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 uΔn (t )+ n−1 j=1 Mj uΔj (t ) = f t,u(t) , t∈ [a, b], uΔi (a) = uΔi σ(b) , i= 0, . . . , n −1. Here f : [a, b] × R→R is such that f (·,x) is rd-continuous in I for every x ∈ R and f (t, ·) is continuous in R uniformly at t ∈ I , Mj ∈ R are given constants and [a, b] = Tκn 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 α and β. To this end, givenM >0, we study some maximum principles related with operators T ± n [M]u(t ) ≡ uΔn (t )+ n−1 j=1 Mj uΔj (t )±Mu(t ), in the space of periodic functions.  2003 Elsevier Inc. All rights reserved.