Some existence theorems for functional equations arising in dynamic programming, II

On Solvability for Certain Functional Equations Arising in Dynamic Programming

“…We construct some nontrivial examples to explain our results. The results presented here unify the results of Bellman [1], Bhakta and Mitra [3], Bhakta and Choudhury [4], Liu and Ume [10], Liu et al [11], Liu et al [13], Liu and Kang [9], Jiang et al [7], and Liu [8].…”

“…It follows from Theorem 2.2 in [4] that H has a unique fixed point w ∈ BB(S) and {H n h} n≥1 converges to w for each h ∈ BB(S). Obviously, w is also a unique solution of the functional equation (10).…”

Existence and uniqueness of solutions for certain functional equations and system of functional equations arising in dynamic programming

“…To show how our results can be used in practice we also illustrate some nontrivial examples. The results presented here generalize and unify the results of Bellman [1], Bhakta and Mitra [3], Bhakta and Choudhury [4], Liu and Ume [11], Liu et al [12], Liu et al [16], Liu and Kang [10], Jiang et al [8], Liu [9], Liu et al [14], Liu et al [13], Pathak and Deepmala [17] Chang [6], Chang and Ma [7], and Liu et al [15].…”

“…The example below demonstrates that Theorem 5.3 generalizes and unifies the results in [1,3,4,8,10,11,12,13,15,16,17].…”

“…(14) f (x)= opty∈D opt{u(x, y), pi(x, y) + Ai(x, y, f (ai(x, y))) : i = 1, 2}, and f (x)= opty∈D opt{u(x, y), pi(x, y) + qi(x, y)f (ai(x, y)) : i = 1, 2, 3}, ∀x ∈ S. (15) Unauthenticated Download Date | 5/12/18 11:37 AM These equations have been studied by Bellman [2], Bhakta and Mitra [3], Bhakta and Chuodhary [4], Liu and Ume [11], Liu et al [12,16], Liu and Kang [10], Jiang et al [8], Liu [9], Liu et al [14], Liu et al [13], and Pathak and Deepmala [17], respectively. They established the existence of solutions of functional equations (4)- (15) in the spaces BC(S), B(S) or BB(S) (defined in Section 2), respectively.…”