Quasilinearization And Monotone Iteration Methods For Second-order Nonlinear Boundary Value Problems On Time Scales

A time scale T is an arbitrary nonempty closed subset of the real numbers.In the paper,we study quasilinearization and the monotone iteration methods for second-order nonlinear boundary value problems on time scales.In section l,We use Leggett-Williams fixed point theorem to establish the existence of solutions to boundary value problemThe main conclusion is the following:Theorem 1 Suppose the following conditions are satisfied:f : [a, b] x [0, oo) -4 [0, oo) is continuous,f(t,u) (t) (u) on [a.b] [0, ), with : [a,a(b)] - (0, ) continuous and nonnegative continuous and nondecreasing on [0, oo). Assume that [a, 2(b)] is such that = min{ } exists and satisfies (b),f(t,u) (t) (u) on [ ,b] x [0, ) with r : [ ,6] - (0, ) continuous, r > 0 with ( ) sup G(t,s) (s) s < r,L > r with (L) min G(t, s)r(s) s > L,Then (l.l)has three solutions y1,y2,y3 C[a, 2(b)] with yi Crd[a,b], i = 1,2,3 and |y1|0 < r,y2(t) > L on t [ , 2(b)], |y3|0 > r, min y3(t) < L. We generalize the work in [8],where f is one dimension.In section 2,the quasilinearization method is used to approach to the unique solution of the second-order separated boundary value problemon time scales from below and above by monotone convergent sequence of upper and lower solutions.The rate of convergence is also determined. The main conclusion is the following:Theorem 2 Assume that 0, 0 are respectively lower and upper solutions of the SBVP(2.1)- 0, then there exist monotone sequences { n},{ n} converging uniformly to the unique solution x(t) of the SBVP(2.1)-(2.2) and exist positive constants C1,C2,C3,C4 satisfyWe weak the conditions in [21],we can also change the conditions of Theorem 2 with weaker conditions that such that (k+1) 0, (k+1) 0 and (f + )(k+1) 0, (g + )(k+1) 0. In section 3,we consider second order periodic boundary value problems on tune scales. Only a few have studyed family of problems.We extend the work in [26]-[27].First,under the classical assumption that (t) (t),we consider the periodic boundary value problem, when / is independent of x (t).Second,under the case that (t) (t),we consider the periodic boundary value problem and describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to periodic boundary value problems,when / depends on x (t).