Periodic boundary value problems on time scales
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We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale, a condition containing a Lipschitz constant of right-hand side function and the supremum of the graininess function is introduced. 1. Introduction We recall the basic definitions concerning calculus on time scales. Further details can be found in the survey monography upon this topic [2], its second part [3], or in the paper [6]. Time scale T is an arbitrary nonempty closed subset of the real numbers R. The natural numbers N, the integers Z, or the union of intervals [0,1] ∪ [2,3] are examples of time scales. For t ∈ T we define the forward jump operator σ : T → T and the backward jump operator ρ : T → T by σ(t) := inf {s ∈ T : s > t },
ρ(t) := sup{s ∈ T : s < t },
(1.1)
where we put inf ∅ = sup T and sup ∅ = inf T. We say that a point t ∈ T is right scattered, left scattered, right dense, left dense if σ(t) > t, ρ(t) < t, σ(t) = t, ρ(t) = t, respectively. A point t ∈ T is isolated if it is right scattered and left scattered. A point t ∈ T is dense if it is right dense and left dense. Finally, we define the forward graininess function µ : T → [0, ∞) by µ(t) := σ(t) − t.
(1.2)
Similarly, we define the backward graininess function ν : T → [0, ∞) by ν(t) := t − ρ(t). Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:1 (2005) 81–92 DOI: 10.1155/ADE.2005.81
(1.3)
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Periodic boundary value problems on time scales
We define the function f σ (t) = f (σ(t)) and the modified time scale Tκ as follows: if T has a left-scattered maximum m, then Tκ = T \ {m}, otherwise Tκ = T. We endow T with the topology inherited from R. The continuity of the function f : T → R is defined in the usual manner. The function f is said to be delta differentiable (the word delta is omitted later) at t ∈ Tκ if there exists a number (denoted by f ∆ (t)) with the property that given any > 0 there is a neighbourhood U of t (in the time-scale topology) such that σ f (t) − f (s) − f ∆ (t) σ(t) − s ≤ σ(t) − s
∀s ∈ U.
(1.4)
If f is differentiable at every t ∈ T, then f is said to be differentiable on T. The second derivative is defined by f ∆∆ (t) = ( f ∆ (t))∆ . Similarly, one can define nabla derivative f ∇ with the aid of the backward jump operator ρ instead of σ. See [3] for more details. In the following we often use formula f σ (t) = f (t) + µ(t)x∆ (t).
(1.5)
We also define the interval with respect to the time scale T denoted by [a,b]T := {t ∈ T : a ≤ t ≤ b}.
(1.6)
We assume further that a,b ∈ T. A function F : Tκ → R is called an antiderivative of f : T → R provided that F ∆ (t) = f (t)
∀t ∈ Tκ .
(1.7)
We then define the Cauchy integral by s r
f (t)∆t = F(s) − F(r)
∀r,s ∈ T.
(1.8)
We define an rd
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