Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

and Applied Analysis 3 where the kernel K t, s is an n × n Cauchy matrix defined in the square a, b × a, b being, for every fixed s ≤ t , a solution of the matrix Cauchy problem LK ·, s t : ∂K t, s ∂t −A t Sh0K ·, s t Θ, K s, s I, 1.8 where K t, s ≡ Θ if a ≤ t < s ≤ b, Θ is n × n null matrix and I is n × n identity matrix. A fundamental n × n matrix X t for the homogeneous φ ≡ θ equation 1.5 has the form X t K t, a , X a I 2 . Throughout the paper, we denote by Θs an s × s null matrix if s / n, by Θs,p an s × p null matrix, by Is an s × s identity matrix if s / n, and by θs an sdimensional zero column-vector if s / n. A serious disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find the Cauchy matrix K t, s 3, 4 . It exists but, as a rule, can only be found numerically. Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly. Below we consider the case of a system with so-called single delay 5 . In this case, the problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayedmatrix exponential defined below. 1.1. A Delayed Matrix Exponential Consider a Cauchy problem for a linear nonhomogeneous differential system with constant coefficients and with a single delay τ ż t Az t − τ g t , 1.9 z s ψ s , if s ∈ −τ, 0 , 1.10 with an n × n constant matrix A, g : 0,∞ → n , ψ : −τ, 0 → n , τ > 0 and an unknown vector-solution z : −τ,∞ → n . Together with a nonhomogeneous problem 1.9 , 1.10 , we consider a related homogeneous problem ż t Az t − τ , z s ψ s , if s ∈ −τ, 0 . 1.11 4 Abstract and Applied Analysis Denote by e τ a matrix function called a delayed matrix exponential see 5 and defined as e τ : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Θ if −∞ < t < −τ, I if − τ ≤ t < 0, I A t 1! if 0 ≤ t < τ, I A t 1! A2 t − τ 2 2! if τ ≤ t < 2τ, · · · I A t 1! · · · A t − k − 1 τ k k! if k − 1 τ ≤ t < kτ, · · · . 1.12 This definition can be reduced to the following expression: e τ t/τ 1 ∑ n 0 A t − n − 1 τ n n! , 1.13 where t/τ is the greatest integer function. The delayed matrix exponential equals the unit matrix I on −τ, 0 and represents a fundamentalmatrix of a homogeneous systemwith single delay. Thus, the delayed matrix exponential solves the Cauchy problem for a homogeneous system 1.11 , satisfying the unit initial conditions z s ψ s ≡ e τ I if − τ ≤ s ≤ 0, 1.14 and the following statement holds see, e.g., 5 , 6, Remark 1 , 7, Theorem 2.1 . Lemma 1.1. A solution of a Cauchy problem for a nonhomogeneous system with single delay 1.9 , satisfying a constant initial condition z s ψ s c ∈ n if s ∈ −τ, 0 1.15