Positivity 7: 335–346, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
On the Existence Problem of Solutions for a Class
of Functional Equations Arising in Multistage
Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Received 13 July 2000; accepted 1 July 2002
Abstract. The purpose of this paper is to study the existence problems of solutions for some
classes of functional equations and systems of functional equations arising in multistage allocation
1991 AMS Mathematics Subject Classiﬁcation: 90C39; 90D20.
Key words: Multistage allocation process, dynamic programming, functional equations
1. Introduction and Preliminaries
The method of dynamic programming is one of the most important optimiza-
tion methods. In recent years dynamic programming has its wide applications
in multistage allocation processes and multistage decision processes appeared in
various branches of operation research.
Recently, some optimization problems and existence problem of solution for
some kinds of functional equations arising in decision processes have been con-
sidered by Baskaran , Bellman et al [2, 3], Bhakta et al , Chang [5, 6] and
Wang [7–10]. The purpose of this paper is in the setting of Banach space to study
the existence problems of solutions for some kinds of functional equations and
systems of functional equations arising in multistage allocation processes.
Let X, Y be two real Banach spaces, R = (−∞, +∞), R
=[0, ∞), S ⊂
X × X and D ⊂ Y × Y be the state space and decision space, respectively, S
be the projections of S and D on X and Y , respectively.
In a class of multistage games, at each stage of the game, the resources of
players A and B are represented by x and y ∈ S
and the pair of (x, y) ∈ S
is called the state vector. The decisions of A and B are represented by u and
v ∈ D
, respectively. And the pair of (u, v) ∈ D is called the decision vector. As