Stability, boundedness, and square integrability of solutions to third order neutral differential equations with delay
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Stability, boundedness, and square integrability of solutions to third order neutral differential equations with delay Anes M. Khatir1 · John R. Graef2
· Moussadek Remili1
Received: 8 April 2019 / Accepted: 25 July 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, the asymptotic behavior of solutions of a class of non-autonomous third order neutral delay differential equations is studied. Results on the stability, boundedness, and the square integrability of solutions and their derivatives are obtained. An example to illustrate the results is included. Keywords Lyapunov functional · Delay differential equations · Third-order · Neutral differential equations Mathematics Subject Classification 34C11 · 34D20
1 Introduction Asymptotic properties of solutions of delay and neutral delay differential equations have received considerable attention in recent years; see, for example, [1,3–10,12–34] and the references therein. Due to the applicability of neutral type equations to problems in the natural sciences, technology, and population dynamics, there is a continuing interest in obtaining new sufficient conditions for the stability, boundedness, and square integrability of the solutions of various forms of neutral differential equations. Based on the Lyapunov functional approach, here we give new results on asymptotic behavior of solutions of a broad class of neutral equations. In particular, sufficient conditions are given for the asymptotic stability, boundedness, and square integrability of solutions of the third order neutral delay differential equation
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John R. Graef [email protected] Anes M. Khatir [email protected] Moussadek Remili [email protected]
1
Department of Mathematics, University of Oran, 1 Ahmed Ben Bella, 31000 Oran, Algeria
2
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
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p(t) x(t) + β(t)x(t − r )
+a(t) Q(x(t))x (t) + b(t)R(x(t))x (t) + c(t) f (x(t − σ )) = h(t),
(1)
for all t ≥ t1 = t0 + r¯ , where r¯ = sup{r , σ } and the functions a, b, c, h, p, β : [t0 , ∞) → R and Q, R, f : R → R are continuous with f (0) = 0 and |β(t)| ≤ β1 < 1. In addition, we also assume that the derivatives Q , p , β , and β exist and are continuous. By a solution of (1) we mean a continuous function x : [tx , ∞) → R such that x(t) + β(t)x(t − r ) ∈ C 3 ([tx , ∞), R) and which satisfies (1) on [tx , ∞). Equation (1) is a quite general third-order nonlinear differential equation. With β(t) ≡ β, σ = r , and p ≡ 1, the stability and the square integrability of solutions of Eq. (1) was studied by Graef et al. [9]. In Sects. 2 and 3 we present our main results. Section 4 contain an example illustrating the results.
2 Asymptotic stability Asymptotic stability of solutions of Eq. (1) will be studied in the case h ≡ 0. Before proceeding further, we start by presenting some notations and conditions on the coefficient functions that will be needed later in the paper. Assume
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