On Bellman's allocation processes (Q1081109)

The author considers the class of k-parametrized functional equations \[ f(x)=\max_{kx\leq y_ 1+y_ 2\leq x;\;y_ 1,y_ 2\geq 0}[g(y_ 1)+h(y_ 2)+f(x-(1-a)y_ 1-(1-b)y_ 2)],\quad x\geq 0,\;f(0)=0,\;0\leq k\leq 1, \] and their inversions \[ u(z)=\min_{kx\leq y_ 1+y_ 2\leq x;\;y_ 1,y_ 2\geq 0,\;x=f^{-1}(z)}[(1- a)y_ 1+(1-b)y_ 2+u(z-g(y_ 1)-h(y_ 2))],\quad z\geq 0,\;u(0)=0. \] Existence and uniqueness of the solution \(f\) resp. \(f^{-1}\) are discussed, and two successive approximation methods are suggested for computing the unique solution \(f^{-1}\).