On the solvability of a functional equation

Abstract

This article studies properties of solutions of a functional equation arising in dynamic programming of multistage decision processes. A sufficient condition for the existence, uniqueness and iterative approximation of solutions of the functional equation is established. A few other behaviours of solutions for certain functional equations which are particular cases of the functional equation are discussed. The results presented in this article extend, improve and unify some known results in literature.

Keywords:

• functional equation
• non-expansive mapping
• iterative approximation
• non-negative solution
• dynamic programming

AMS Subject Classifications::

• 49L20
• 49L99
• 90C39

Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions. This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2009A419) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-313-C00042).