Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations
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We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general nth problem in time scales with linear dependence on the ith ∆derivatives for i = 1,2,...,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f (t,x) which is rd-continuous in t and continuous in x uniformly in t. To do that, we obtain the expression of the Green’s function of a related linear operator in the space of the antiperiodic functions. 1. Introduction The theory of dynamic equations has been introduced by Stefan Hilger in his Ph.D. thesis [12]. This new theory unifies difference and differential equations and has experienced an important growth in the last years. Recently, many papers devoted to the study of this kind of problems have been presented. In the monographs of Bohner and Peterson [5, 6] there are the fundamental tools to work with this type of equations. Surveys on this theory given by Agarwal et al. [2] and Agarwal et al. [1] give us an idea of the importance of this new field. In this paper, we study the existence and uniqueness of solutions of the following nthorder dynamic equation with antiperiodic boundary value conditions: (Ln ) u∆ (t) + n
n −1
j
j =1 ∆i
M j u∆ (t) = f t,u(t) ,
u (a) = −u
∆i
σ(b) ,
∀t ∈ I = [a,b],
(1.1)
0 ≤ i ≤ n − 1. n
Here, n ≥ 1, M j ∈ R are given constants for j ∈ {1,...,n − 1}, [a,b] = Tκ , with T ⊂ R an arbitrary bounded time scale and f : I × R → R satisfies the following condition: Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:4 (2004) 291–310 2000 Mathematics Subject Classification: 39A10 URL: http://dx.doi.org/10.1155/S1687183904310022
292
Higher-order antiperiodic dynamic equations
(H f ) for all x ∈ R, f (·,x) ∈ Crd (I) and f (t, ·) ∈ C(R) uniformly at t ∈ I, that is, for all > 0, there exists δ > 0 such that |x − y | < δ =⇒ f (t,x) − f (t, y) < ,
∀t ∈ I.
(1.2)
n A solution of problem (Ln ) will be a function u : T → R such that u ∈ Crd (I) and satn isfies both equalities. Here, we denote by Crd (I) the set of all functions u : T → R such i that the ith derivative is continuous in Tκ , i = 0,...,n − 1, and the nth derivative is rdcontinuous in I. It is clear that for any given constant M ∈ R, problem (Ln ) can be rewritten as
u∆ (t) + n
n −1
M j u∆ (t) + Mu(t) = f t,u(t) + Mu(t), j
j =1 ∆i
u (a) = −u
∆i
∀t ∈ I,
(1.3)
0 ≤ i ≤ n − 1.
σ(b) ,
n n Defining the linear operator Tn [M] : Crd (I) → Crd (I) for every u ∈ Crd (I) as
Tn [M]u(t) := u∆ (t) + n
n −1
M j u∆ (t) + Mu(t), j
for every t ∈ I,
(1.4)
j =1
and the set
n Wn := u ∈ Crd (I) : u∆ (a) = −u∆ σ(b) , 0 ≤ i ≤ n − 1 , i
i
(1.5)
we can rewrite the dynamic equation (Ln ) as
Tn [M]u(t) = f t,u(t) + Mu(t),
t ∈ I, u ∈ Wn .
(1.6)
From this fact, we deduce that to ensure the existence and uniqueness of solutions of the dynamic equation (Ln ), we must determine the real values M,M1 ,...,Mn−1 for
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