Abstract :
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.
