On the existence and uniqueness of solutions to boundary value problems on time scales
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This work formulates existence, uniqueness, and uniqueness-implies-existence theorems for solutions to two-point vector boundary value problems on time scales. The methods used include maximum principles, a priori bounds on solutions, and the nonlinear alternative of Leray-Schauder. 1. Introduction This paper considers the existence and uniqueness of solutions to the second-order vector dynamic equation
y ∆∆ (t) = f t, y σ(t)
+ P(t)y ∆ σ(t) ,
t ∈ [a,b],
(1.1)
subject to any of the boundary conditions y(a) = A, αy(a) − βy ∆ (a) = C, αy(a) − βy ∆ (a) = C, y(a) = A,
y σ 2 (b) = B,
(1.2) ∆
∆
γy σ 2 (b) + δ y σ(b) = D,
y σ 2 (b) = B,
(1.3) (1.4)
γy σ 2 (b) + δ y σ(b) = D,
(1.5)
where f : [a,b] × Rd → Rd ; P(t) is a d × d matrix; A,B,C,D ∈ Rd ; and α,β,γ,δ ∈ R. The problems (1.1), (1.2); (1.1), (1.3); (1.1), (1.4); and (1.1), (1.5) are known as boundary value problems (BVPs) on “time scales.” To understand the notation used above and the idea of time scales, some preliminary definitions are useful. Definition 1.1. A time scale T is a nonempty closed subset of the real numbers R. Since a time scale may or may not be connected, the concept of jump operators is useful. Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:2 (2004) 93–109 2000 Mathematics Subject Classification: 39A12 URL: http://dx.doi.org/10.1155/S1687183904308071
94
Systems of BVPs on time scales
Definition 1.2. Define the forward (backward) jump operator σ(t) at t for t < sup T (resp., ρ(t) at t for t > inf T) by σ(t) = inf {τ > t : τ ∈ T},
ρ(t) = sup τ < t : τ ∈ T ,
∀t ∈ T.
(1.6)
Also define σ(sup T) = sup T if sup T < ∞, and ρ(inf T) = inf T if inf T > −∞. For simplicity and clarity denote σ 2 (t) = σ(σ(t)) and y σ (t) = y(σ(t)). Define the graininess function µ : T → R by µ(t) = σ(t) − t. Throughout this work the assumption is made that T has the topology that it inherits from the standard topology on the real numbers R. Also assume throughout that a < b are points in T with [a,b] = {t ∈ T : a ≤ t ≤ b}. The jump operators σ and ρ allow the classification of points in a time scale in the following way: if σ(t) > t, then call the point t right-scattered; while if ρ(t) < t, then we call t left-scattered. If t < sup T and σ(t) = t, then call the point t right-dense; while if t > inf T and ρ(t) = t, then we call t left-dense. If T has a left-scattered maximum at m, then define Tk = T − {m}. Otherwise Tk = T. Definition 1.3. Fix t ∈ T and let y : T → Rd . Define y ∆ (t) to be the vector (if it exists) with the property that given > 0 there is a neighbourhood U of t such that, for all s ∈ U and each i = 1,...,d, yi σ(t) − yi (s) − y ∆ (t) σ(t) − s ≤ σ(t) − s.
(1.7)
i
Call y ∆ (t) the (delta) derivative of y(t) at t. Definition 1.4. If F ∆ (t) = f (t), then define the integral by t a
f (s)∆s = F(t) − F(a).
(1.8)
The following theorem is due to Hilger [12]. Theorem 1.5. Assume that f : T → Rd and let t ∈ Tk . (i) If f i
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