Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem [GRAPHICS] u(Delta i)(a)=u(Delta i)(sigma(b)), i=0,...,n-1 Here f:[a,b] x R-->R is such that f (., x) is rd-continuous in I for every x is an element of R and f (t, .) is continuous in R uniformly at t is an element of I, M-j is an element of R are given constants and [a,b]=T-kappa n for an arbitrary bounded time scale T. We obtain sufficient conditions in f to guarantee the existence and approximation of solutions lying between a pair of ordered lower and upper solutions alpha and beta. To this end, given M>0, we study some maximum principles related with operators [GRAPHICS] in the space of periodic functions. (C) 2003 Elsevier Inc. All rights reserved.