Abstract
This paper studies the existence, uniqueness and iterative approximations of solutions, bounded solutions and continuous bounded solutions for a functional equation arising in dynamic programming of multistage decision processes by using fixed point theorems and the iterative methods in complete metric spaces and Banach spaces, respectively. Four examples are given to illustrate the validity and applications of the results presented in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellman R.: Dynamic Programming. Princeton University Press, Princeton, New Jersey (1957)
Bellman R.: Methods of Nonlinear Analysis, vol. 2. Academic Press, New York (1973)
Bellman R., Roosta M.: A technique for the reduction of dimensionality in dynamic programming. J. Math. Anal. Appl. 88, 543–546 (1982)
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)
Bhakta P.C., Mitra S.: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348–362 (1984)
Jiang G.J., Kang S.M., Kwun Y.C.: Solvability and algorithms for functional equations originating from dynamic programming. Fixed Point Theory Appl. 2011 Article ID 701519, 30 pages (2011)
Liu Z.: Existence theorems of solutions for certain classes of functional equations arising in dynamic programming. J. Math. Anal. Appl. 262, 529–553 (2001)
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)
Liu Z., Li X., Kang S.M., Cho S.Y.: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Fixed Point Theory Appl. 64, 18 (2011)
Liu Z., Kang S.M.: Existence and uniqueness of solutions for two classes of functional equations arising in dynamic programming. Acta Math. Appl. Sini. 23, 195–208 (2007)
Liu Z., Kang S.M.: Properties of solutions for certain functional equations arising in dynamic programming. J. Global Optim. 34, 273–292 (2006)
Liu Z., Kang S.M., Ume J.S.: Solvability and convergence of iterative algorithms for certain functional equations arising in dynamic programming. Optimization 59, 887–916 (2010)
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)
Liu Z., Ume J.S., Kang S.M.: On properties of solutions for two functional equations arising in dynamic programming. Fixed point Theory Appl. 2010 Article ID 905858, 19 pages (2010)
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)
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, 1517–1536 (2010)
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)
Liu Z., Zhao L.S., Kang S.M., Ume J.S.: On the solvability of a functional equation. Optimization 60, 365–375 (2011)
Pathak H.K.: Applications of fixed point technique in solving certain dynamic programming and variational inequalities. Nonlinear Anal. 63, e309–e319 (2005)
Singh S.L., Mishra S.N. Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl. 2010 Article ID 898109, 14 pages (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, Z., Zhu, J., Kang, S.M. et al. Solvability and iterative approximations for a functional equation. J Glob Optim 57, 969–995 (2013). https://doi.org/10.1007/s10898-012-9986-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9986-0