CC BY
7DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N 3, 2008 Electronic Journal, reg. N P2375 at 07.03.97 ISSN 1817-2172
http://www.neva.ru/journal http://www.math.spbu.ru/diffjournal/ e-mail: [email protected]
New results on the stability and boundedness of solutions of certain third order nonlinear vector differential equations
Mathew Omeike1
Abstract
We investigate in this paper, the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. Our results revise and improve those results obtained by Tunc and Ates [Tunc C., Ates, M., Stability and boundedness results for solutions of certain third order nonlinear vector differential equations,Nonlinear Dynamics 45 (2006); 273-281].
Keywords: boundedness, differential equation of third order, Lyapunov function, stability
2000 Mathematics Subject Classification. 34C10, 34C11.
1. Introduction
1 Department of Mathematics, University of Agriculture, Abeokuta, Nigeria. E-mail: [email protected]
Recently, Tunc and Ates [11] considered the differential equation
X +F(X, X, X)X + B(t)X + H(X) = P(t, X, X, X), (1.1) or the equivalent system form
X = Y
Y = Z (1.2)
Z = —F(X,Y,Z)Z — B(t)Y — H(X) + P(t,X,Y,Z) where F and B are n x n-symmetric continuous matrix functions, H and P are continuous vector functions, t £ [0, to) and X £ lRn,lRn denotes the real n-dimensional Euclidean space IR x IR x • • • x IR (n factors). It is also assumed that the Jacobian matrix Jh(X) and the matrix B(t) exist, and are symmetric and continuous. Hence the following theorems were proved. In the case P = 0, the following result was established.
Theorem A (Tunc and Ates[11]). In addition to the fundamental assumptions on F, B and H suppose that:
(i) there exists an n x n-real continuous operator A(X, Y) for any vectors X, Y in lRn such that
H (X) = H (Y) + A(X, Y )(X — Y), (H (0) = 0), whose eigenvalues \(A(X,Y)), (i = 1, 2, • • • ,n), satisfy
0 <6h < Xi(A(X,Y)) < Ah for fixed constants 5h and Ah;
(ii) there exists a real n x n-constant symmetric matrix A such that the matrices A, B(t), B(t), (F(X, Y, Z) — A) have positive eigenvalues and pairwise commute with themselves as well as with operator A(X, Y) for any X, Y in lRn, and that
da = min {Ai(A),Ai(F(X,Y,Z))}, Aa = max {Ai(A), Ai(F (X, Y, Z))},
1<i<n 1<i<n
db = min (Ai(B(t))), Aa = max (Ai(B(t)))
Ah < kdadb (where k is a positive constant),
0 < Ai (F(X,Y,Z) - A) and e = max |Ai(B(t))|, (i = 1, 2, ••• ,n),
where e < - min i f ^ , f—d^d^^ , d^, 1
- 2 |V4Ab + 4) , V6Aa + 7J , 4 ,
Then, the zero solution of system (1.2) is asymptotically stable. In the case P = 0, the following result was established.
Theorem B (Tunc and Ates[11]). Let all the conditions of Theorem A be satisfied, and in addition we assume that there exist a finite constant K > 0 and a non-negative and continuous function 0 = 0(t) such that the vector P satisfies
||P(t,X,Y,Z)|| < 0(t) + 0(t)(||X|| + ||Y|| + ||Z||),
where 0(s)ds < K < to for all t > 0. Then the exists a constant D > 0 0
such that any solution (X(t), Y(t),Z(t)) of (1.2) determined by
X (0) = Xo, Y (0) = Yo, Z (0) = Zo
satisfies
||X|| < D, ||Y|| < D, ||Z|| < D
for all t > 0.
These are very interesting results obtained by the authors [11]. However, these results contain certain conditions which are not necessary for the stability and boundedness of (1.2). Our aim in this paper is to further study the stability (when P = 0) and boundedness (when P = 0) of solutions of Eq. (1.1). In the
next section, we establish criteria for the stability of the zero solution of Eq. (1.1) when P = 0, and the boundedness of solutions of Eq. (1.1) when P = 0, which extend and improve Theorems A and B Respectively. An effective method for studying the stability and boundedness of nonlinear differential equations is the second method of Lyapunov (See [1-11]). 2. Statement of the results
Let H(0) = 0 and Jh = Jh(X) denote the Jacobian matrix (dhi/dxj) derived from the vector H(X) in (1.1). Our first theorem is given for the case in which P = 0.
Theorem 1. Assume that F(X,Y,Z),B(t),B(t) and Jh(X) are symmetric for all X, Y, Z in lRn and t £ [0, to), and let 5a, 5b, Sh, Aa, Ab, Ah and e be positive constants.
(i) The matrices F(X,Y,Z),B(t),B(t) and Jh(X) are associative and commute pairwise. The eigenvalues Xi(F(X,Y,Z)),Ai(B(t)), Ai(B(t)), and Xi (Jh (X)) (i = 1, 2, ••• ,n) of F (X, Y, Z ),B (t),B(t) and Jh(X) satisfy
0 < 5a < Ai(F(X, Y, Z)) < Aa (2.1)
0 < 56 < Ai(B(t)) < A6 (2.2)
0 < 5h < Ai(Jh(X)) < Ah (2.3)
e = max |Ai(B(t))| (2.4)
with 5a5b — Ah > e.
Then, the zero solution of system (1.2) is asymptotically stable.
In the case P = 0 we have the following result. Theorem 2. Let all the conditions of Theorem 1 be satisfied, and in addition we assume that there exists a finite constant K > 0 and a non-negative and continuous function 0 = 0(t) such that the vector P satisfies
||P(t,X,Y,Z)|| < 0(t) + 0(t)(||X|| + ||Y|| + ||Z||), (2.5)
where 0(s)ds < K < to for all t > 0. Then there exists a constant D > 0 Jo
such that any solution (X(t),Y(t),Z(t)) of (1.2) determined by
X (0) = Xo, Y (0) = Yo, Z (0) = Zo
satisfies
||X(t)|| < D, ||Y(t)|| < D, ||Z(t)|| < D
for all t > 0.
3. Some Preliminaries
The following results will be basic to the proofs of Theorems 1 and 2. We do not give the proofs since they are found in [1-7,9,10,11]. Lemma 1. Let D be a real symmetric n x n matrix, then for any X in lRn, we have
Ad||X||2 > (DX,X) > 5d||X||2,
where 5d, Ad are the least and greatest eigenvalues of D, respectively. Lemma 2. Let Q,D be any two real n x n commuting symmetric matrices. Then
(i) The eigenvalues Ai(QD)(i = 1, 2, • • • ,n) of the product matrix QD are real and satisfy
max Aj(Q)Ak(D) > Ai(QD) > min Aj(Q)Ak(D).
1<j,k<n 1<j,k<n
(ii) The eigenvalues Ai(Q + D) (i = 1, 2, • • • ,n) of the sum of matrices Q and D are real and satisfy
{
where Aj (Q) and Ak(D) are, respectively, the eigenvalues of Q and D.
max Aj(Q) + max Ak(D) > > Ai(Q+D) > < min Aj(Q) + min Ak(D)
l1<j<n 1<k<n I I1<j<n 1<k<n
4. The Function V
Our main tool in the proof of our result is the Lyapunov function V = V(t,X, Y,Z) defined by
2V = 2da i (H (jX ),X >dj + 5a[ (jF (X,jY,Z )Y,Y>dj
(4.1)
+affdb(X, X> + (Z, Z> + (B(t)Y, Y> + 2affda(X, Y>
+2aff(X, Z> + 2da(Y, Z> + 2(Y, H(X)> - aff(Y, Y> where ff = dadb and a satisfies
.n dh ff - dh - e | (42)
a< min da , ff(Aa - da) ' ff[da + d-1(Ab - db)2] J (4.2)
The function V above can be written thus,
2V = ||Z + daY + affX||2 + da/ (jF(X, jY, Z)Y,Y>dj - d;2(Y,Y>
+ (B(t)Y, Y> - ffd-1 (Y, Y> + aff(db - aff)(X,X>
+2da / (H(jX),X>dj - ff-1da|H(X)| 0
(4.3)
+ff||da 2Y + ff-1dlH(X)||2 The following result is immediate from (4.3).
Lemma 3. Assume that all the hypotheses on matrices F(X, Y, Z), B(t) and vector H(X) in Theorems 1 and 2 are satisfied. Then there exists a positive constant d1 such that
V(t,X, Y,Z) > d1(||X||2 + ||Y||2 + ||Z||2), (4.4)
for arbitrary X, Y, Z in
Proof of Lemma 3. We shall make use of the result:
H(X)= f Jh(j1X)Xdj1 (4.5)
0
2
for arbitrary X in iRn, which follows from integrating the equality
d
—H (j1 X) = Jh(j1X )X
dj 1
with respect to j1 and then using the fact that H(0) = 0. By (4.5), we can rewrite (4.3) thus,
2V = ||Z + daY + affX ||2 + da / a ({F(X, aY, Z) - da/}Y,Y>da
0
+ ({B (t) - ffda-1/}Y,Y > + aff (db - aff )(X,X >
+2da / / J1({/ - Jh(j1X)ff-1}Jh(j1J2X)X,X>dj1dj2 00
+ff ||d- 2 Y + ff-1da2 H (X )||2. By (2.1), (2.2) and (2.3) of Theorem 1, and Lemma 1, we have that
2V > ||Z + daY + affX ||2 + aff (db - aff )||X ||2 + 2da(1 - Ahff-1)dh||X ||2.
By (2.5) and (4.2), we have that there is a constant d2 > 0 such that
2V > ||Z + daY + affX||2 + d2||X||2.
Hence we can find a positive number d1 small enough such that (4.4) holds. This completes the proof of Lemma 3.
The following lemma is instrumental in the proof of the next result. Lemma 4. Subject to earlier conditions on F and H the following are true.
d r1
(i) dt Jo (aF(X, JY,Z)Y,Y>da = (F(X, Y, Z)Y,Z>,
d r1
(ii) -J (H(jX),X>da = (H(X),Y>.
Proof. See [4,5,9].
Lemma 5. Assume that all the conditions of Theorem 1 are satisfied. Then
v(t) < 0 for all t > 0 (4.6)
and especially d
v(t) = —V (t, X, Y,Z ) < 0 provided ||X ||2 + || Y y2 + ||Z ||2 > 0 (4.7) dt
Proof of Lemma 5. A straightforward calculation from (4.1), (1.2) and Lemma 4 give
d
v = —V(t, X(t),Y(t),Z(t)) = - Vi - V - V3
where
1 f1
Vi = - - (X, Jh(aX )X )da 2 Jo
-(Y, {6aB(t) - B(t) - {[Jh(X) + aßöa]}Y} -(Z, {F(X, Y, Z) - öal}Z) 1 f1
V2 = - - aW {(Jh(vX )X,X ) +4(X, [B (t) - öbI]Y)}da 4o
1 f1
V3 = - -¡aß J {(Jh(aX)X,X) + 4(X, {F (X,Y,Z) - ^aI]}Z)}da.
Since Jh(X) is symmetric and positive definite, we have that (Jh(aX)X,X) + 4(X, [B(t) - 6bI]Y) = || j\X + 2J-1 [B(t) - öbI]Y||2 - ||2[B(t) - öbl] J-2Y||2 and
( Jh(aX)X, X) + 4(X, {F(X, Y, Z) - öaI}Z)
= || j\X + 2J-2[F(X,Y,Z) - öaI]Z||2 - ||2[F(X,Y,Z) - öaI] J-1Z||2. Using the fact that
i ||2[B(t) - öbI]Jh2Yfda = ^ (Jh-1[B(t) - öbI]Y, [B(t) - ö}jI]Y)da
oo
and
i 1
I ||2[F(X, Y, Z) - daZ||2da
0a
= 4 / (J-1[F(X, Y,Z) - da/]Z, [F(X, Y,Z) - da/]Z>da, 0
we have,
1 f1
v(t) < -1 aff / (X,Jh(aX)X>da 20
- / (Y, {daB(t) - Jh(X) - [B(t) + affda/] - aff J-1 [B(t) - db/]2}Y>da 0
- /1(Z, [F(X, Y,Z) - da/]{/ - aff J-1[F(X, Y,Z) - da/]}Z>da 0
< -2affdh|X||2
-{dadb - dh - e - affda - affd-1(Ab - db)2}||Y||2 -Y{1 - affd-1(Aa - da)}|Z||2 < -ds||X||2 - d4|Y||2 - ds|Z||2
where d3 = 1 affdh, d4 = dadb - dh - e - aff [da + d-1(Ab - db)2] and ds = 1 - affd-1(Aa - da).
By (4.2),d3,d4 and d5 are positive. This completes the proof.
Proof of Theorem 2. Consider the function V defined by (4.1). Then, under
the assumptions of Theorem 2 the conclusion of Lemma 3 can be obtained, that
is,
V > d1(||X||2 + ||Y||2 + ||Z||2) (4.8)
and since P(t,X, Y, Z) = 0, then the conclusion of Lemma 5 can be revised as follows
d
v = tV < (affX + daY + Z, P(t, X, Y, Z)>. dt
Next, by noting the assumption of Theorem 2 on PY, Z) and using Schwarz's inequality, we obtain
v < (affl|XII + da||Y|| + ||ZII) x ||P(t,X,Y,Z)||
< (aff||X|| + da||Y|| + ||Z||) x (0(t) + 0(t)(||X|| + ||Y|| + ||Z||))
< do(||X|| + ||Y|| + ||Z||) x (0(t) + 0(t)(||X|| + ||Y|| + ||Z||)) where d6 = maxjaff, da, 1}.
Hence, by using the inequalities
||X|| < 1 + ||X||2, ||Y|| < 1 + ||Y||2, ||Z|| < 1 + ||Z||2 and (4.8), we obtain
vv < dr^(t) + (4.9)
where d7 = 3d6 and d8 = 4d6d-1.
Integrating both sides of (4.9) from 0 to t(t > 0), leads to the inequality
v(t) - v(0) < d7 / 0(s)ds + 5% i v(s)0(s)ds. Jo Jo
On putting dg = v(0) + d7K, it follows that
v(t) < d9 + d8 /* v(s)0(s)ds. o
Gronwall-Bellman inequality yields
v(t) < dg exp ^d8 J 0(s)ds^ .
The proof of the theorem is now complete. 5. Remarks
i). Clearly, Theorems 1 and 2 are improvement and extension of Theorems A and B respectively. Particularly, from Theorems 1 and 2, we see that hypothesis (i) of Theorems A and B is not necessary since H(X) is assumed differentiable.
(ii). Also, from Theorems 1 and 2, it is clear that we do not need any symmetric matrix A (as assumed in Theorems A and B), thus the condition 0 <
Xi(F(X, Y, Z) — A) < (i = 1, 2,... ,n), is not necessary.
2
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