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.
stability; boundedness; Lyapunov functional; differential equations of third-order with delay
Stability and boundedness of solutions of a kind of third-order delay differential equations* * This research was supported by University of Antioquia Research Grant CODI through SUI No. IN10095CE.
A.U. AfuwapeI; M.O. OmeikeII
IDepartmento de Matemáticas, Universidad de Antioquia Calle 67, No. 53-108, Medellín AA 1226, Colombia. E-mail: aafuwape@yahoo.co.uk
IIDepartment of Mathematics, University of Agriculture, Abeokuta, Nigeria. E-mail: moomeike@yahoo.com
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 : CH →n is a continuous mapping, f(0) = 0, CH := {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 {tn}, tn → ∞, as n → ∞, with ║xtn(Φ) - ψ║→ 0 as n → ∞ where xtn(Φ) = x(tn + θ, 0, Φ) for -r < θ < 0.
Definition 2.2 (See [17]). A set Q ⊂ CH is an invariant set if for any Φ ∈ Q, the solution of (2.1), x(t, 0, Φ), is defined on [0, ∞), and xt(Φ) ∈ Q for t ∈ [0,∞).
Lemma 2.1 (See [13]). If Φ ∈ CH is such that the solution xt(θ) of (2.1) with x0(Φ) = Φ is defined on [0, ∞) and ║xt(Φ)║ < H1 < H for t ∈ [0,∞), then ω(Φ) is a nonempty, compact, invariant set and
Lemma 2.2 (See [13]). Let V(Φ): CH → 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 < CH.
The the zero solution of (2.1) is uniformly stable. If we define Z = {Φ ∈ CH: 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(xt, yt, zt) defined as
where λ and δ are positive constants which will be determined later.
The Lyapunov functional V = V(xt, yt, zt) 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(xt, yt, zt) 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(xt,yt,zt) ≡ 0 if and only if yt = zt = 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(xt, yt, zt) is positive definite. Hence it is evident from the terms contained in (2.7), that there exist sufficiently small positive constant δi,(i = 1, 2, 3) such that
where δ4 = min{δ1, δ2, δ3}.
Now, the time derivative of the functional V(xt, yt, zt) 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(xt, yt, zt) ≡ 0 if and only if zt = yt = 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
satisfies the inequalities
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(xt, yt, zt) 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(xt, yt, zt) 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(xt, yt, zt) <D2(2 + y2 + z2)q(t).
By (2.3), we have
(x2 + y2 + z2) <V(xt, yt, zt)
hence
V(xt, yt, zt)< D2(2 + V(xt, yt, zt))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(xt, yt, zt) < K3,
where K3 = K2. Thus, we conclude that
for all t > t0. That is
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(xt, yt, zt) 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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Publication Dates
-
Publication in this collection
22 Nov 2010 -
Date of issue
2010
History
-
Received
05 Dec 2008 -
Accepted
03 Jan 2010