF FUNCTIONAL EQUATIONS ARISING IN DYNAMIC PROGRAMMING

Zeqing Liu, Jeong Sheok Ume, and Shin Min Kang

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Abstract

In this paper we study the existence, uniqueness and iterative approximation of solutions for a few classes of functional equations arising in dynamic programming of multistage decision processes. By using of monotone iterative technique, we also establish the existence and iterative approximation of coincidence solutions for certain kinds of system of functional equations. The results presented in this paper extend, improve and unify the results due to Bhakta and Mitra [7], Chang [10] and Liu [12] and others.

Article information

Source
Taiwanese J. Math., Volume 14, Number 4 (2010), 1517-1536.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405965

Digital Object Identifier
doi:10.11650/twjm/1500405965

Mathematical Reviews number (MathSciNet)
MR2663929

Zentralblatt MATH identifier
1215.49008

Subjects
Primary: 49L20: Dynamic programming method 49L99: None of the above, but in this section 90C39: Dynamic programming [See also 49L20]

Keywords
dynamic programming functional equation system of functional equations solution coincidence solutions fixed point nonexpansive mapping

Citation

Liu, Zeqing; Ume, Jeong Sheok; Kang, Shin Min. SOME EXISTENCE THEOREMS FOR FUNCTIONAL EQUATIONS AND SYSTEM OF FUNCTIONAL EQUATIONS ARISING IN DYNAMIC PROGRAMMING. Taiwanese J. Math. 14 (2010), no. 4, 1517--1536. doi:10.11650/twjm/1500405965. https://projecteuclid.org/euclid.twjm/1500405965


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References

  • [1.] R. Baskaram and P. V. Subrahmanyam, A Note on the Solution of a Class of Functional Equations, Appl. Anal., 22 (1986), 235-241.
    Zentralblatt MATH: 0604.39006
    Digital Object Identifier: doi:10.1080/00036818608839621
  • [2.] S. A. Belbas, Dynamic Programming and Maximum Principle for Discrete Goursat Systems, J. Math. Anal. Appl, 161 (1991), 57-77.
    Zentralblatt MATH: 0749.49019
    Digital Object Identifier: doi:10.1016/0022-247X(91)90362-4
  • [3.] R. Bellman and E. S. Lee, Functional Equations Arising in Dynamic Programming, Aequationes Math., 17 (1978), 1-18.
    Zentralblatt MATH: 0397.39016
    Digital Object Identifier: doi:10.1007/BF01818535
  • [4.] R. Bellman and M. Roosta, A Technique for the Reduction of Dimensionality in Dynamic Programming, J. Math. Anal. Appl., 88 (1982), 543-546.
    Mathematical Reviews (MathSciNet): MR667077
    Zentralblatt MATH: 0498.90079
    Digital Object Identifier: doi:10.1016/0022-247X(82)90212-8
  • [5.] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.
  • [6.] R. Bellman, Methods of Nonlinear Analysis, II, Academic Press, New York, NY, 1973.
    Zentralblatt MATH: 0265.34002
  • [7.] P. C. Bhakta and S. Mitra, Some Existence Theorems for Functional Equations Arising in Dynamic Programming, J. Math. Anal. Appl., 98 (1984), 348-362.
    Zentralblatt MATH: 0533.90091
    Digital Object Identifier: doi:10.1016/0022-247X(84)90254-3
  • [8.] P. C. Bhakta and S. R. Choudhury, Some Existence Theorems for Functional Equations Arising in Dynamic Programming II, J. Math. Anal. Appl., 131 (1988), 217-231.
    Mathematical Reviews (MathSciNet): MR934442
    Zentralblatt MATH: 0662.90085
    Digital Object Identifier: doi:10.1016/0022-247X(88)90201-6
  • [9.] D. W. Boyd and J. S. W. Wong, On Nonlinear Contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464.
    Zentralblatt MATH: 0175.44903
    Digital Object Identifier: doi:10.1090/S0002-9939-1969-0239559-9
  • [10.] S. S. Chang, Some Existence Theorems of Common and Coincidence Solutions for a Class of Functional Equations Arising in Dynamic Programming, Appl. Math. Mechanics, 12 (1991), 31-37.
  • [11.] S. S. Chang and Y. H. Ma, 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, J. Math. Anal. Appl., 106 (1991), 468-479.
    Zentralblatt MATH: 0753.47029
    Digital Object Identifier: doi:10.1016/0022-247X(91)90319-U
  • [12.] Z. Liu, Existence Theorems of Solutions for Certain Classes of Functional Equations Arising in Dynamic Programming, J. Math. Anal. Appl., 262 (2001), 529-553.
    Zentralblatt MATH: 1018.90064
    Digital Object Identifier: doi:10.1006/jmaa.2001.7551
  • [13.] Z. Liu, Coincidence Theorems for Expansive Mappings with Applications to the Solutions of Functional Equations Arising in Dynamic Programming, Acta Scientiarum Mathematicarum $($Szeged$)$, 65 (1999), 359-369.
    Mathematical Reviews (MathSciNet): MR1702183
    Zentralblatt MATH: 0929.54036
  • [14.] Z. Liu, Compatible Mappings and Fixed Points, Acta Scientiarum Mathematicarum $($Szeged$)$, 65 (1999), 371-383.
    Mathematical Reviews (MathSciNet): MR1702179
    Zentralblatt MATH: 0928.54040
  • [15.] Z. Liu and J. S. Ume, On Properties of Solutions for a class of Functional Equations Arising in Dynamic Programming, J. Opti. Theory Appl., 117 (2003), 533-551.
    Zentralblatt MATH: 1045.90075
    Digital Object Identifier: doi:10.1023/A:1023945621360

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