The functional equations of undiscounted denumerable state Markov renewal programming

Abstract

In this paper we want to establish conditions for the existence of a finite solution of the functional equations

$$ g\left( i \right) = \mathop {\max }\limits_{a \in a\left( 1 \right)} \mathop {\;\Sigma }\limits_{j \in I} \;p\left( {i,a,j} \right)g\left( j \right),\;i \in I,\;$$

(1)

$$ v\left( i \right) = \mathop {\max }\limits_{a \in {A_o}\left( i \right)} \;\left[ {r\left( {i,a} \right) - t\left( {i,a} \right)g\left( i \right) + \mathop \Sigma \limits_{j \in I} p\left( {i,a,j} \right)v\left( j \right)} \right],i \in I$$

(2)

where I is a denumerable set, A(i) is a compact metric space for all i ϵ I, 0 < t(i,a) < ∞, − ∞ < r (i,a) < ∞, p (i,a,j) ⩾ 0 for all i, j ϵ I, a ϵ A(i) and $\sum\limits_{j \in I} {p\left( {i,a,j} \right){\text{ }} = {\text{ }}1}$ for all i ϵ I, a ϵ A(i).