Existence and Approximation of Solutions of Boundary Value Problems on Time Scales

Existence and approximation of solutions for a class of boundary value problems on time scales of the type − [p(t)y∆(t)]∇ + q(t)y(t) = f(t, y(t)), t ∈ [a, b]T, c1y(ρ(a))− c2y(ρ(a)) = 0, d1y(b) + d2y(b) = 0, is established. For the existence theory, we develop the method of upper and lower solutions. To approximate the solution, we develop the generalized method of quasilinearization. We show that under suitable conditions on f , there exists a bounded monotone sequence of solutions of linear problems which converges monotonically and rapidly to solution of the original problem. AMS Subject Classifications: 39A12, 34A45, 34B15.