Asymptotic stability of nonlinear functional differential equations

Nonlinear Analysis, Theory, Methods & Applications, Vol. 28, No. 12, pp. 1997-2003, 1997 @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

Pergamon PII: S0362-546X(96)00029-6 ASYMPTOTIC

STABILTY OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

T. S E N G A D I R Department of Mathematics, Indian lnstitue of Science, Bangalore-560012, India (Received 31 May 1995; received in revisedform 20 November 1995; receivedfor publication 19 January 1996) Key words and phrases: Functional differentialequations, asymptotic stability.

1. INTRODUCTION In this paper, we study asymptotic stability of solutions of the following functional differential equation: x'(t) = a(t)x(t) + b ( t ) x ( p ( t ) ) + f ( t , x(t), x(p(t))) (1)

x(O) = xo.

Here a,b : [0, oo) ~ C, f : [0, oo) × C × C ~ C are continuous functions, x0 E C and p : [0, oo) ~ [0, oo) is a continuously differentiable function such that the following conditions hold: (i) limt_oo p ( t ) = oo (ii) p ( t ) < t , t E [0, co) (iii) 0 * limt_oo p' (t) < oo. The technical condition (iii) holds, for example, for the case of proportional delays. Our results can be generalised for equations with f i n i t e number of delays and also for systems. To achieve the latter, use of logarithmic norm (cf. [1]) of a matrix is essential. For the ease of presentation, however, we consider only the simplest case. Earlier, asymptotic stability of solutions of the neutral functional differential equation x' (t) = f ( t , x ( t ) ) + g(t, x ( p ( t ) ) + C ( t ) x ' ( p ( t ) )

(2),

x(O) = xo

where f , g : [0, co) × C" ~ cm are locally Lipschitz functions, C is a matrix valued continuous function, p : [0, oo) ~ [0, co) is a continuous function satisfying (i), (ii) above, and x0 c (m is investigated in [3]. In [2], global behaviour of solutions of functional differential equations of the form x' (t) = h(x(t), x(pt) ) x(O) = xo 1997

(3)

1998

T. SENGADIR

where h : C × C ~ (2 is a polynomial and p > 0 is studied using Dirichlet series expansions. Sufficient conditions for stability of solutions o f neutral functional differential equations are obtained in [4] and [5]. Consider Problem (3) with p ~ (0, 1). Assume that h is differentable and h(O, 0) -- 0. Let a and b be the values of partial derivatives of h with respect to the first and second variable respectively at (0, 0). T h e o r e m 6 of [2] states the following: If (a) there is Q > 0 such that any solution of (3) with Ix0[ < Q satisfies lim,_oo x ( t ) = 0 then (b) ~ ( a ) _< 0 and Ibt -< [a[ where ~ ( z ) denotes the real part of the complex n u m b e r z. As a special case of T h e o r e m 2.3 of this paper, we can show that if h is a polynomial then (c) ~ ( a ) < 0 and Ibl < - ~ ( a ) . is a sufficient condition for (a). Thus, if (c) holds, the restrictions imposed on the coefficients of higher degree terms by T h e o r e m 15 o f [2] are not necessary. Next we state two earlier results for future reference. T h e o r e m 1.1 is a special case of T h e o r e m 3 o f [3] and T h e o r e m 1.2 is a special case of T h e o r e m 4.8 of [4]. THEOREM 1.1 ([3]). Let p : [0, oo) ~ [0, oo) be a continuous function satisfying conditions (i) and (ii). Assume that a, b : [0, oo) ~ (2 are continuous functions. Further, suppose that the following hypotheses hold: (iv) there is 0 > 0 such that ~ ( a ( t ) ) + (1 + 0) {supT~[o.tllb('r)l} <- O, t ~ [0, oo)

(v) ~o ~ ( a ( s ) d s = - o o . Then, for arbitrary x0 E q2, the solution x of (1) with f - 0 satisfies lim x( t ) = O.

f~oo

THEOREM 1.2 [4]. Let p : [0, oo) ~ [0, co) be a continuous function satisfying the condition (ii). Assume that a , fl : [0, ~ ) ~ (2 are continuous functions such that the following condition holds: (vi) ~(~x(t)) + I/3(t)] < - y , for some y > 0.

t ~ [0, oo)

Further, suppose that g : [0, oo) x (2 x (2 ~

(2 is a continuous function satisfying the hypothesis

(vii) there exists c >_ 0 a n d / J > 0 such that I g ( t , x , y ) l <-- c (max{lxl, lyl}) l+u.

Then there exists a P > 0 such that ly01 < P implies that every local solution of the problem y' = a ( t ) y ( t ) + f l ( t ) y ( p ( t ) ) + g(t, y(t), y ( p ( t ) ) )

y(0) = Y0

(4)

can be continued as a solution to the whole of [0, ~ ) . Also, given ¢ > 0, there is a 6 > 0 such that ly01 < min{P, 6} ~ ly(t)l < ¢, t ~ [0, oo), where y is a solution of Problem (4).

A s y m p t o t i c stabilty

1999

2. M A I N R E S U L T S

The following theorem is an easy consequence o f T h e o r e m 1.2. THEOREM 2.1. L e t p : [0, 0o) ~ [0, co) be as in T h e o r e m 1.2. Assume that qo • [0, oo) ~ is a continuously differentiable function satisfying the following conditions:

(0, 0o)

(viii) limt_oo qg(t) = oo ~(t) < 0o (ix) suPt~[0.~) ¢p(p(t)) Further, suppose that a, b " [0, e~) ~ hypothesis holds: ~o'(t)

qo(t)

(x) ~ ( a ( t ) ) + - ~ y + ~ l b ( t ) D Moreover, let f " [0, co) × C × C ~ c >_ 0 and t~ > 0 satisfying

C are continuous functions such that the following

-< - y ,

t ~ [0, co) for some ¥ > 0.

C be a continuous function for which there are constants

(xi) I f ( t , x , y ) l <- c (max{Ixl, lyl}) l+u Then, there is Q > 0 such that Ix01 < Q implies that every solution o f Problem (1) exists in [0, oo) and satisfies lim x(t) = O.

I~00

Besides, g i v e n E > 0 t h e r e i s a 6 > 0 s u c h t h a t l x 0 1 where x is a solution o f Problem (1).

< m i n { Q , 6}

~

Ix(t)l < c , t ~ [0, eo),

Proof. N o t e that ~z is well defined, continuous and b o u n d e d on [0, co). It is easy to see that a function x" [0, 0o) ~ C is a solution o f Problem (1) if a n d only if y " [0, oo) ~ o f the following problem

y'(t)=

qo'(t)] [a(t)+--~jy(t)+

C is a solution

b(t)cp(t) ((t)) (y(t) y(p(t))] ~ y p + e p ( t ) f t, q g ( t ) , ~ p ( t ) ) , ] (5)

y(O) = Yo with Y0 = qo(0)x0. It is easy to see that a and ~ " [0, oo) ~

£ defined as qg'(t)

oe(t) = a(t) + - qg(t) b(t)cp(t) ~(t) = qg(p(t)) satisfy (vii) o f T h e o r e m 1.2. Let

1

Ol =

1

sup - sup t~[0,oo) qo(t) t~[0.oo)cp(p(t))

and 02 =

qo(t) sup t~to, oo) q)(p(t))

2000

T. S E N G A D I R

N o w the map g" [0, co) × (~ × C ~

C defined as

g(t, u, v) = q ) ( t ) f

t, q)(t)" qo(fi(t))

is such that

[g(t, u, v)[ -< ccp(t)

( 1 ,~1+~ \~-~7 '

max

1

cp(p(t))

1

(max{lul, lvl}) 1+~

_< c (max{ 1, 02}) ~ (max{lul, Ivl}) L+~. Thus g satisfies (vii) of T h e o r e m 1.2 and the conclusion of the same theorem holds for Problem (5). N o w take Q = ~ e and let Ix01 < Q. Then Iqg(0)x01 < P. Therefore there is a solution y of (5) withy0 = qg(0)x0. Hence x(t) act = ~y(t) S is a solution o f (1). Since y is a b o u n d e d solution o f (5), lim x ( t ) = lira y(t____}_) = O. t-oo t-~o qg(t) Let 6 > 0 be given. There is a 6* > 0 such that lyo[ < min{P, 6*} •

•

~

ly(t)l < T~ where y is a

d~*

6

solution of(5). N o w let Ixol < mm{Q, ~-76i}" Then Iqo(0)xol < min{P, 6*} and hence ly(t)l < o. where y is a solution of Problem (5) with Y0 = qo(0)xo. Therefore,

I, t,L ~

< 01q)(t) ' t ~ [0, ~ )

i.e. Ix(t)l < e , t E [0, oo) and the p r o o f is complete.

•

In the light of Theorem 1.1, to deduce 'good' asymptotic stability results from the above theorem, we should be able to get examples of q9 with ~cp(t) - ~ 'close' to zero and ep(p(t) qo~t)) 'close' to 1. The following proposition gives one such example• PROPOSITION 2.2• F o r a given h > 1, define qoa : [0, oo) ~ (0, oo) as q~a(t) = ln(A + t). Assume that p : [0, oo) ~ [0, oo) is a continuously differentiable function satisfying the conditions (i), (ii) and (iii). Then each qoA satisfies (viii) and (ix) of Theorem 2.1. Further, given 0 > 0, there is A > 1 with the properties that ,e~ct) < 1 + 0 . (xii) supt~[o, oo) ~~A(f) - ~ < 0 (xiii) suPte[0,oo ) ¢PA(p(t)) P r o o f The condition (viii) is obvious. Since q~(t) ~pa(t) = ( h + t ) l nl ( A + t ) -<- n-]n-]n---~ h 1 given 0 > O, we can choose h large enough so that (xii) holds. N o t e that

lira p(t) t-oo t = l i m p ' ( t ) < co. Therefore,

'

t E [0, oo) for a

(6)

2001

Asymptotic stabilty

lim

¢pa(t)

A + p(t)

t~oo qox(p(t))

_ l i m p, (t)(h + t) _a + (e22 = lim t t t - ~ p ' ( t ) ( ~ + 1)

which by (6) implies that lira

qgA(t)

t-~ epa(p(t))

=1.

Hence, (ix) follows. N o w we shall show that for large enough h , (xiii) is true. By differentiating with respect to h, we can show that qoa)(t)

<

qoa2(t)

cpA) ( p ( t ) ) - cPa2 ( p ( t ) ) '

t ~ [0, ~ )

whenever At > h2. It is also easy to see that for every fixed t ~ [0, co), lima_oo qga(p(t)) q,~(t) = 1. Let [0, oo] be the one point compactification o f [0, co). Defining qb • [0, co] ~ [0, oo) as ln(h + t) if t ~ [ O , eo) in(h + p ( t ) ) ' 4,~(oo) = 1 qba (t) -

(7)

we note the following: (xiv) qba are continuous functions on [0, oo] (xv) qbx.(t) < qba2(t), t E [0, ~ ] whenever hi > A2 (xvi) lima_~ qba(t) = 1, t E [0, oo]. Therefore, by Dini's Theorem, qbA ---, 1 uniformly on [0, oo] as h -- ~ . In particular, as h ~ o% (sup,el0 = ] qba(t)) ~ 1. Thus, for large enough h, (xiii) holds and the p r o o f is complete. • THEOREM 2.3. Let a and b : [0, co) ~ conditions

C be continuous functions such that the following

(xvii) ~ ( a ( t ) ) + Ib(t)l -< - y , t ~ [0, oo) for some y > 0; (xviii) b is b o u n d e d on [0, ~ ) hold. Further, let p be as in Proposition 2.2. Assume that f : [0, oo) × C × £ ~ of T h e o r e m 2.1. Then the zero solution o f (1) is asymptotically stable.

£ satisfies (xi)

P r o o f N o t e that the function x - 0 is a solution o f (1) with x0 = 0. The p r o o f follows from T h e o r e m 2.1 , once we show that (x) holds for some ¢p : [0, ~ ) ~ [0, oo) satisfying (viii) and (ix) o f T h e o r e m 2.1. Choose 0 > 0 so that 0 (sup,~[0 ~)([b(t)l + 1)) < ~ and A as in Proposition 2.2 such that qoa satisfies (xii) and (xiii). t

2002

T. SENGADIR

Now, ~ ( a ( t ) ) + qg'A( t-----~)+ cp A ( t__...~) Ib(t)l <__~,(a(t)) + 0 + (1 + O)lb(t)l qgA(t) qga(p(t)) < ~ ( a ( t ) ) + Ib(t)l + 0(1 + Ib(t)l) ¥ _ < - y + -2

¥ 2 and the p r o o f is complete.

•

R e m a r k . For the linear case ( f -~ 0), if Ilbli is increasing, it is clear that (xvii) a n d (xviii) together are stronger than the hypotheses of T h e o r e m 1.1. But we do not k n o w whether (iv) and (v) are sufficient for a s y m p t o t i c stability of the nonlinear problem covered by T h e o r e m 2.3. Besides, even in the linear case, E x a m p l e 2.4 shows that T h e o r e m 2.1 covers situations not covered by T h e o r e m 1. I. E x a m p l e 2.4. Here we give an example o f a linear F D E for which the condition (iv) o f T h e o r e m 1.1 does not hold but conditions o f T h e o r e m 2.1 holds. Choose h > 1 large enough so that < 1 and ¥ > 0 small enough so that lyl < 1 - Aln(a-----~" Define a as a(t) = - ( t + 1). Let b : [0, ~ ) ~ (0, ~ ) be defined by the equation

b(t)ln(2~ + t) ln(h + p ( t ) )

1 - t +

1 (h + t)ln(h + t)

=

-y.

Clearly (x) of T h e o r e m 2.1 is satisfied. Since for large enough t, b is increasing, it is easy to see that lim {supT~[°'rlb(r)} = iim b(t___~)= - 1 . t-~ a(t) t - ~ a(t) But if (iv) holds, then -

1 =

b(t) -1 lim >_ - t-oo a(t) 1 + O'

a contradiction. We omit the p r o o f of the following theorem which can be proved along the lines o f T h e o r e m 2.3. THEOREM 2.5. Let a and b : [0, oo) - - . C be continuous functions such that (iv) of T h e o r e m 1.1 and the condition (xix) there is a ¥ > 0 such that ~(a(t))

<_ - ¥ ,

t C [0, oo)

hold. Further, let p be as in Proposition 2.2. Assume that f : [0, oo) × C × C ~ o f T h e o r e m 2.1. Then the zero solution o f (1) is asymptotically stable.

C satisfies (xi)

Acknowledgements--1 wish to acknowledge with thanks financial support from the National Board for Higher Mathematics of Department of Atomic Energy through grant of a Post-doctoral Fellowship. The author is grateful to Professor S. Kesavan, Institute of Mathematical Sciences, Madras for useful comments and suggestions. He would also like to thank Professor A. Iserles, Cambridge University for having communicated his results.

Asymptotic stabilty

2003

REFERENCES 1. ISERLES, A. and LIU, Y., On neutral functional differential equations with proportional delays. Cambridge University Tech. Rep. DAMPT, 1993/NA3. 2. ISERLES, A., On nonlinear delay differential equations. Trans. Amer. Math. Soc., 1994, 344, 441-477. 3. ISERLES, A. and TERJEKI, J., Stability and asymptotic stability of functional-differential equations. J. London Math. Soc., 1995, 51, 559-572. 4. SENGADIR, "IF.,Existence and stability of nonlinear functional differential equations. Y Math. Anal. Appl., 1996, 197, 890-907. 5. SENGAD1R, T., Stability of neutral functional differential equations (preprint). Presented at the National Symposium on Nonlinear Analysis and Applications to Real World Problems held at N.E.S. Science College, Nanded, Maharashtra, India, 1994.