Dynamic Programming
Solutions to two functional equations arising in dynamic programming
Journal of Computational and Applied Mathematics
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In this paper, we introduce and study properties of solutions for the following functional equation arising in dynamic programming of multistage decision processes$$\eqalign{f(x) =\mathop{\hbox{opt}}\limits_{y \in D}\{u(x,y)\max\{p(x,y),f(a(x,y))\} +v(x,y)\min\{q(x,y),f(b(x,y))\}\cr +w(x,y)[r(x,y)+f(c(x,y))]\}, \quad \forall x\in{S}.}$$ A sufficient condition which ensures the existence, uniqueness and iterative approximation of solution for the functional equation is provided. A few other behaviors of solutions for certain functional equations which are particular cases of the functional equation are discussed. The results presented in this paper extend, improve and unify the results due to Bellman, Bhakta and Choudhury, Bhakta and Mitra, Liu, and Liu and Ume. Several examples which dwell upon the importance of our results are also included.