We explore the sign properties of eigenvalues and the basis properties of eigenvectors for a special quadratic matrix polynomial and use the results obtained to solve the corresponding linear system of differential equations on the half line subject to an initial condition at t = 0 and a condition at t = ∞. | Turk J Math (2016) 40: 317 – 332 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Solving an initial boundary value problem on the semiinfinite interval Ferihe ATALAN∗, Gusein Sh. GUSEINOV Department of Mathematics, Atılım University, Ankara, Turkey • Received: Accepted/Published Online: • Final Version: Abstract: We explore the sign properties of eigenvalues and the basis properties of eigenvectors for a special quadratic matrix polynomial and use the results obtained to solve the corresponding linear system of differential equations on the half line subject to an initial condition at t = 0 and a condition at t = ∞ . Key words: Quadratic eigenvalue problem, eigenvalues, eigenvectors 1. Introduction For various types of infinite interval problems we refer to the book [1] by Agarwal and O’Regan. In this paper, we deal with the existence and uniqueness and the explicit form of solution u(t) to the problem C d2 u(t) dt2 = J du(t) + Ru(t), 0 ≤ t 0 (0 ≤ n ≤ N − 1), (0 ≤ n ≤ N − 1). rn > 0 (9) (10) In the present paper, we replace condition (10) by the condition r0 = 0, rn > 0 (1 ≤ n ≤ N − 1), (11) allowing one of the rn ’s to be zero, and study the consequences of the condition r0 = 0 . It turns out that if r0 = 0 , then λ = 0 is an eigenvalue of Eq. (4), and, moreover, this eigenvalue is defective if b0 = 0 . For solving the problem (1), (2) it will also be important to investigate the sign properties of the nonzero eigenvalues. To fix the terminology used in the paper let us recall some concepts related to the quadratic eigenvalue problems. Let N be a positive integer and M , L , and K be N × N complex matrices. The quadratic matrix polynomial (quadratic matrix pencil) Q(λ) = λ2 M + λL + K (12) is called regular when det Q(λ) is not identically zero for all values of λ , and nonregular otherwise. We assume that Q(λ) is regular. By the .
