On the asymptotic behavior of solutions of certain second-order differential equations

doi:10.1016/j.jfranklin.2006.02.011

Journal of the Franklin Institute

Volume 344, Issue 5, August 2007, Pages 391-398

https://doi.org/10.1016/j.jfranklin.2006.02.011 Get rights and content

Abstract

In this paper, the second order non-linear differential equation $\ddot{x} + a (t) f (x, \dot{x}) \dot{x} + b (t) g (x) = p (t, x, \dot{x})$

is considered, and Lyapunov's second method is used to show that uniform boundedness and convergence to zero of all solutions of this equation together with their derivatives of the first order.

Introduction

In the present paper, we are interested in obtaining a result on the uniform boundedness and convergence of solutions of second order nonlinear differential equations of the form: $\ddot{x} + a (t) f (x, \dot{x}) \dot{x} + b (t) g (x) = p (t, x, \dot{x}),$ in which the functions a, b, f, g and p are continuous for the arguments displayed explicitly in Eq. (1), and the dots denote differentiation with respect to t. It is assumed that the functions b and g are differentiable on $ℜ^{+} (ℜ^{+} = [0, \infty))$ and $ℜ$ , respectively, and the functions a, b, f, g and p are so constructed such that the uniqueness theorem is valid.

As we know, the investigation of qualitative properties of differential equations of Lienard type, in particular, the study of stability, instability of solutions in the case $p = 0$ , boundedness of solutions, convergence of solutions and existence of periodic solutions of differential equations of Lienard type is a very important problem in the theory and applications of differential equations. It is also well-known that, in the relevant literature up to now, a large number of works have been done and many interesting results have been obtained concerning the qualitative behavior of solutions for the particular cases of Eq. (1) or different forms of differential equations of Lienard type. The interested reader is advised to look up the references cited in the listed publications [1], [2], [3], [4], [5], [6], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [25], [26], [27], [28] and the references quoted therein for some works. In [8], Hatvani established sufficient conditions for stability of the zero solution of the differential equation: $\ddot{x} + a (t) f (x, \dot{x}) \dot{x} + b (t) g (x) = 0 .$ More recently, Jitsuro and Yusuke [13] have studied the global asymptotic stability of solutions of non-autonomous systems of Lienard type as follows: $\dot{x} = y - F (x), (f (x) = d F (x) / d x),$ $\dot{y} = - p (t) g_{1} (x) - q (t) g_{2} (x) .$ Our aim in this paper is to establish sufficient conditions under which all solutions of the differential Eq. (1) are uniformly bounded and tend to zero together with their derivatives of the first order as $t \to \infty$ . The main tool for proving the result here will be a Lyapunov function. It should be noted that the motivation for the present work has come from the papers mentioned above and exist in the relevant literature. The considered equation and the established assumptions and the result here are going to be different from the observations in the relevant literature.

In what follows, in place of Eq. (1), we consider the equivalent system $\dot{x} = y,$ $\dot{y} = - a (t) f (x, y) y - b (t) g (x) + p (t, x, y)$ which is obtained from Eq. (1).

Section snippets

Main result

The main result of this paper is the following:

Theorem

In addition to the basic assumptions imposed on the functions a, b, f, g and p, suppose the following assumptions are valid (α₁, α₂-some arbitrary positive constants and ε₀, ε₁ and ε₂ are some sufficiently small positive constants):

(i)
$A ⩾ a (t) ⩾ a_{0} ⩾ 1, B ⩾ b (t) ⩾ b_{0} ⩾ 1$ for all $t \in ℜ^{+}$ (where A, B, a₀, b₀ are some constants).
(ii)
$ε_{0} ⩽ f (x, y) - α ⩽ ε_{1}$ for all x and y.
(iii)
$g (0) = 0, g (x) sgn x > 0 (x \neq 0), G (x) \equiv \int_{0}^{x} g (ξ) d ξ \to \infty$ as $| x | \to \infty$ and $0 ⩽ α_{2} - g^{'} (x) ⩽ ε_{2}$ for all x.
(iv)
$\int_{0}^{\infty} γ_{0} (t) d t < \infty, b^{'} (t) \to 0$ as $t \to \infty$ , where $γ$

The Lyapunov function $V_{0} (t, x, y)$

The proof of the theorem depends on some fundamental properties of a certain continuously differentiable function $V_{0} = V_{0} (t, x, y)$ defined as follows: $V_{0} = b (t) \int_{0}^{x} g (ξ) d ξ + \frac{y^{2}}{2} + k,$ where k is a positive constant to be specified below.

In the subsequent discussion we require the following lemmas.

Lemma 1

Suppose that conditions (i)–(iii) of the theorem are fulfilled. Then there exist positive constants D₁ and D₂ such that $D_{1} [G (x) + y^{2} + k] ⩽ V_{0} ⩽ D_{2} [G (x) + y^{2} + k] .$

Proof

In view of the conditions of (i) and (iii) of the theorem, we

Completion of proof of the theorem

Consider the function $V_{0} = V_{0} (t, x, y)$ defined as follows: $V (t, x, y) = e^{- \int_{0}^{t} γ (τ) d τ} V_{0} (t, x, y),$ where $γ (t) = D_{4} γ_{0} + \frac{4}{D_{1}} {p_{1} (t) + p_{2} (t)} .$ Next, it can be confirmed that there exist two functions $ψ_{1}$ and $ψ_{2}$ satisfying $ψ_{1} (‖ \bar{x} ‖) ⩽ V (t, x, y) ⩽ ψ_{2} (‖ \bar{x} ‖)$ for all $\bar{x} \in ℜ^{2}$ and for all $t \in ℜ^{+}$ ; where $ψ_{1}$ is a continuous increasing positive definite function, $ψ_{1} (r) \to \infty$ as $r \to \infty$ and $ψ_{2}$ is a continuous increasing function. Differentiating the function (8) along any solution $(x, y)$ of the system (2) we obtain $\dot{V} = e^{- \int_{0}^{t} γ (τ) d τ} [{\dot{V}}_{0} - γ (t) V_{0}] ⩽ e^{- \int_{0}^{t} γ (τ) d τ} [-$

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On the asymptotic behavior of solutions of certain second-order differential equations

Abstract

Introduction

Section snippets

Main result

The Lyapunov function V0(t,x,y)

Completion of proof of the theorem

Quart. J. Math.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Systems Sci. Math. Sci.

Ann. Differential Equations

On the stability of the Liénard equation with unsteady perturbations

Differ. Uravn.

The generalized Lienard equation

J. Soc. Indust. Appl. Math. Ser. A Control

On the equation x″+f(x)h(x′)x′+g(x)=e(t)

Ann. Mat. Pura Appl.

Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation