Abstract
We introduce and study two new functional equations, which contain a lot of known functional equations as special cases, arising in dynamic programming of multistage decision processes. By applying a new fixed point theorem, we obtain the existence, uniqueness, iterative approximation, and error estimate of solutions for these functional equations. Under certain conditions, we also study properties of solutions for one of the functional equations. The results presented in this paper extend, improve, and unify the results according to Bellman, Bellman and Roosta, Bhakta and Choudhury, Bhakta and Mitra, Liu, Liu and Ume, and others. Two examples are given to demonstrate the advantage of our results over existing results in the literature.
Highlights
Introduction and PreliminariesThe existence, uniqueness, and successive approximations of solutions for the following functional equations arising in dynamic programming:f x max p x, y q x, y f a x, y, ∀x ∈ S, y∈D f x max p x, y f a x, y, ∀x ∈ S, y∈D f x min max p x, y, f a x, y, ∀x ∈ S, y∈DFixed Point Theory and Applications f x min max p x, y, q x, y f a x, y, ∀x ∈ S, y∈D f x sup p x, y y∈D m qi x, y f ai x, y i1, ∀x ∈ S, 1.1 were first introduced and discussed by Bellman 1, 2
We introduce and study two new functional equations, which contain a lot of known functional equations as special cases, arising in dynamic programming of multistage decision processes
The existence, uniqueness, and successive approximations of solutions for the following functional equations arising in dynamic programming: f x max p x, y q x, y f a x, y, ∀x ∈ S, y∈D
Summary
The existence, uniqueness, and successive approximations of solutions for the following functional equations arising in dynamic programming:. Afterwards, further analyses on the properties of solutions for the functional equations 1.1 and 1.2 and others have been studied by several authors in [3,4,5,6,7] and [8,9,10,11] by using various fixed point theorems and monotone iterative technique, where 1.2 are as follows:. By applying the new fixed point theorem, we establish the existence, uniqueness, iterative approximation, and error estimate of solutions for the functional equation 1.3 and 1.4. Φ4 φ, ψ : φ, ψ ∈ Φ3, ψ t > 0, ψ φn t < ∞ for t > 0
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