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7DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES № 3, 2007 Electronic Journal, reg. № P2375 at 07.03.97 ISSN 1817-2172
http://www.newa.ru/journal http://www.math.spbu.ru/user/diffiournal e-mail: jodiff@mail.ru
Ordinary differential equations
Stability and boundedness of solutions of nonlinear differential equations of third-order with delay
Cemil Tunç
Department of Mathematics, Faculty of Arts and Sciences Yüzüncü Yil University, 65080, Van -Turkey E-mail:cemtunc@yahoo.com
Abstract
In this paper we investigate stability and boundedness of solutions of some nonlinear differential equations of third order with delay. By constructing a Lyapunov functional, sufficient conditions for the stability and boundedness of solutions for equations considered are obtained.
Keywords: Stability, boundedness, Lyapunov functional, differential equations of third order with delay.
AMS (MOS) Subject Classification: 34K20. 1. Introduction
The results existing in the literature on the stability and boundedness of solutions of nonlinear differential equations of third order with bounded delay have been developed over the last several decades. After a literature survey about nonlinear equations of third order with bounded delay, one can conclude that there are not so many results on the stability and boundedness of solutions. Up to this moment, the investigations concerning stability and boundedness of solutions of nonlinear equations of third order with bounded delay have not been fully developed. Certainly, these results should be obtained to be able to benefit from the applications of the theory of stability and boundedness of solutions. At the same time, we should recognize that some significant theoretical results concerning the stability and boundedness of solutions of third order nonlinear differential equations with delay have been achieved, see for example the papers of Sadek [9], Tejumola and Tchegnani [10], Tun9 ([11], [12]), Zhu [14] and the references citied in these papers. It should be noted that, in 1969,
Palusinski et al. [8] applied an energy metric algorithm for the generation of a Lyapunov function for third order ordinary nonlinear differential equation of the form:
x " + a1 x" + f2( x') x ' + a3 x = 0. They found some conditions for the stability of zero solution of this equation as follows:
a1 > 0, f2 (x') > a3 > 0 .
In this paper we are concerned with the third order ordinary nonlinear delay differential equations of the type
x (t) + aj x'(t) + f2( x '(t - r (t)) + a3 x(t) = p(t, x(t), x '(t), x(t - r (t)), x '(t - r (t)), x'(t)) (1) or its equivalent system
x ' (t )= y(t), y'(t )= z (t),
t
z'(t)= - az(t) - f2(y(t)) - a3x(t)+ J f2(y(s))z(s)ds
t-r (t)
+ p(t, x(t), y(t), x(t - r (t)), y(t - r (t)), z(t)) , (2)
where r is a bounded delay, 0 < r(t) < y, r'(t) < fi, 0 < fi < 1, fi and y are some positive constants, Y which will be determined later; a1 and a3 are some positive constants; the functions f2 and p depend only on the arguments displayed explicitly and the primes in equation (1) denote differentiation with respect to t. It is principally assumed that f2(0) = 0 and the functions f2 and p are continuous
for all values their respective arguments on ^ and x^5, = (0, ro), respectively. This fact guarantees the existence of the solution of delay differential equation (1). Besides, it is supposed that
the derivative f2'(y) = —- exists and is continuous. In addition, it is also assumed that the functions
dy
f2(y(t - r(t))) and p(t, x(t), y(t), x(t - r(t)),y(t - r(t)), z(t)) satisfy a Lipschitz condition in x(t), y(t), x(t - r(t)) , y(t - r(t)) and z(t); throughout the paper x(t), y(t) and z(t) are, respectively, abbreviated as x, y and z . Then the solution is unique (See [2, pp.14].)
The motivation for the present work has been inspired basically by the paper of Palusinski et al. [8] and the papers mentioned above. Our aim here is to discuss the result verified by Palusinski et al. [8] on the stability of the solutions to the equation (1) for the stability and boundedness of solutions of this equation in the case p = 0 and p ^ 0, respectively.
2. Preliminaries
In order to reach our main result, we give some important basic information for the general non-autonomous delay differential system (see also Burton [1], El'sgol'ts [2], El'sgol'ts and Norkin [3], Hale [4], Kolmanovskii and Myshkis [5], Kolmanovskii and Nosov [6], Krasovskii [7] and Yoshizawa [13].
Now, we consider the general non-autonomous delay differential system
x = f (t, xt), xt (0) = x(t + 0), - r <0< 0, t > 0, (3)
where f : [0, &)x CH ^ ^n is a continuous mapping, f (t,0) = 0, and we suppose that f takes closed bounded sets into bounded sets of ^n. Here (c, ||. ||) is the Banach space of continuous function p: [- r, 0]^^n with supremum norm, r > 0, CH is the open H -ball in C; CH := {p e (c[- r,0], ^n): (( < H}. Standard existence theory, see Burton [1, pp.312], shows that if pe CH and t > 0, then there is at least one continuous solution x(t,10,p) such that on [t0, t0 + a) satisfying equation (3) for t > 10, xt (s, t,p) = ( (s) and a is a positive constant. If there is a closed subset B e CH such that the solution remains in B, then a = &. Further, the symbol I. I will denote
the norm in ^n with x = max1£i£n|x^ .
Definition 1: (See [1, pp.223].) A continuous function W : [0, &) ^ [0, &) with W(0) = 0, W(s) > 0 if s > 0, and W strictly increasing is a wedge. (We denote wedges by W or Wi, where i an integer.)
Definition 2: (See [1, pp. 260].) Let V(t,p) be a continuous functional defined for t > 0, pe CH . The derivative of V along solutions of (3) will be denoted by V(3) and is defined by the following relation
(>(t,P) = limsup V (t + xt +hfc()) - V xt0'p)),
h^0 h
where x(t0,p) is the solution of (3) with xt (t0,p) = p.
Definition 3: (See [13, pp.184].) A function x(t0,p)is said to be a solution of (3) with the initial condition p e CH at t = 10, 10 > 0, if there is a constant A > 0 such that x(t0,p) is a function from [[0 - h, t0 + A] into ^n with the properties:
(i) xt (t0, p) e CH for t0 < t < 10 + A,
(ii) xt0(t 0,p) = p,
(iii) x(t0,p) satisfies (3) for 10 < t < t0 + A .
Theorem 1: (See [13, pp.184].) If f (t,p) in (3) is continuous in t, p, for every p e CH , Hx < H, and t0, 0 < t0 < c, where c is a positive constant, then there exist a solution of (3) with initial value p at t = 10, and this solution has a continuous derivative for t > t0.
For the general autonomous delay differential system x = f (xt), which is a special case of (3), the following lemma is given.
Proposition: (See [7].) Suppose f (0) = 0 . Let V be a continuous functional defined on CH = C with V(0) = 0, and let u(s) be a function, non-negative and continuous for 0 < s < & , u(s) ^ & as u ^ & such that for all p e C
(i) u(|((0)|) < V((), V(() > 0,
(ii) V(3)(p) < 0 for (* 0.
Then all solutions of x = f (xt) approach zero as t ^ ro and the origin is globally asymptotically stable.
Note that CH = C when H = ro ; and that the set R of p in C for which V(3)(p) = 0 has a largest invariant set M = {0} by the condition V(3) (p) < 0 for p * 0 .
3. Main results
First for the case p(t, x(t), y(t), x(t - r(t)), y(t - r(t)), z(t)) = 0 the following result is established.
Theorem 2: In addition to the basic assumptions imposed on the functions f 2 and p that appeared in equation (1) , we assume that there are positive constants a1, a2, a3,s0, L and / such that the following conditions are satisfied
a1a2 -a3 > 0, f2(0) = 0, ^^-a2 > e0, (y * 0), and |f2'(y)| < L for all y .
y
Then for sufficiently small y the zero solution of (1) is globally asymptotically stable provided that
2e0 2(a1a2 - a3) I
Y < min<
L a2 L + 2/
Proof: The proof of this theorem depends on a scalar differentiable Lyapunov functional. V = V(xt, yt, zt). The idea of Lyapunov's method is to impose some conditions on the functional V and
its time derivative —V (xt, yt, zt) which both imply the stability of the zero solution of equation (1). dt
We introduce the Lyapunov's functional V = V(xt,yt,zt):
11 y
V(xt,yt,zt) =—a32x2 + a2a3xy +—a2z2 + a3yz + a2 Jf2(^)d^
2 2 J
^ ^ 0
1 0 t +—a1a3 y2 + u J J z2 (d)d6ds, (4)
2 1 3
-r(t) t+ s
where a1, a2, a3 and / are some positive constants and the constant / which will be determined later in the proof. Now, the Lyapunov functional V = V (xt, yt, zt) defined in (4) can be rearranged in the form:
1
V (xt, yt, zt)=-a3
f a2 ^ x + — y
. a3 j
1
+ a2 2 2
f a3 V z +— y
a2 j
+
_ 2 _ f2(^)
1 3 2 I 2
a
2
£
V (
£d£ + uJ J z \6)d8ds. (5)
_r(t) t+s
/ / \
In view of the assumption —-> a + , it is clear that
y
2 a^ f2(£)
a a _ a22 —- + a 2
a
2
£
£d£>J
y
y
a1a3 _ a2 _ — + a2 (a2 +s0) 1 3 2 a 2 2 0
2
a
1 3 a 2 0
22 2 3 3 \ 2 ^ 0
2a,,
y2 > 0.
Hence, it is evident, from the terms contained in (5), that there exist sufficiently small positive constants Di, (i = 1, 2, 3), such that
0t
V > Djx2 + D2y2 + D3z2 + j J Jz2 (d)d6ds
_r(t) t+s
> D4 (x2 + y2 + z2) , (6)
0t
since the integral u J Jz2(6)d6ds is non-negative, whereD4 = min{D1, D2, D3}.
_r(t) t+ s
Now, calculating the time derivative of the functional V(xt, yt, zt) with respect to t along a solution (x(t), y(t), z(t)) of the system (2), we have
d_ dt
V(xt, yt, zt )= _(a1a2 _ a3 )z 2 _
a
f2 ( y)
_a
y ;
y
+ a2 z J f2( y( s)) z(s)ds + a3 y J f2( y( s)) z(s)ds
t_r (t) t_r(t)
t
+ jr(t)z2- j(1 _ r '(t)) J z2 (s)ds
t _r (t)
= _(a2 _ a3 _jr (t))z2 _ a3
^ f2 (y) ^ v y ;
_ a2
y
+ a2 z J f2( y( s)) z(s)ds + a3 y J f2( y( s)) z(s)ds
t _r (t)
t _r (t)
- u(1 _ r '(t)) J z2 (s)ds.
(7)
t _r (t)
By noting the assumption |f2'(y)| < L and the inequality 2|ab| < a2 + b2, we obtain the following relations:
and
a2 z J f2'( y (s)) z(s)ds < a2L r (t)z 2(t)+ ^ J z 2(s)ds
t-r (t )
t-r (t )
tt a3 y Jf'(y(s))z(s)ds < aa- r (t ) y 2(t )+ -3- Jz2(s)ds .
t-r(t)
t-r(t)
r / \
Hence, using the assumptions 2
y
- a2 >s0, 0 < r (t) <y, r'(t ) <P, 0 <P< 1, and the above
discussion, we get from (7) that
d_ dt
V(Xt, yt, z) <- a3
f2 ( y) . y
A
- a,,
L ^
--Jr (t )
y
(aia2 - a3 a2 L + ^ (t )
+
(a2 + a3)) Jz2(s)ds -¡u(1 - r'(t)) Jz2 (s)ds
t-r(t)
t-r(t)
< - a.
L
s0 -~ZY
y
(a1a2 - a3 ))
a2 L + 2^
Y
+
(a2 + a3)L - ^(1 - P) 2
L
Jz2(s)ds .
t-r(t)
(8)
(a + a )L
If we choose n = —2-3—, then we have from (8) that
1 -P
d
—V(xt, yt, zt) < - a3 dt
L
s0 --Y
y
(a1a2 - a3 )- ^
a2 L + 2^
Y
(9)
Therefore, in view of (9), one can conclude for some positive constants a and p that
d_ dt
V(xt,yt,zt) <-ay2-Pzz
(10)
provided
Y < min
2s0 2(a1a2 - a3) L a2 L + 2^
Finally, it is followed that dV(xt,yt, zt) = 0 if and only if yt = zt =0, dV(p) < 0 for p^ 0 and
dt t t t t t dt
V(p) > u(|p(0)|) > 0. Thus, in view (6), (10) and the last discussion, it is seen that all the conditions of
2
z
2
z
2
z
the above Proposition are satisfied. This shows that the trivial solution of equation (1) is globally asymptotically stable. Hence, the proof of Theorem 2 is complete.
Example 1: Consider the third order nonlinear delay differential equation
x (t) + 3x "(t) + 4 x'(t _ r (t)) + sin x '(t _ r (t)) + 2 x(t) = 0
(11)
Equation (11) is equivalent to the system x ' (t )= y(t), y'(t )= z (t),
z'(t)= _3z(t) _4y(t)_siny(t) _2x(t) + J(4 + cosy(s))z(s)ds
(12)
t _r (t)
where we suppose that 0 < r(t) < y, r'(t) < fi, 0 < fi < 1, fi and y are positive constants, y which will be determined later, t e [0, &). It is obvious that
3 < 4 +
sin y
y
for all y, (y ^ 0).
Our main tool is the Lyapunov functional
V(xt, yt, zt) = 2Vx + yj + ^ (z + 2 y )2 + J
0 t
+ j J J z2 (6)d0ds,
5 +
sin£
(13)
~r(t) t+ s
where u is a positive constant 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 (13), that there exist sufficiently small positive constants Di, (i = 5, 6, 7), such that
0t
V(xt, yt, zt) > D5 x 2 + D6y 2 + D7 z 2 + j J J z2 (6)d0ds
_r(t) t+ s
t t t 5 6
> D5 x 2+ D6 y 2+ D7 z2
> D8 (x2 + y2 + z2 2 ,
where D8 = min{D5, D6, D7}.
Now, the time derivative of the functional V (xt, yt, zt) in (13) with respect to the system (12) can be calculated as follows:
d
—V(xt,yt,zt) = _(1 _jr(t))z2 _2
f
dt
3 +
sin y
A
y J
y
t-r (t)
+ z J (4 + cos y(s))z(s)ds + 2y J (4 + cos y(s))z(s)ds
t-r(t)
t
- /(1 - r'(t)) J z2 (s)ds.
(14)
t-r(t)
Making use of the facts |4 + cos y| < 5
sin y
y
< 1, 0 < r (t) < y , r '(t) <fi, 0 <fi< 1 and the
inequality 2|uv| < u2 + v2, we obtain the following inequalities for all terms contained in the equality (14), respectively:
-(1 -/r (t))z2 <-(1 -uy)z 2,
(
- 2
f sin y ) 3 +--
y J
y2 <-4y2,
5
z J(4 + cosy(s))z(s)ds < - r(t)z2(t)+- Jz2(s)ds
t-r(t) 2 2
t-r(t)
<
5Y z2(t)+2 Jz2(s)ds,
t-r(t)
t-r(t)
2y J(4 + cosy(s))z(s)ds < 5 r(t)y2(t) + 5 Jz2(s)ds
t-r(t)
t
< 5y y2(t) + 5 Jz2 (s)ds
t-r(t)
and
- /(1 - r' (t)) J z2 (s)ds < - /(1 -fi) J z2 (s)ds.
t-r(t)
t-r(t)
Gathering all of these inequalities into (14), we have
j-V(xt, yt, zt) <- 2(2 - f) y2-^1 -(/ + \ ^Y^ z2- /-fi) -125'J Jz 2(s)ds.
t-r(t)
Let us choose / =
15
2(1 -fi)
. Then, it easy to see that
d
5Y
f
dV(Xt,yt) <- 21 2y2-h-I/ + 2 JY
5
A
(15)
Now, in view of (15), one can conclude for some positive constants a and p that
2
z
dV(xt,yt,Zt) < — ay2— pz2. (16)
dt
provided
.J 2 41
Y < mini- — >.
[ 2/ + 5 5 J
It is also easy to see that dV(xt,yt, zt) = 0 if and only if zt = xt =0, dV(<) < 0 for <<* 0 and
dt t t t t t dt
V(<) > u(|<(0)|) > 0. Thus all the conditions of the above Proposition are satisfied. This shows that the trivial solution of equation (11) is globally asymptotically stable.
For the case p(t, x(t), y(t), x(t — r(t)), y(t — r(t)), z(t)) * 0 the following result is established.
Theorem 3: In addition to the basic assumptions imposed on the functions f2 and p that appeared in equation (1) , we assume that there are positive constants a1, a2, a3,s0, L, /, H and H1 such that the following conditions are satisfied for every x, y and z in
Q := {(x,y, z) e^3 : |x| < H1, |y| < H1, |z| < H1, H1 < H}:
(i) a,a2 — a3 > 0, f2(0) = 0, ^^ — a2 > (y * 0), and |f2'(y)| < L .
y
(ii) |p(t, x(t), y(t), x(t — r (t)), y(t — r (t)), z(t))| < q(t),
where max q(t) < ro and q e L (0, , L (0, ro) is space of integrable Lebesgue functions.
Then, there exists a finite positive constant K such that the solution x(t) of equation (1) defined by the initial functions
x(t) = <(t), x'(t) = <(t), x'(t) = < (t)
satisfies the inequalities
|x(t) < K , |x' (t) < K , |x'(t) < K
for all t > 10 , where <pe C2 ([[0 — r, t0 ], ,provided that
. J2s0 2(a,a2 — a3) 1 Y < mini—-, -— ^.
[ L a2 L + 2/ J
Proof: As in the Theorem 2, the proof of this theorem also depends on the scalar differentiable Lyapunov functional V = V (xt, yt, zt), which is defined in (4). Now, since p(t,x(t),y(t),x(t — r(t)),y(t — r(t)),z(t)) * 0, in view of (4), (2) and (10), it can be easily followed that the time derivative of the functional V (xt, yt, zt) satisfies the following inequality:
—V^,yt,zt) < -ay2-pz2 + ay + a2z|.|p(t,x(t),y(t),x(t-r(t)),y(t-r(t)),y(t))|
2 ~2 ' 'a3y + a2z|<
< - ay2 - pz2 + |a3y + a2z|q(t).
Hence, it follows that
—V(xt,yt,zt) < - ay2 -pz2 + D9(y| + |z|)q(t)
D9 (( + |z| )(t)
<- 9
for a constant D9 > 0, where D9 = max{a2, a3}.
Making use of the inequalities lyl < 1 + y2 and Izl < 1 + z2, it is clear that
—V (xt, yt, zt) < D9 (2 + y2 + z2) q(t).
By (6), we have
(y2 + z2)< D- V(xt,yt,zt).
Hence
4-V(xt,yt, zt) <D9(2 + D-V(xt,yt, zt))q(t) .
Now, integrating the last inequality from 0 to t, using the assumption q e L:(0, ro) and Gronwall-Reid-Bellman inequality, we obtain
t
V (xt, yt, zt) < V (x0, y 0, z 0) + 2 D9 A + D9 D- J (V (xs, ys, zs ))(s)ds
0
( t
<
(((x0, y0, z0) + 2D9 A) exp D9D4-1 J q(s)ds I
I 0 J
<(, y 0, z 0) + 2D9A) exp^D- a)= K <ro, (17)
ro
where K1 > 0 is a constant,K1= (((x0,y0,z0) + 2D9A) exp(D9D4-Ja) and A = Jq(s)ds .
0
Now, the inequalities (6) and (17) together yield that
x 2(t) + y 2(t) + z 2(t) < D- V (xt, yt, zt) < K, where K = K1D-. Thus, we conclude that
|x(t)| < K , |y(t) < K , |z(t) < K
for all t > 10. That is,
|x(t) < K , |x'(t) < K , |x"(t) < K
for all t > t0 .
The proof of the theorem is now complete.
Example 2: Consider the third order nonlinear delay differential equation
xm(t) + 3x"(t) + 4 x'(t - r (t)) + sin x'(t - r (t)) + 2 x(t)
=_2_
1 +12 + x2 (t) + x '2 (t) + x 2(t - r (t)) + x ' 2(t - r (t)) + x ' 2(t)
Clearly, equation (18) is equivalent to the system x ' (t )= y(t), y'(t )= z (t),
t
z'(t)= -3z(t) - 4y(t) - siny(t) - 2x(t) + J(4 + cosy(t))z(s)ds
t-r (t)
+ -
2
1 +12 + x2 (t) + y 2(t) + x 2(t - r (t)) + y 2(t - r (t)) + z 2(t)
22
(18)
(19)
Observe that
2
1 +12 + x (t) + y2(t) + x2(t-r(t)) + y2(t-r(t)) + z2(t) 1 + t
2
<--r = q(t)
for all t , x(t), y(t), x(t - r(t)), y(t - r(t)), z(t), and
Jq(s)ds = J1 2 2 ds = n < ro , that is, q e L (0, ro) .
To show the boundedness of the solutions we use as a main tool the Lyapunov functional in (13). Now, in view of (16), the time derivative of the functional V (xt, yt, zt) with respect to the system (19) can be revised as follows:
—v(X, yt, zt )=-ay2-pz 2+ dt
4 y + 2 z
1 +12 + x 2(t) + y 2(t) + x 2(t - r (t)) + y 2(t - r (t)) + z 2(t)
(20)
Making use of the fact
1
<.
1
we get
1 +12 + x2(t) + y2(t) + x2(t - r(t)) + y2(t -r(t)) + z2(t) 1 + t2
d 2 2 2I2 y + z|
-V(xt, yt, zt) < -ay2 -pz dt 1 +1
Hence, it is obvious that
aj
U.i
dT„ . 2|2y + z| < 4(y| + \z\)
—V (xt, yt, Zt ) < 1 1 < dt 1 +1 1 +1
(y2 + z
4(2 + y2 + z2) < 8 +4(y2 + z2)
---<-+ -
1 +12 1 +12 1 +12 8 4 D
< 7+V + ^V(Xt,yt,Zt). (21)
1 +12 1 +12
Now, integrating (21) from 0 to i, using the fact -- e Li(0, œ) and Gronwall-Reid-Bellman
1 +1
inequality, it can be easily concluded the boundedness of all solutions of equation (18).
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