Stability and boundedness of solutions of a kind of third-order delay differential equations

Stability and boundedness of solutions of a kind of third-order delay differential equations^*

A.U. Afuwape^I; M.O. Omeike^II

^IDepartmento de Matemáticas, Universidad de Antioquia Calle 67, No. 53-108, Medellín AA 1226, Colombia. E-mail: [e-mail]

IIDepartment of Mathematics, University of Agriculture, Abeokuta, Nigeria. E-mail: [e-mail]

ABSTRACT

This paper studies the stability and boundedness of solutions of certain nonlinear third-order delay differential equations. Sufficient conditions for the stability and boundedness of solutions for the equations considered are obtained by constructing a Lyapunov functional.

Mathematical subject classification: 34K20.

Key words: stability, boundedness, Lyapunov functional, differential equations of third-order with delay.

1 Introduction

This paper deals with the stability and boundedness of solution of the delay differential equation

or its equivalent system

where 0 < r(t) < γ, r'(t) < β, 0 < β < 1, β and γ are some positive constants, γ will be determined later, f(x), g(y), h(y), p(t, x, y, x(t - r(t)), y(t - r(t)), z) are continuous in their respective arguments. Besides, it is supposed that the derivatives f'(x), g'(y) are continuous for all x, y with f(0) = g(0) = 0. In addition, it is also assumed that the functions f(x(t - r(t))), g(y(t - r(t))) and p(t, x, y, x(t - r(t)), y(t - r(t)), z) satisfy a Lipschitz condition in x, y, x(t - r(t)), y(t - r(t)) and z ; throughout the paper x(t), y(t) and z(t) are, respectively, abbreviated as x, y and z. Then the solution is unique. (See [5, pp. 14]).

In recent year, many books and papers dealt with the delay differential equation and obtained many good results, for example, [1, 2, 3, 18, 19, 21], etc. In many references, the authors dealt with the problems by considering Lyapunov functions or functionals and obtained the criteria for the stability and boundedness. (See [1-21]).

In particular, recently, Tunç [15], obtained sufficient conditions which ensure the stability and the boundedness of systems

x"'+ a₁x" + f₂(x'(t - r(t))) + a₃x = 0

and

x"'+ a₁x" + f₂ ( x' (t - r(t))) + a₃x = p(t, x, x', x(t - r(t)), x'(t - r(t)), x"),

where r(t) is as defined above, a₁ and a₃ are some positive constants.

Our objective in this paper is to establish some sufficient conditions for the stability and for the boundedness of solutions of (1.1) in the cases p ≡ 0, 0, respectively.

2 Stability

First, we will give the stability criteria for the general autonomous delay differential system. We consider

where f : C_H →ⁿ is a continuous mapping, f(0) = 0, C_H := {Φ ∈ (C[-r, 0], ⁿ): ║Φ║ < H} and for H₁ < H, there exists L(H₁) > 0, with |f(Φ)| < L(H₁) when ║Φ║ < H₁.

Definition 2.1. An element ψ ∈ C is in the ω-limit set of Φ, say ω(Φ), if x(t, 0, Φ) is defined on [0,∞) and there is a sequence {t_n}, t_n → ∞, as n → ∞, with ║xt_n(Φ) - ψ║→ 0 as n → ∞ where xt_n(Φ) = x(t_n + θ, 0, Φ) for -r < θ < 0.

Definition 2.2 (See [17]). A set Q ⊂ C_H is an invariant set if for any Φ ∈ Q, the solution of (2.1), x(t, 0, Φ), is defined on [0, ∞), and x_t(Φ) ∈ Q for t ∈ [0,∞).

Lemma 2.1 (See [13]). If Φ ∈ C_H is such that the solution x_t(θ) of (2.1) with x₀(Φ) = Φ is defined on [0, ∞) and ║x_t(Φ)║ < H₁ < H for t ∈ [0,∞), then ω(Φ) is a nonempty, compact, invariant set and

dist(x_t(Φ), Ω(Φ)) → 0, as t → ∞

Lemma 2.2 (See [13]). Let V(Φ): C_H → be a continuous functional satisfying a local Lipschitz condition. V(0) = 0 and such that

(i) W₁(|Φ(0)|) < V(Φ) < W₂(║Φ║) where W₁(r), W₂(r) are wedges.

(ii) V'_(2.1)(Φ) < 0, for Φ < C_H.

The the zero solution of (2.1) is uniformly stable. If we define Z = {Φ ∈ C_H: V^'_(2.1)(Φ) = 0}, then the zero solution of (2.1) is asymptotically stable, provided that the largest invariant set in Z is Q = {0}.

The following will be our main stability result for (1.1).

Theorem 2.1. Consider system (1.2) with

p(t, x, y, x(t - r(t)), y(t - r(t)), z) ≡ 0, f(x), f'(x), g(y), g'(y), h(y)

continuous in their respective arguments. Suppose further that

(i) for some a > 0, ₀ > 0, h(y) > a+ ₀for all y;

(ii) for some b > 0, > b for all y ≠ 0;

(iii) for some c₀, > c₀for all x ≠ 0;

(iv) for some c > 0,f'(x) < c for all x, where ab - c > 0;

(v) for some constants L, M,|f'(x)| < L,|g'(y)| < M, for all x, y.

Then the zero solution of (1.2) is asymptotically stable, provided that

Proof. Using the equivalent system form (1.2), our main tool is the following Lyapunov functional V(x_t, y_t, z_t) defined as

where λ and δ are positive constants which will be determined later.

The Lyapunov functional V = V(x_t, y_t, z_t) defined in (2.2) can be arranged in the form

On using (i), (ii), (iii) and (iv) of Theorem (2.1), we obtain

Since the integrals

are non-negative,

Thus, we can find a positive constant D₁, small enough such that

Next, our target is to show that V(x_t, y_t, z_t) satisfies the conditions of Lemma 2.2. First, by (1.2) and (2.2), we obtain

By (v) and using 2uυ < u² + υ², we obtain

since r'(t) < β,0 < β < 1.

If we choose λ = > 0, and δ = > 0, and using (i), (ii), (iv) and r(t) < γ, we obtain

choosing

we have

Finally, it follows that

V(x_t,y_t,z_t) ≡ 0 if and only if y_t = z_t = 0, V(Φ) < 0 for Φ ≠ 0 and V(Φ) > u(|Φ(0)|) > 0. Thus, in view of (2.3), (2.4) and the last discussion, it is seen that all the conditions of Lemma 2.2 are satisfied. This shows that the trivial solution of Eq. (1.1) is asymptotically stable. Hence the proof of Theorem 2.1 is complete.

Remark 2.1. If h(x') = a in (1.1), then Theorem 2.1 reduces to Theorem 1 of [13] and a result of [1].

Remark 2.2. If h(x') = a, f(x(t - r(t))) = cx(t) in (1.1), then Theorem 2.1 reduces to Theorem 2 of [15].

Example 1.1. Consider the third order nonlinear delay differential equation

or its equivalent system form

where we suppose that 0 < r(t) < γ, r'(t) < β, β and γ are positive constants, γ will be determined later, t ∈ [0, ∞). It is obvious that

for all

1 < y² + y + 2 for al y.

Our main tool is the Lyapunov functional

where λ and δ are some positive constants which will be determined later.

It is clear that the functional V(x_t, y_t, z_t) is positive definite. Hence it is evident from the terms contained in (2.7), that there exist sufficiently small positive constant δ_i,(i = 1, 2, 3) such that

where δ₄ = min{δ₁, δ₂, δ₃}.

Now, the time derivative of the functional V(x_t, y_t, z_t) in (2.7) with respect to the system (2.6) can be calculated as follows:

Making use of the fact that

and the inequality 2|u υ| < u²+ υ², we obtain the following inequalities for all terms contained in the inequality (2.8), respectively:

and

Gathering all these inequalities into (2.8), we have

Let us choose δ = and λ = . Then, it is easy to see that

Now, in view of (2.9), one can conclude for some positive constants ν and ρ that

provided

It is also easy to see that

V(x_t, y_t, z_t) ≡ 0 if and only if z_t = y_t = 0, V(Φ) < 0 for Φ ≠ 0 and V(Φ) > u(|Φ(0)|) > 0. Thus, all the conditions of Lemma 2.2 are satisfied. This shows that the trivial solution of (2.5) is globally asymptotically stable.

3 The boundedness of solutions

Now, we shall state and prove our main result on boundedness of (1.1) with p(t, x(t), x'(t), x(t - r(t)), x'(t - r(t)), x"(t)) ≠ 0.

Theorem 3.1 Let all the conditions of Theorem 2.1 be satisfied, in addition assume that there are positive constants H and H₁ such that the following conditions are satisfied for every x,y and z in

Ω: = {(x, y, z) ∈ ³:|x| < H₁,|y| < H₁,|z| < H₁,H₁ < H}.

(i) |p(t, x(t), y(t), x(t - r(t)), y(t - r(t)), z(t))| < q(t),

where max q(t) < ∞ and q ∈ L¹(0,∞) the space of integrable Lebesgue functions.

Then, there exists a finite positive constant K₁such that the solution x(t) of (1.1) defined by the initial functions

x(t) = Φ(t), x'(t) = Φ'(t), x"(t) = Φ"(t)

satisfies the inequalities

|x(t)| < K₁, |x'(t)| < K₁, |x"(t)| < K₁

for all t > t₀, where Φ ∈ ²([t₀-r, t₀],R), provided that

Proof. As in Theorem 2.1, the proof of this theorem also depends on the scaler differentiable Lyapunov functional V = V(x_t, y_t, z_t) defined in (2.2). Now, since p(t, x(t), y(t), x(t - r(t)), y(t - r(t)), z(t)) ≠ 0,in view of (2.2), (1.2) and (2.4), it can be easily followed that the derivative of the functional V(x_t, y_t, z_t) along (1.2) satisfies the following inequality,

Hence it follows that

for a constant D₂ > 0, where D₂ = max{1, a^-1}.

Making use of the inequalities |y| < 1 + y² and |z| < 1 + z², it is clear that

V(x_t, y_t, z_t) <D₂(2 + y² + z²)q(t).

By (2.3), we have

(x² + y² + z²) <V(x_t, y_t, z_t)

hence

V(x_t, y_t, z_t)< D₂(2 + V(x_t, y_t, z_t))q(t).

Now, integrating the last inequality from 0 to t, using the assumption q ∈ L(0, ∞) and Gronwall-Reid-Bellman inequality, we obtain

where K₂ > 0 is a constant, K₂ = (V(x₀, y₀, z₀)+2D₂A) exp (D₂A) and

A = q(s)ds.

Now, the inequalities (2.3) and (3.1) together yield that

x² + y² + z²<V(x_t, y_t, z_t) < K₃,

where K₃ = K₂. Thus, we conclude that

|x(t)| < K₃, |y(t)| < K₃, |z(t)| < K₃

for all t > t₀. That is

|x| < K₃, |x'(t)| < K₃, |x"(t)| < K₃

for all t > t₀.

The proof of the theorem is now complete.

Example 3.1. Consider the third order nonlinear delay differential equation

or its equivalent system form

Observe that

for all t ∈ ⁺, x, y, x(t - r(t)), y(t - r(t)), z and

q(s)ds =

ds = π < ∞, that is q ∈ L¹(0, ∞)

To show the boundedness of solutions we use as a main tool the Lyapunov functional (2.7). Now, in view of (2.10), the time derivative of the functional V(x_t, y_t, z_t) with respect to the system (3.3) can be revised as follows:

Making use of the fact

we get

Hence it is obvious that

Now, integrating (3.4) from 0 to t, using the fact ∈ L¹(0, ∞) and Gronwall-Reid-Bellman inequality, it can be easily concluded the boundedness of all solutions of (3.2).

Received: 05/XII/08.

Accepted: 03/I/10

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*

This research was supported by University of Antioquia Research Grant CODI through SUI No. IN10095CE.

Publication Dates

• Publication in this collection
  22 Nov 2010
• Date of issue
  2010

History

• Received
  05 Dec 2008
• Accepted
  03 Jan 2010