Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 159–162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ERROR ESTIMATES FOR ASYMPTOTIC SOLUTIONS OF DYNAMIC EQUATIONS ON TIME SCALES

GRO HOVHANNISYAN

Abstract. We establish error estimates for first-order linear systems of equations and linear second-order dynamic equations on time scales by using calculus on a time scales [1, 4, 5] and Birkhoff-Levinson’s method of asymptotic solutions [3, 6, 8, 9].

1. Results

Asymptotic behavior of solutions of dynamic equations and systems on time scales was investigated in [5]. In this paper we establish error estimates of such asymptotic representations, which may be applied to the investigation of stability of dynamic equations (see f.e. [9]).

Consider the system of ordinary differential equations on time scales

a^{Δ}(t) = A(t)a(t), t > T,  (1.1)

where a^{Δ} is delta (Hilger) derivative, a(t) is a n-vector function, and A(t) is a n×n matrix function from C_{rd}(T, ∞) (definition of rd-continuous functions see in [4]). A time scale is an arbitrary nonempty closed subset of the real numbers. Let 𝕋 be a time scale. For t ∈ 𝕋 we define the forward jump operator σ : 𝕋 → 𝕋 by

σ(t) = inf{s ∈ 𝕋 : s > t}.

The graininess function μ : 𝕋 → [0, ∞] is defined by

μ(t) = σ(t) − t.

We assume that sup 𝕋 = ∞.

Suppose we can find the exact solutions of the auxiliary system

ψ^{Δ}(t) = A_{1}(t)ψ(t), t > T,  (1.2)

with the matrix function A_{1}(t) ∈ C_{rd}(T, ∞) close to the matrix function A(t), which means that condition (1.5) below is satisfied. Let Ψ(t) be the fundamental matrix of the system (1.2). Then the solutions of (1.1) can be represented in the form

a(t) = Ψ(t)(C + ε(t)),  (1.3)

2000 Mathematics Subject Classification. 39A10.

Key words and phrases. Time scale; asymptotic representation; error estimates;

second order differential equation; first order system .

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published February 28, 2007.

where a(t), ε(t), C are the vector columns. We can consider (1.3) as a definition of the error vector function ε(t). Denote

H(t) ≡ (1 + μ(t)Ψ^{−1}(t)Ψ^{Δ}(t))^{−1} Ψ^{−1}(t) (A(t)Ψ(t) − Ψ^{Δ}(t)) .  (1.4)

Theorem 1.1. Assume there exist an invertible and differentiable matrix function Ψ(t) ∈ C_{rd}(T, ∞) such that 1 + μ(t)Ψ^{−1}(t)Ψ^{Δ}(t) is invertible and

∫_{t}^{∞} ( lim_{m↘μ(s)} \frac{log(1 + m∥H(s)∥)}{m}Δs) < ∞.  (1.5)

Then every solution of (1.1) can be represented in form (1.3) and the error function ε(t) can be estimated as

∥ε(t)∥ ≤ ∥C∥ (−1 + e_{∥H∥}(∞, t)) ,  (1.6)

where ∥.∥ is the Euclidean vector (or matrix) norm: ∥C∥ = \sqrt{C_{1}^{2} + ⋯ + C_{n}^{2}}, and expression in (1.5) usually is used to define the exponential function on time scales (see [1, 4]):

e_{∥H∥}(∞, t) = exp (∫_{t}^{∞} lim_{m↘μ(s)} \frac{log(1 + m∥H(s)∥)}{m}Δs).  (1.7)

Remark 1.2. Comparing with the similar result from [5] advantage of Theorem 1.1 is that it not only proves that error vector function approaches to zero as t approaches to infinity, but inequality (1.6) also estimates the speed of that approach to zero.

¿From the estimate (1.6) it follows also that the error vector function ε(t) is small when ∫_{t}^{∞} lim_{m↘μ(s)} \frac{log(1+m∥H(s)∥)}{m}Δs is small.

Proof of Theorem 1.1. Let a(t) be a solution of (1.1). The substitution a(t) = Ψ(t)u(t) transforms (1.1) into

u^{Δ} = H(t)u(t), t > T,

where H is defined by (1.4). By integration we get

u(t) = C − ∫_{t}^{b} H(s)u(s)Δs, t < s < b,  (1.8)

where the constant vector C is chosen as in (1.3). Estimating u(t)

∥u(t)∥ ≤ ∥C∥ + ∫_{t}^{b} ∥H(s)∥ · ∥u(s)∥Δs,

and applying Gronwall’s lemma (see [4]) we have

∥u(t)∥ ≤ ∥C∥e_{∥H∥}(b, t).  (1.9)

From representation (1.3) and expression (1.7), we have

ε(t) = Ψ^{−1}(t)a(t) − C = u(t) − C = − ∫_{t}^{b} H(s)u(s)Δs.

Then using (1.9) we obtain

∥ε(t)∥ ≤ ∫_{t}^{b} ∥H(s)u(s)∥Δs
≤ ∥C∥ ∫_{t}^{b} ∥H(s)∥ · e_{∥H∥}(b, s)
= ∥C∥ [−1 + e_{∥H∥}(b, t)]
≤ ∥C∥ [−1 + e_{∥H∥}(∞, t)] .

Note that from (1.5), it follows that

lim_{t→∞} [−1 + e_{∥H∥}(∞, t)] = lim_{t→∞} [ − 1 + exp ∫_{t}^{∞} lim_{m↘μ(s)} \frac{log(1 + m∥H(s)∥)}{m}Δs] = 0.

□

Consider the second-order dynamic equation on time scales

L[x(t)] = x^{ΔΔ} + p(t)x^{Δ}(t) + q(t)x(t) = 0, t > t_{0} > 0, t ∈ 𝕋.  (1.10)

¿From the functions φ_{1,2}(t) ∈ C_{rd}^{2}(T, ∞) let us construct auxiliary matrix-functions

\begin{matrix}Φ(t) = (\begin{matrix}φ_{1}(t) & φ_{2}(t) \\ φ_{1}^{Δ}(t) & φ_{2}^{Δ}(t) \end{matrix}) , H(t) = (1 + μ(t)Φ^{−1}(t)Φ^{Δ}(t))^{−1}B(t), \\ B(t) = (\begin{matrix}B_{21}(t) & B_{22}(t) \\ −B_{11}(t) & −B_{12}(t) \end{matrix}) , B_{kj}(t) ≡ \frac{φ_{k}(t)L[φ_{j}(t)]}{W (φ_{1}, φ_{2})}, j = 1, 2. \end{matrix} (1.11)

Theorem 1.3. Let φ_{1,2}(t) ∈ C_{rd}^{2}(T, ∞) be complex-valued functions such that

∫_{T}^{∞} ( lim_{m↘μ(s)} \frac{log (1 + m∥ (1 + mΦ^{−1}(t)Φ^{Δ}(t))^{−1} B(t)∥)}{m})Δt < ∞, k, j = 1, 2,

(1.12)

where ∥.∥ is Euclidean matrix norm. Then for arbitrary constants C_{1},C_{2} there exist solution of (1.1) that can be written in the form

x(t) = [C_{1} + ε_{1}(t)] φ_{1}(t) + [C_{2} + ε_{2}(t)] φ_{2}(t),  (1.13)
x^{Δ}(t) = [C_{1} + ε_{1}(t)] φ_{1}^{Δ}(t) + [C_{2} + ε_{2}(t)] φ_{2}^{Δ}(t).  (1.14)

The error vector-function ε(t) = (ε_{1}(t), ε_{2}(t)) is estimated as

∥ε(t)∥ ≤ ∥C∥( − 1 + e_{∥H(t)∥}(∞, t)),  (1.15)

where C = (C_{1}, C_{2}) is an arbitrary constant vector, and the matrix function H(t) is defined in (1.11).

Proof. Rewrite equation (1.10) in form (1.1):

a^{Δ}(t) = A(t)a(t),  (1.16)

where

a(t) = (\begin{matrix}x(t) \\ x^{Δ}(t) \end{matrix}) , A(t) = (\begin{matrix}0 & 1 \\ −q(t) & −p(t) \end{matrix}) .

By substitution

a(t) = Φ(t)w(t),  (1.17)

in (1.16) we get

w^{Δ} = H(t)w(t),  (1.18)

where H(t) defined by (1.11). To apply Theorem 1.1 to system (1.16) we choose A(t) = H(t) and A_{1} ≡ 0. Then the identity matrix is fundamental solution of (1.2), so conditions (1.5) turns to (1.12). ¿From Theorem 1.1 we have

w(t) = C + ε(t), or a(t) = Φ(t)w(t) = Φ(t)(C + ε(t)).

Representations (1.13),(1.14) and estimates (1.15) follow from Theorem 1.1. □

Example 1.4. For solutions of the equation

x^{ΔΔ}(t) + (γ^{2} + \frac{1}{t^{2}} )x(t) = 0, t > t_{0},

we get representations (1.13), (1.14), where

φ_{1} = cos_{γ}(t, t_{0}), φ_{2} = sin_{γ}(t, t_{0})

are trigonometric functions on time scales [4]. By direct calculations H = O(t^{−2}) as t → ∞, and

|ε_{j}(t)| ≤ ∥C∥[ − 1 + exp ( ∫_{t}^{∞} lim_{m↘μ(s)} \frac{log (1 + C_{1}ms^{−2})}{m}Δs)], j = 1, 2.

Acknowledgments. The author wants to thank the anonymous referee for his/her comments that helped improving the original paper.

References

[1] R. Aulbach and S. Hilger. Linear dynamic processes with inhomogeneous time scales. In Non-linear Dynamics and Quantum Dynamical systems. Academie Verlag, Berlin, 1990.

[2] Z. Benzaid and D. A. Lutz. Asymptotic representation of solutions of perturbed systems of linear difference equations. Studies in Applied Mathematics, 1987, 77, 195-221.

[3] J. D. Birkhoff Quantum mechanics and asymptotic series. Bull. Amer. Math., Soc. 32 1933, 681-700

[4] M. Bohner, A. Peterson. Dynamic equations on time scales: An Introduction with Applications. Birkhauser, Boston, 2001.

[5] M. Bohner, D. A. Lutz. Asymptotic behavior of Dynamic Equations on Time Scales. J. Differ. Equations Appl, 2001, 7 (1): 21-50.

[6] N. Levinson The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J., 1948, 15, 111-126.

[7] W.A. Harris and D.A. Lutz. A unified theory of asymptotic integration. J. Math. Anal. Appl., 1977, 57 (3), 571-586.

[8] G. Hovhannisyan. Estimates for error functions of asymptotic solutions of ordinary linear differential equations. Izv. NAN Armenii, Matematika, [English translation: Journal of Contemporary Math. Analysis - (Armenian Academy of Sciences) 1996, v. 31, no.1, 9-28.

[9] G. Hovhannisyan. Asymptotic stability for second-order differential equations with complex coefficients Electron. J. Diff. Eqns., Vol. 2004 (2004), No. 85, 1-20.

Gro R. Hovhannisyan Kent State University, Stark Campus, 6000 Frank Ave. NW, Canton, OH 44720-7599, USA

E-mail address: ghovhannisyan@stark.kent.edu