The dynamic programming problem (example 2.32) subject to xt+1 G(xt), t = 0, 1, 2, . . . , x0 X gives rise to an

The dynamic programming problem (example 2.32)

subject to xt+1 ˆŠ G(xt), t = 0, 1, 2, . . . , x0 ˆŠ X
gives rise to an operator

on the space B(X) of bounded functionals (exercise 2.16). Assuming that
€¢ f is bounded and continuous on X × X
€¢ G(x) is nonempty, compact-valued, and continuous for every x ˆŠ X
show that T is an operator on the space C(X) of bounded continuous functionals on X (exercise 2.85), that is Tv ˆŠ C(X) for every v ˆŠ C(X).

Bf(x, +) (To) (x-sup If(x, y) B(y)) SUl yeG(x)