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Chapter IV Nonlinear Delay-Differential Equations
CHAPTER IV
+
Nonlinear Delay-Dzflerential Equations
Having in the previous two chapters studied first the general theory of delay-differential equations and then the linear theory, we shall in this chapter briefly discuss some nonlinear equations. For the sake of simplicity, we will restrict ourselves to scalar equations. 4.1. Approximation of Continuous Functionals by Functional Polynomials: Theorem of M. Frechkt
I n Section 3.1 we mentioned the theorem of F. Riesz which gives an integral representation for bounded linear functionals defined over the space of continuous functions. T h e following theorem, due to M. FrechCt, is an extension of Weierstrass’ theorem on polynomial approximations of continuous functions to the continuous functionals. As we have already noted, all the quantities mentioned in the sequel are scalars. Theorem 4.1 (M. FrechCt [2]). Let G [ x ( t ) ]be a continuous functional deJined for x ( t ) E C(I[a,83, D ) , where D is a compact set on the real line and I[a,p ] is a finite interval. Then
(4.1.1)
1 . .
113
(4.1.2)
114
IV.
NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
the functions G 7 L , j ( ~..., l , T ~ ) being continuous and determined for the functional G independently of the variable function x(t). The expansion (4.1.1) is uniformly convergent in every compact set of continuous functions in C(I[a,PI, 0). Now let V ( t ,x( be a continuous functional over I[a,/3]x C(I[u,/3], D). Then, by virtue of the above theorem, we may write 0
)
)
T h e functions V n , i ( ~,l ..., T~ ; t ) , j = 0, 1, 2, ..., k, , are continuous and determined by the functional V(t,x ( - ) )independently of the variable function x(t). T h e limit (4.1.3) is uniform in every compact set of functions x(i) E C(l[or,/3], 0). T he functional Vn(t,x(.)) is analytic in the functional sense and is called a “polynomial of degree k, .”
4.2. Approximations of Solutions
Let V ( t , be a continuous functional with respect to t and x ( t ) on l [ a ,83 x C(l[ a ,PI, D) and locally Lipschitzian with respect to x. Given E > 0, let n be so chosen such that .(a))
for t E I [ a , /I]. L e t L be the Lipschitz constant for the functional V ( t ,x(.)). Consider the equations (4.2.2) and (4.2.3)
4.3. Functional Polynomials of Second Order
115
for t E ] [ t o , /3], 01 < to < /3, and their solutions x ( t , to ,4)and xn(t, t o , #J), whose existence and uniqueness are assured by the results of Chapter 11. According to the inequality (2.1 1.4), we may write
1 x(t)
- x,(t)
1 < (E/L)(eL(t-to)- 1 )
(4.2.4)
lim .%(t, t o +),
(4.2.5)
for t E ] [ t o , /3], i.e., x(t, to
,4)
=
f
n
uniformly in I [ t o ,/3]. Thus, the solution x ( t , t o , 4) of the original equation (4.2.2) may be well approximated by the solution xn(t, t o , #J)of the approximate equation (4.2.3) in the intervall[t, , /3], with a sufficiently large n. 4.3. Equations Whose Right-Hand Sides are Functional Polynomials of Second Order
This section is devoted to the study of an equation of the form m
44
=
2 A,(t)x(t - h") f f KdT; "=a
+(T)
dT
+ f(t)
t0
f
+ J-: 0
Kd71 7 72; t)'471)X(T,)
dTI d72,
(4.3.1)
to
where A is a real parameter, the functions AY(t),v = 0, 1, ..., m, f(t), K,(T;t), and K,(T, , T , ; t ) are given functions continuous for t E ] [ t o , PI, 7,T, , 7, E ] [ t o , 81. We shall denote by x(t, to , A) the solution of Eq. (4.3.1) corresponding to the initial condition
+,
"V(4
a
t o , 4, A) = +(t>
for
t
E
qa,t o ] ,
(4.3.2)
< to < /3. We assume that the interval I[a, to] is sufficiently large.
Since Eq. (4.3.1) depends analytically upon the parameter A, then, by Theorem 2.12, the solution x ( t , t o , +, A) is an analytic function of the parameter A. We now construct the Maclaurin expansion of the function x(t, t o , 4,A) with respect to A. For this purpose, we define the functions
subject to the conditions zo(t) = +(t),
for t E I[.,
to].
z,(t)
= 0,
n
=
I , 2, ...,
(4.3.4)
116
I V . NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
By Theorem 2.10, the function z,(t) satisfies the linear equation
Let M(u, t ) and N(o, t ) be the kernel functions of the first and second kind associated with Eq. (4.3.5). By Theorem 3.11, the solution of Eq. (4.3.5) satisfying the first condition of (4.3.4) may be represented in the form (4.3.6) z,(t) = f0 M(o, t)$(u) do + N ( u , t)f(o)do
Lo
, for t ~ 1 [ t ,PI. Differentiating Eq. (4.3.1) with respect to A, putting X = 0, and making use of the results of Section 2.14, we find the following linear equation: 7P
%(t> =
2 A.(l)%(t
-
A,) -k
”=O
f Kl(7; t)zl(~)dT t0
where the function z,(t) is given by (4.3.6). Since we are seeking the solution of the linear equation (4.3.7) which satisfies the initial condition zl(t) = 0 for t EI[CI, to], Theorem 3.11 gives the following integral representation of the solution zl(t): zl(t)=
j”‘ N(u, t) do j”ro Jyo
K2(71
I
T 2 ; u)z0(71)20(T2)
dT1 d72
(4*3.8)
t0
for t E 1[t, , PI. Differentiating twice Eq. (4.3.1) with respect to h and putting X we obtain the following linear equation:
-
=
0,
J
to
(4.3.10)
4.3. Functional Polynomials of Second Order
117
+
h
Differentiating (n 1) times Eq. (4.3.1) with respect to h and putting 0, we obtain the following linear equation:
=
for t E I[to,PI, where
and Cak are binomial coefficients
- n(n - 1 ) ... ( n - k
Cn -
+ 1)
(4.3.13)
k!
Therefore, taking into account the initial condition (4.3.4), Theorem 3.1 1 yields zn+l(t)
f
.-.,zn)
t ) ~ + l ( o"0; 9
~ ' ( 0 ,
do
(4.3.14)
10
for t € l [ t oPI. , Since we have already found the functions zo(t),zl(t), z,(t), the recurrence formula (4.3.14) holds indefinitely. We now obtain some estimates for the functions z,(t). For this purpose, assume that
I z&t) I < A ,
< B,
I N(o, t ) I
I K(71, 72; t ) I
(4.3.15)
for t e I [ t o PI, , A, B, and C being finite constants. Then, Eq. (4.3.8) yields A2BC (4.3.16) I Zl(t) < 3 ( t - t0)3. Similarly, Eq. (4.3.10) yields
.I a&) I
A3B2C2 < 7( t - to)6.
Likewise, from Eq. (4.3.14) for n
I %(t) I
=
(4.3.17)
2, we find
< 5A4B3C3 504 ( t -
(4.3.18)
tO)9.
Suppose now that the inequalities
I z&) 1
< X(k)Ak+'B"Ck(t
-
tO)3k
(4.3.19)
118
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
are true for k = 0, 1, 2, 3, ..., n. Then, by the general recurrence formula (4.3.14), we obtain
<
j ~ " + ~ 1 ( t ) x(n
+ 1)An+2Bn+1Cn+1 (t
- f0)3"+3,
(4.3.20)
where
By the inequalities (4.3.15) we have x(0) = 1.
(4.3.22)
A successive application of the recurrence formula (4.3.21) yields with ten figures
x(0)= 1 , X( 1) = *, x(3) = 0.0099206349, ~ ( 5= ) 0.0004481850, x(7) = 0.0000590912, x(9) = 0.0000034308,
x(2) x(4) x(6) x(8)
= 0.0555555556,
0.0019841270, = 0.0000227121, = 0.0000089009, ~ ( 1 0= ) O.OOO0011399. =
(4.3.23)
Contrary to the evidence of the early elements, the sequence x(n) tends to infinity due to the large values of Cnkwhen n is large. T o see this, put (4.3.24) x(k) = ( 3 k 1)
+
Then (4.3.25) Writing aI;
=
K! a"lpk,
(4.3.26)
we find n
(4.3.27) I n (4.3.26) we may choose a arbitrarily, since there is a modified homogeneity in (4.3.25). According to (4.3.22) we have x(0)= mo = 1. Therefore mp0 = 1.
4.3. Functional Polynomials of Second Order First, we choose
a =
119
1/54, Po = 54. We may easily show that pk
>, 54(K
+
(4.3.28)
for all integer k >, 0. This is true when K = 0. Suppose it holds for n. We deduce from (4.3.27) and (4.3.28) that
k
<
=
54(n
l)(n + 3) > 54(n + 2) . 9(3n(n++3)(3n + 4) *
Combining (4.3.24), (4.3.26) and (4.3.28) with
x(4 >
(n
+ I)! (3n + 1) , Po = 18. We
pk
for all integer k
=
2),
1/54, we obtain
(n 2 I),
54"
which shows that x(n) -+ 00. Now we choose 01 = 1/18,
01
+
(4.3.29)
may similarly show that
< 18(K + 1 )
(4.3.30)
2 0. Thus we have I[n
> 1).
(4.3.31)
Consider now the following power series m
P(5) = cc?P,
(4.3.32)
n=O
T h e radius of convergence of the series (4.3.32) is R = 18/ABC. Hence P(5) represents an analytic function of 5 in j 5 I < R. Now consider the series (4.3.34)
which is Maclaurin expansion of the function x ( t , to ,4, A) with respect to A. By the inequalities (4.3.19) and (4.3.31), this series is majorated
120
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
by the power series P(<).Since P(<) is uniformly and absolutely convergent for all A, t and t, satisfying
I A(t
-
I < R,
(4.3.35)
the series (4.3.34) is also uniformly and absolutely convergent for A, t and to satisfying (4.3.35). Therefore it represents the function x ( t , to , +, A). Thus we have (4.3.36)
for t E I [ t o , 83, where (4.3.37)
provided that the functions f(t), A , ( t) , v = 0, 1, 2, ..., m, K,(T;t) and & ( T ~ , T ~ t); are continuous for t to and A, B, C are given by (4.3.15). It should be noticed that the above results are natural consequences of Theorem 2.12 and 2.14. 4.4. Equations Whose Right-Hand Sides Are Functional Power Series
The method of the preceding section may be used for the study of equations whose right-hand sides are power series in the functional sense of the form
(4.4.1)
where the functions AY(t),v = 0, 1, ..., m , f ( t ) , and K,(T,, ..., 7, ; t), n = 1, 2, ..., are given continuous functions for T~ , t E I [ t o , 81, j = 1,2, ..., n. We denote again by x ( t , to ,(5, A) the solution of Eq. (4.4.1) satisfying the initial condition x(t, t o ,
4, A)
= +(t)
for
We shall consider once more the functions
t
E 1[a,t,].
(4.4.2)
121
4.4. Functional Power Series subject to the initial conditions zo(t)= $(t),
zn(t) = 0,
n = 1, 2,
...,
t
E I[a,to].
(4.4.4)
Since the right-hand side of Eq. (4.4.1) is analytic with respect to in the functional and ordinary sense, respectively, we may apply Theorem 2.12. By successive differentiation with respect to h and putting h = 0, we obtain
x ( t ) and h
and, in general,
where
with G,(t) - f ( t ) .
122
IV.
NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
All the equations above are linear. Let M(u, t ) and N(a, t ) be the common kernel functions of the first and second kind of Eqs. (4.4.5). Then, owing to the initial functions (4.4.4), we may write
(4.4.8)
and, in general, z,(t)
=
f N ( u , t)G,(u; z o ,z , , ..., zk-,)do,
(4.4.10)
'
,I
where G,(t; z o , ..., zkPl)is defined by (4.4.6). By Theorem 2.12 we have (4.4.1 1)
+
for h in some neighborhood of h = 0 and for t e I [ t o , to A], h being sufficiently small. Thus, according to (4.4.3),for these A and t , we have (4.4.12)
where z,(t) are found by the above inductive process.
4.5. Separable Kernels: 1. Equations of Second Degree T h e analysis of Section 4.3 may be further developed in the case in which the kernel function K,(T,, T~ ; t ) has the following form: 7
K*(T~ T ,~ t ;)
=
CG ; ~ ) ( Tt)Gj2)(~,; ~; t), J=1
(4.5.1)
4.5. I . Equations of Second Degree
123
where G ; ~ ) (tT) ; and G ; ~ ’ ( tT) ;, j = 1, 2, ..., r , are given functions, continuous for T , t 3 t o . Any kernel function K 2 ( ~ 1T~, ; t ) of the form (4.5.1) will be called separable, if it is bounded. Consider the equation in
A,(t)x(t - h,)
*(t) =
+f
“=O
K1(7; t)X(T)
dT
+ f(t)
t0
(4.5.2)
where t 3 t o , AY(t),v = 0, 1, ..., m,f ( t ) , K1(7;t ) , and K 2 ( ~ T~ 1 , ; t ) are supposed to be continuous for t 3 t o , T , T ~ T~, 3 t o , and the kernel K 2 ( ~ T~ 1 ,; t ) is separable. In this case Eq. (4.5.2) may be written in the form
I n Section 4.3 we have shown that the solution x(t, t o , 4, A) of Eq. (4.5.2) has the series representation
where zo(t)=
4, A)
An
27 rO
x(t, to >
(4.5.4)
zn(t>,
=
n=o 71.
r0
M(u, t)+(u)do
+ f N(u, t)f(o)do
(4.5.5)
to
and zn+l(t>
= (n
+ 1) 2 f Cnk
k=l
~
(
t ),doo:J
0
Jio
~
(
7 > 17 2 ; t)Zk(Tl)Zn-k(Tz) dT1
dTz
7
(4.5.6)
t0
where N(a, t ) and M(a, t ) are the kernel functions of the first and second kind associated with the linear equation
c m
44
=
A”(t)X(t - h,)
+f
“-0
and Cnkare the binomial coefficients.
$0
u 7 ;
t)x(.) d-r
+ f(t)
124
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
Owing to Eq. (4.5.1), the formula (4.5.6) yields
where Z$"(t) =
It G:F)(T;
d ~ , p
~ ) z ~ ( T )
1 , 2.
(4.5.8)
p = 1 , 2,
(4.5.9)
=
to
Let M;P)(u,t ) =
f G?)(T;t)M(a,
T)
dT,
10
and Nj")(u, t )
=
It
G ; ~ ) ( t)N(u, T; T ) d ~ , p = 1 , 2.
(4.5.10)
,7
Now, let us multiply the function X , + ~ ( T ) by G ~ ' ( T t ) ; and integrate between tn and t . Using the formula (4.5.7) and applying the rule of Dirichlet, we find
T o use the recurrence formula (4.5.11) we need the functions ZgJt), 1, 2. These functions may be computed easily. T o do so, we multiply Eq. (4.5.5) by G~.")(T; t ) and integrate between to and i . Applying Dirichlet's rule we find
p =
Starting from the formulas (4.5.12), by successive application of the recurrence formula (4.5.1 I), we may compute all the functions ZkJt), and, by successive application of the formula (4.5.7), we obtain all the functions z,(t) which we need to form the series (4.5.4). Clearly the special case G ~ ) ( tT) ;= G~'")(T; t),
j
=
1, 2,
is particularly simple for the computation. It should be noticed that Pincherle-Goursat separable kernels.
..., Y ,
(4.5.13)
kernels are special
4.6. II. Equations with Analytic Right-Hand Side
125
4.6. Separable Kernels: II. Equations with Analytic Right-Hand Side
T h e method of Section 4.5 can be extended to the equations whose right-hand sides are functional power series of the form
where
We assume that the functions A,(t), v G%;(T;t), i = 1 , 2 , ..., r, , p = 1, 2, ..., n, n for T, t 3 to and
I K ( 7 1 ..., T
~
t ); I
= 0, 1 , ..., m , f ( t ) , K1(7;t ) , = 2, 3, ..., are continuous
(4.6.3)
for all n = 1 , 2, ..., and for T ~ ..., , T,, t 3 t o , where C is a finite positive number. Any kernel K , ( T , , ..., r, ; t ) o f the form (4.6.2) with the property (4.6.3) will be called separable. I n Section 4.4, we have shown that the solution x(t, t o ,4, A) of Eq. (4.6.1) is of the form (4.6.4)
where (4.6.5)
and
the functions G,(t; zo , zl,..., z ~ - ~being ) defined by (4.4.6). These functions may be computed easily when all the kernels K , , n = 1, 2, ..., are separable.
126
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
To do so, we first define the following functions: (4.6.7)
Multiplying the function . z ~ ( T )by G%;(T; t ) , integrating from to to t with respect to T , and changing the order of integration, we find
where (4.6.9)
and (4.6.10) Now, let us multiply the function z~(T)by G$)l,i(~;t ) and integrate from to to t with respect to T . Using the formulas (4.4.6), (4.6.2), (4.6.6), and (4.6.7) and applying Dirichlet’s rule, we find
where (4.6.12)
On the other hand, from the formulas (4.6.2), (4.6.6), and (4.6.7), we obtain
Starting from the formulas (4.6.8), by successive application of the recurrence formula (4.6.1 l), we can compute all the functions ZeA+l,i(t), and successive applications of the formula (4.6.13) yield all the functions zn(t)which we need to form the series (4.6.4).
4.7. Approximation of Continuous Kernels
127
4.7. Approximation of Continuous Kernels by Separable Kernels
Let
where to < fi < GO. By the approximation theorem of Weierstrass, any function K , ( T , , ..., T , ; t ) , continuous in I , , is the limit of a uniformly convergent sequence of polynomials in I,. Therefore, given E > 0, there exists a polynomial P , ( T ~ ..., T , ; t ) such that
K , ( T ~..., , rn;t )
= Pn(rl,
...,
7,;
t)
+ Q , ( T ~ ..., ,
t),
7 ;,
(4.7.2)
where Qn(rl, ..., T , ; t ) is a continuous function defined in I , and
I Q n ( r 1 , ..-,rn; in I ,
. Clearly
t) I
(4.7.3)
each polynomial P,(T~, ..., 1% ; t ) is separable; i.e.,
Pn(r,, ..., r n ; t)
= i+i,+.
2
..+i,=O
aii,,,.i,t irli i ... r2
,
(4.7.4)
aiil.. being constants. Assuming that each K , ( T ~..., , T , ; t ) , n = 2, 3, ..., is written in the form (4.7.2), consider the nonlinear functionals
and
128
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
Since the solutions x ( t ) = x ( t , to ,4, A) and Z ( t ) = Z(t, to , 4, A), with the usual notation, are defined and continuous in I [ t o ,j?], there exists a positive number A such that
I x(t) I
< A,
I “ q t )I
(4.7.9)
for t E I [ t o , is]. Let L be the Lipschitz constant of Eq. (4.7.7) for t € I [ t O j,?] and 1 x I A. Then, we have
<
I w t , 4 . )) - W t , C( . )) I
< L 144 - q t ) I
(4.7.10)
and
1 W ( t ,2( . ))
- @ ( t , 2(
.)) I < c A ( t - to)[eAA(t-to)
-
I]
(4.7.11)
for t E ] [ t o , is]. Since
I W t , 4 . )) - @P> a( . 1) I
< I W(t,4 . 1)
-
w, q )) I ’
+ I w, q . )) - w, q . )) I,
the inequalities (4.7.10) and (4.7.1 1) yield
I W ( t ,x( . 1)
-
@ ( t , a( . 1) 1
< L 1 x ( t ) - ~ ( tI )+ c ~ ( t
- to)[eAA‘t-to) -
11
(4.7.12)
for t d [ t o j,?]. We now integrate both sides of Eqs. (4.7.7) and (4.7.8) with respect to t, from to to t . Putting u(t) = I x ( t ) - q
and using the inequality (4.7.12), we find u(t)
<(c/~)(t
- t,,)eAA(t-to)
(4.7.13)
t)I
+L
st
(4.7.14)
u(s) ds
to
for t E I [ t o ,is]. Therefore, by Gronwall’s lemma, the function u ( t ) , i.e., 1 x ( t ) - Z ( t ) l , is an infinitesimal with E for every finite interval I [ t o ,j?]. Thus, the solutions x(t, to , 4, A ) of Eq. (4.7.8) with separable kernels approximate well the solutions x ( t , to ,4, A) of the original equation (4.7.7). 4.8. Extension
In this section we shall deal with an equation of the form n(t) = V(t, X( . ))
+ @ ’ ( ~ ( t ) ,~
(-t hj), ..., x(t
-
hm), t ) ,
(4.8.1)
4.8. Extension
129
where p is a parameter,
and F(u, , u2 , ..., urn+,, t ) is an analytic function of u1 , ..., urn+,, t in a compact set D of the ( m 1)-dimensional u space and for t € I [ t o ,PI, where ,B may be arbitrarily large. We suppose that the parameter h is fixed, and the functions A,(t), v = 0, 1, ..., m , f ( t ) ,K n ( ~,l..., 7, ; t ) , n = 1,2, ..., are continuous for t E I [ t o ,PI. Since Eq. (4.8.1) depends analytically upon the parameter p, by Theorem 2.12, the solution x ( t ) = x ( t , t o , 4, p), with the usual notation, is also an analytic function of p. T h e method of Section 4.3 may be easily modified for the present equation. For simplicity, we shall work on the equation
+
k(t) = f(t)
t A(t)x(t)t B(t)x(t - W)
+ f KI(T; t)x(T) $0
+ f (" + W t ) ,x(t
&(TI
to
where
w
, 72;
&4T,)x(Tz)
d71d72
to
-
w),
0,
is a positive constant. Let
with the initial conditions
(4.8.3)
130
I V . NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
Hence, zo(t)can be computed by the method of Section 4.3. Therefore, we may assume that zo(t)is known. Differentiating Eq. (4.8.3) with respect to p and letting p = 0, we find
which is linear with respect to z,(t). Since z,(t) = 0 for t E I[to ,/3], we can calculate zl(t) by the method of Chapter 111. Differentiating twice Eq. (4.8.3) with respect to p and putting p = 0, we find %(t)
=
A(t)z,(t)
+ B(t)z,(t
-
w)
+f
&(7;
t ) 4 4 d7
t0
+ A J"; o
It
KZ(T1 Y 72;
t"O(T1)~2(72)
+ 2~1(7,)~1(~2) + %(71)X0(72)1d71d72
*n
+ 2[I;,(zo(t), zo(t
-
w),
t)z,(t)
+ Fz(zo(t), zo(t
-
w),
t)zl(t
-
w)I,
(4.8.8)
which is linear with respect to zz(t).I n (4.8.8) F, and F, denote the partial derivatives of the function F(u, , u 2 , t ) with respect to u1 and u2 respectively. Since z2(t) = 0 for t E I[t, , /3], we may easily find the function zz(t)for t E ][to, 81. Following this process we obtain all zn(t)as solutions of linear equations satisfying the initial conditions (4.8.5). Having then constructed the series (4.8.9)
we can show, making use of the analyticity of F(u, , ug , t ) and the continuity of A(t),B(t),K1(7; t ) , and K z ( r l ,r 2 ; t ) ,that the series (4.8.9) converges absolutely and uniformly for t E I [ t o ,81. Hence the series (4.8.9) represents the solution x ( t , t o ,4, p). T h e above method may be applied to the general equation (4.8.1) without any difficulty.
4.9. Boundedness and Stability
131
4.9. Boundedness and Stability
I n Sections 2.19-2.22, we have investigated stability and boundedness of delay-differential equations in general. I n this section we shall briefly discuss, with a different approach, some stability and boundedness properties of some nonlinear equations. We shall consider in particular equations of the form
(4.9.1)
where X is a constant parameter and the functions f ( t ) , AY(t),Y = 0, 1, 2, ..., m,K,(T;t ) , and K , ( T , ,T~ ; t ) are supposed to be continuous for T , T , , T ~ t ,3 t o . Equations of the form (4.4.1) can be investigated similarly. Let M(o, t ) and N(o, t ) be the kernel functions of the first and second kind of the linear equation m
(4.9.2) “=O
Consider the solution x(t, t o , 6,A) of Eq. (4.9.1) corresponding to the initial condition x(t, t o , 4, 4 = + ( t ) for t E I[%t o ] , (4.9.3) where $(t) E C(l[a,to],D), D being a closed bounded set on the real line. I n addition to the above continuity hypotheses, we assume that
s:
I M(o, t ) I do
s’ I
No,
t ) I do
for
t
for
t >, t o ,
for
t >, t o ,
for t >, t o ,
t0
where a,8, y , 6, , and 6, are positive constants.
E I[a,to],
132
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
I n Section 4.3 we have shown that (4.9.5) where
the function zo(t)is given by
and the functions zJt), n
=
1, 2, ..., satisfy the recurrence formula
for t 2 t o . Using the inequalities (4.9.4) and the formula (4.9.7) we find (4.9.9)
I zo(t) I G $0 for 1
2 t o , where 9.0
= 61B
+ 6,Y.
(4.9.10)
Making use of the general recurrence formula (4.9.8),we obtain by induction (4.9.11) I zn(t) 1 < 8 n for t 3 t o , where $n -__ n2n-l$n+lan
y n,
n = l , 2 ,....
(4.9.12)
Therefore, the series (4.9.5)is majorated by the series (4.9.13) Thus: Theorem 4.2. Let Eq. (4.9.1) be given and assume that the inequalities (4.9.4)are satisfied, all functions involved in (4.9.1)being continuous f o r
133
4.9. Boundedness and Stability
t 2 t o . Then, the solution x(t, t o , 4, A) of Eq. (4.9.1) is bounded by B, y , 4 f o r t >, to ,
4 6 0 , %
I x(t,
to
9
474 I < 4 8 0 , B, Y , 4,
(4.9.14)
where A(9, , a, p, y, A) is the sum of the series (4.9.13);i.e.,
+ 8oayX e~p(28~ayA)l.
a, B, y, A) = $.,[I
(4.9.15)
Remembering that zo(t) = x ( t , tn 3 $7
O),
the above argument also shows that: Under the hypotheses of Theorem 4.2, we have
Theorem 4.3.
I x(t, to 4, A) - x(t, t o , A 0 ) I < A(80, a, B, y , A) 9
- 80
(4.9.16)
for t 2 t o . We now compare the solutions x(t, t o , 4, A) and 2(t, t o , A) of the same equation (4.9.1) corresponding to the initial functions + ( t ) and +(t) in the same initial interval ][a,to]. Let
4,
(4.9.18)
and
lp, = max I zn(t)I ,
= 0,
I , 2,
t>to
... .
(4.9.19)
Since Zo(t) is the solution of the linear equation (4.9.2) with the initial function &t), we may write Zo(t) =
J
t0
M(u, t)&u) du
+
(4.9.20)
Thus, by virtue of Eqs. (4.9.7) and (4.9.20), we have
I for t
- %(t)
!
<
(4.9.21)
t o . Therefore, f o ( t ) = zo(t)
70 1
$0
+
= 81)
3
(4.9.22)
134
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
where (4.9.23) Since
and
making use of the equality (4.9.22) and the inequalities (4.9.4) and (4.9.23), we find
I zi(t) - zi(t) I
< a2+Y8n + 8,)
(4.9.24)
for t 3 t o . Therefore, %(t> = 4
4
+-
71 >
where
I 71 !, I q 1 I Similarly, from rt
and
we obtain
"U
"(1
$1
< a"Y9.0
= 9.1
+
+
$0).
7j1,
(4.9.25) (4.9.26)
4.9. Boundedness and Stability
135 (4.9.29)
By induction, we can easily show that (4.9.30)
(4.9.31) (4.9.32) Consider now the function
(4.9.27),and (4.9.30)yield T h e inequalities (4.9.21),(4.9.24), 9.3 ) Since the right-hand side of (4.9.33)is infinitesimal with 6 , we may state the following: Theorem 4.4. Let the inequalities (4.9.4) be satisfied. If all the functions involved in Eq. (4.9.1)are continuous and if
I B ( t ) - 4(t) I < 6,
(4.9.34)
where 6<
41
+ 4 a o + $,)A
c
exp(2v(80
+ 80)A)l’
(4.9.35)
then
I Z ( t , t o , 6,A)
- x ( t , t o , $ , A)
I
(4.9.36)
for t 3 t o .
It should be noticed that Theorem 4.4expresses the uniform stability of the system (4.9.1) under small variation of the initial functions. Similar results can be established if to and h also vary. T h e method described above can also be extended to the systems of the form (4.4.1).
+
Nonlinear Delay-Dzflerential Equations
Having in the previous two chapters studied first the general theory of delay-differential equations and then the linear theory, we shall in this chapter briefly discuss some nonlinear equations. For the sake of simplicity, we will restrict ourselves to scalar equations. 4.1. Approximation of Continuous Functionals by Functional Polynomials: Theorem of M. Frechkt
I n Section 3.1 we mentioned the theorem of F. Riesz which gives an integral representation for bounded linear functionals defined over the space of continuous functions. T h e following theorem, due to M. FrechCt, is an extension of Weierstrass’ theorem on polynomial approximations of continuous functions to the continuous functionals. As we have already noted, all the quantities mentioned in the sequel are scalars. Theorem 4.1 (M. FrechCt [2]). Let G [ x ( t ) ]be a continuous functional deJined for x ( t ) E C(I[a,83, D ) , where D is a compact set on the real line and I[a,p ] is a finite interval. Then
(4.1.1)
1 . .
113
(4.1.2)
114
IV.
NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
the functions G 7 L , j ( ~..., l , T ~ ) being continuous and determined for the functional G independently of the variable function x(t). The expansion (4.1.1) is uniformly convergent in every compact set of continuous functions in C(I[a,PI, 0). Now let V ( t ,x( be a continuous functional over I[a,/3]x C(I[u,/3], D). Then, by virtue of the above theorem, we may write 0
)
)
T h e functions V n , i ( ~,l ..., T~ ; t ) , j = 0, 1, 2, ..., k, , are continuous and determined by the functional V(t,x ( - ) )independently of the variable function x(t). T h e limit (4.1.3) is uniform in every compact set of functions x(i) E C(l[or,/3], 0). T he functional Vn(t,x(.)) is analytic in the functional sense and is called a “polynomial of degree k, .”
4.2. Approximations of Solutions
Let V ( t , be a continuous functional with respect to t and x ( t ) on l [ a ,83 x C(l[ a ,PI, D) and locally Lipschitzian with respect to x. Given E > 0, let n be so chosen such that .(a))
for t E I [ a , /I]. L e t L be the Lipschitz constant for the functional V ( t ,x(.)). Consider the equations (4.2.2) and (4.2.3)
4.3. Functional Polynomials of Second Order
115
for t E ] [ t o , /3], 01 < to < /3, and their solutions x ( t , to ,4)and xn(t, t o , #J), whose existence and uniqueness are assured by the results of Chapter 11. According to the inequality (2.1 1.4), we may write
1 x(t)
- x,(t)
1 < (E/L)(eL(t-to)- 1 )
(4.2.4)
lim .%(t, t o +),
(4.2.5)
for t E ] [ t o , /3], i.e., x(t, to
,4)
=
f
n
uniformly in I [ t o ,/3]. Thus, the solution x ( t , t o , 4) of the original equation (4.2.2) may be well approximated by the solution xn(t, t o , #J)of the approximate equation (4.2.3) in the intervall[t, , /3], with a sufficiently large n. 4.3. Equations Whose Right-Hand Sides are Functional Polynomials of Second Order
This section is devoted to the study of an equation of the form m
44
=
2 A,(t)x(t - h") f f KdT; "=a
+(T)
dT
+ f(t)
t0
f
+ J-: 0
Kd71 7 72; t)'471)X(T,)
dTI d72,
(4.3.1)
to
where A is a real parameter, the functions AY(t),v = 0, 1, ..., m, f(t), K,(T;t), and K,(T, , T , ; t ) are given functions continuous for t E ] [ t o , PI, 7,T, , 7, E ] [ t o , 81. We shall denote by x(t, to , A) the solution of Eq. (4.3.1) corresponding to the initial condition
+,
"V(4
a
t o , 4, A) = +(t>
for
t
E
qa,t o ] ,
(4.3.2)
< to < /3. We assume that the interval I[a, to] is sufficiently large.
Since Eq. (4.3.1) depends analytically upon the parameter A, then, by Theorem 2.12, the solution x ( t , t o , +, A) is an analytic function of the parameter A. We now construct the Maclaurin expansion of the function x(t, t o , 4,A) with respect to A. For this purpose, we define the functions
subject to the conditions zo(t) = +(t),
for t E I[.,
to].
z,(t)
= 0,
n
=
I , 2, ...,
(4.3.4)
116
I V . NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
By Theorem 2.10, the function z,(t) satisfies the linear equation
Let M(u, t ) and N(o, t ) be the kernel functions of the first and second kind associated with Eq. (4.3.5). By Theorem 3.11, the solution of Eq. (4.3.5) satisfying the first condition of (4.3.4) may be represented in the form (4.3.6) z,(t) = f0 M(o, t)$(u) do + N ( u , t)f(o)do
Lo
, for t ~ 1 [ t ,PI. Differentiating Eq. (4.3.1) with respect to A, putting X = 0, and making use of the results of Section 2.14, we find the following linear equation: 7P
%(t> =
2 A.(l)%(t
-
A,) -k
”=O
f Kl(7; t)zl(~)dT t0
where the function z,(t) is given by (4.3.6). Since we are seeking the solution of the linear equation (4.3.7) which satisfies the initial condition zl(t) = 0 for t EI[CI, to], Theorem 3.11 gives the following integral representation of the solution zl(t): zl(t)=
j”‘ N(u, t) do j”ro Jyo
K2(71
I
T 2 ; u)z0(71)20(T2)
dT1 d72
(4*3.8)
t0
for t E 1[t, , PI. Differentiating twice Eq. (4.3.1) with respect to h and putting X we obtain the following linear equation:
-
=
0,
J
to
(4.3.10)
4.3. Functional Polynomials of Second Order
117
+
h
Differentiating (n 1) times Eq. (4.3.1) with respect to h and putting 0, we obtain the following linear equation:
=
for t E I[to,PI, where
and Cak are binomial coefficients
- n(n - 1 ) ... ( n - k
Cn -
+ 1)
(4.3.13)
k!
Therefore, taking into account the initial condition (4.3.4), Theorem 3.1 1 yields zn+l(t)
f
.-.,zn)
t ) ~ + l ( o"0; 9
~ ' ( 0 ,
do
(4.3.14)
10
for t € l [ t oPI. , Since we have already found the functions zo(t),zl(t), z,(t), the recurrence formula (4.3.14) holds indefinitely. We now obtain some estimates for the functions z,(t). For this purpose, assume that
I z&t) I < A ,
< B,
I N(o, t ) I
I K(71, 72; t ) I
(4.3.15)
for t e I [ t o PI, , A, B, and C being finite constants. Then, Eq. (4.3.8) yields A2BC (4.3.16) I Zl(t) < 3 ( t - t0)3. Similarly, Eq. (4.3.10) yields
.I a&) I
A3B2C2 < 7( t - to)6.
Likewise, from Eq. (4.3.14) for n
I %(t) I
=
(4.3.17)
2, we find
< 5A4B3C3 504 ( t -
(4.3.18)
tO)9.
Suppose now that the inequalities
I z&) 1
< X(k)Ak+'B"Ck(t
-
tO)3k
(4.3.19)
118
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
are true for k = 0, 1, 2, 3, ..., n. Then, by the general recurrence formula (4.3.14), we obtain
<
j ~ " + ~ 1 ( t ) x(n
+ 1)An+2Bn+1Cn+1 (t
- f0)3"+3,
(4.3.20)
where
By the inequalities (4.3.15) we have x(0) = 1.
(4.3.22)
A successive application of the recurrence formula (4.3.21) yields with ten figures
x(0)= 1 , X( 1) = *, x(3) = 0.0099206349, ~ ( 5= ) 0.0004481850, x(7) = 0.0000590912, x(9) = 0.0000034308,
x(2) x(4) x(6) x(8)
= 0.0555555556,
0.0019841270, = 0.0000227121, = 0.0000089009, ~ ( 1 0= ) O.OOO0011399. =
(4.3.23)
Contrary to the evidence of the early elements, the sequence x(n) tends to infinity due to the large values of Cnkwhen n is large. T o see this, put (4.3.24) x(k) = ( 3 k 1)
+
Then (4.3.25) Writing aI;
=
K! a"lpk,
(4.3.26)
we find n
(4.3.27) I n (4.3.26) we may choose a arbitrarily, since there is a modified homogeneity in (4.3.25). According to (4.3.22) we have x(0)= mo = 1. Therefore mp0 = 1.
4.3. Functional Polynomials of Second Order First, we choose
a =
119
1/54, Po = 54. We may easily show that pk
>, 54(K
+
(4.3.28)
for all integer k >, 0. This is true when K = 0. Suppose it holds for n. We deduce from (4.3.27) and (4.3.28) that
k
<
=
54(n
l)(n + 3) > 54(n + 2) . 9(3n(n++3)(3n + 4) *
Combining (4.3.24), (4.3.26) and (4.3.28) with
x(4 >
(n
+ I)! (3n + 1) , Po = 18. We
pk
for all integer k
=
2),
1/54, we obtain
(n 2 I),
54"
which shows that x(n) -+ 00. Now we choose 01 = 1/18,
01
+
(4.3.29)
may similarly show that
< 18(K + 1 )
(4.3.30)
2 0. Thus we have I[n
> 1).
(4.3.31)
Consider now the following power series m
P(5) = cc?P,
(4.3.32)
n=O
T h e radius of convergence of the series (4.3.32) is R = 18/ABC. Hence P(5) represents an analytic function of 5 in j 5 I < R. Now consider the series (4.3.34)
which is Maclaurin expansion of the function x ( t , to ,4, A) with respect to A. By the inequalities (4.3.19) and (4.3.31), this series is majorated
120
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
by the power series P(<).Since P(<) is uniformly and absolutely convergent for all A, t and t, satisfying
I A(t
-
I < R,
(4.3.35)
the series (4.3.34) is also uniformly and absolutely convergent for A, t and to satisfying (4.3.35). Therefore it represents the function x ( t , to , +, A). Thus we have (4.3.36)
for t E I [ t o , 83, where (4.3.37)
provided that the functions f(t), A , ( t) , v = 0, 1, 2, ..., m, K,(T;t) and & ( T ~ , T ~ t); are continuous for t to and A, B, C are given by (4.3.15). It should be noticed that the above results are natural consequences of Theorem 2.12 and 2.14. 4.4. Equations Whose Right-Hand Sides Are Functional Power Series
The method of the preceding section may be used for the study of equations whose right-hand sides are power series in the functional sense of the form
(4.4.1)
where the functions AY(t),v = 0, 1, ..., m , f ( t ) , and K,(T,, ..., 7, ; t), n = 1, 2, ..., are given continuous functions for T~ , t E I [ t o , 81, j = 1,2, ..., n. We denote again by x ( t , to ,(5, A) the solution of Eq. (4.4.1) satisfying the initial condition x(t, t o ,
4, A)
= +(t)
for
We shall consider once more the functions
t
E 1[a,t,].
(4.4.2)
121
4.4. Functional Power Series subject to the initial conditions zo(t)= $(t),
zn(t) = 0,
n = 1, 2,
...,
t
E I[a,to].
(4.4.4)
Since the right-hand side of Eq. (4.4.1) is analytic with respect to in the functional and ordinary sense, respectively, we may apply Theorem 2.12. By successive differentiation with respect to h and putting h = 0, we obtain
x ( t ) and h
and, in general,
where
with G,(t) - f ( t ) .
122
IV.
NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
All the equations above are linear. Let M(u, t ) and N(a, t ) be the common kernel functions of the first and second kind of Eqs. (4.4.5). Then, owing to the initial functions (4.4.4), we may write
(4.4.8)
and, in general, z,(t)
=
f N ( u , t)G,(u; z o ,z , , ..., zk-,)do,
(4.4.10)
'
,I
where G,(t; z o , ..., zkPl)is defined by (4.4.6). By Theorem 2.12 we have (4.4.1 1)
+
for h in some neighborhood of h = 0 and for t e I [ t o , to A], h being sufficiently small. Thus, according to (4.4.3),for these A and t , we have (4.4.12)
where z,(t) are found by the above inductive process.
4.5. Separable Kernels: 1. Equations of Second Degree T h e analysis of Section 4.3 may be further developed in the case in which the kernel function K,(T,, T~ ; t ) has the following form: 7
K*(T~ T ,~ t ;)
=
CG ; ~ ) ( Tt)Gj2)(~,; ~; t), J=1
(4.5.1)
4.5. I . Equations of Second Degree
123
where G ; ~ ) (tT) ; and G ; ~ ’ ( tT) ;, j = 1, 2, ..., r , are given functions, continuous for T , t 3 t o . Any kernel function K 2 ( ~ 1T~, ; t ) of the form (4.5.1) will be called separable, if it is bounded. Consider the equation in
A,(t)x(t - h,)
*(t) =
+f
“=O
K1(7; t)X(T)
dT
+ f(t)
t0
(4.5.2)
where t 3 t o , AY(t),v = 0, 1, ..., m,f ( t ) , K1(7;t ) , and K 2 ( ~ T~ 1 , ; t ) are supposed to be continuous for t 3 t o , T , T ~ T~, 3 t o , and the kernel K 2 ( ~ T~ 1 ,; t ) is separable. In this case Eq. (4.5.2) may be written in the form
I n Section 4.3 we have shown that the solution x(t, t o , 4, A) of Eq. (4.5.2) has the series representation
where zo(t)=
4, A)
An
27 rO
x(t, to >
(4.5.4)
zn(t>,
=
n=o 71.
r0
M(u, t)+(u)do
+ f N(u, t)f(o)do
(4.5.5)
to
and zn+l(t>
= (n
+ 1) 2 f Cnk
k=l
~
(
t ),doo:J
0
Jio
~
(
7 > 17 2 ; t)Zk(Tl)Zn-k(Tz) dT1
dTz
7
(4.5.6)
t0
where N(a, t ) and M(a, t ) are the kernel functions of the first and second kind associated with the linear equation
c m
44
=
A”(t)X(t - h,)
+f
“-0
and Cnkare the binomial coefficients.
$0
u 7 ;
t)x(.) d-r
+ f(t)
124
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
Owing to Eq. (4.5.1), the formula (4.5.6) yields
where Z$"(t) =
It G:F)(T;
d ~ , p
~ ) z ~ ( T )
1 , 2.
(4.5.8)
p = 1 , 2,
(4.5.9)
=
to
Let M;P)(u,t ) =
f G?)(T;t)M(a,
T)
dT,
10
and Nj")(u, t )
=
It
G ; ~ ) ( t)N(u, T; T ) d ~ , p = 1 , 2.
(4.5.10)
,7
Now, let us multiply the function X , + ~ ( T ) by G ~ ' ( T t ) ; and integrate between tn and t . Using the formula (4.5.7) and applying the rule of Dirichlet, we find
T o use the recurrence formula (4.5.11) we need the functions ZgJt), 1, 2. These functions may be computed easily. T o do so, we multiply Eq. (4.5.5) by G~.")(T; t ) and integrate between to and i . Applying Dirichlet's rule we find
p =
Starting from the formulas (4.5.12), by successive application of the recurrence formula (4.5.1 I), we may compute all the functions ZkJt), and, by successive application of the formula (4.5.7), we obtain all the functions z,(t) which we need to form the series (4.5.4). Clearly the special case G ~ ) ( tT) ;= G~'")(T; t),
j
=
1, 2,
is particularly simple for the computation. It should be noticed that Pincherle-Goursat separable kernels.
..., Y ,
(4.5.13)
kernels are special
4.6. II. Equations with Analytic Right-Hand Side
125
4.6. Separable Kernels: II. Equations with Analytic Right-Hand Side
T h e method of Section 4.5 can be extended to the equations whose right-hand sides are functional power series of the form
where
We assume that the functions A,(t), v G%;(T;t), i = 1 , 2 , ..., r, , p = 1, 2, ..., n, n for T, t 3 to and
I K ( 7 1 ..., T
~
t ); I
= 0, 1 , ..., m , f ( t ) , K1(7;t ) , = 2, 3, ..., are continuous
(4.6.3)
for all n = 1 , 2, ..., and for T ~ ..., , T,, t 3 t o , where C is a finite positive number. Any kernel K , ( T , , ..., r, ; t ) o f the form (4.6.2) with the property (4.6.3) will be called separable. I n Section 4.4, we have shown that the solution x(t, t o ,4, A) of Eq. (4.6.1) is of the form (4.6.4)
where (4.6.5)
and
the functions G,(t; zo , zl,..., z ~ - ~being ) defined by (4.4.6). These functions may be computed easily when all the kernels K , , n = 1, 2, ..., are separable.
126
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
To do so, we first define the following functions: (4.6.7)
Multiplying the function . z ~ ( T )by G%;(T; t ) , integrating from to to t with respect to T , and changing the order of integration, we find
where (4.6.9)
and (4.6.10) Now, let us multiply the function z~(T)by G$)l,i(~;t ) and integrate from to to t with respect to T . Using the formulas (4.4.6), (4.6.2), (4.6.6), and (4.6.7) and applying Dirichlet’s rule, we find
where (4.6.12)
On the other hand, from the formulas (4.6.2), (4.6.6), and (4.6.7), we obtain
Starting from the formulas (4.6.8), by successive application of the recurrence formula (4.6.1 l), we can compute all the functions ZeA+l,i(t), and successive applications of the formula (4.6.13) yield all the functions zn(t)which we need to form the series (4.6.4).
4.7. Approximation of Continuous Kernels
127
4.7. Approximation of Continuous Kernels by Separable Kernels
Let
where to < fi < GO. By the approximation theorem of Weierstrass, any function K , ( T , , ..., T , ; t ) , continuous in I , , is the limit of a uniformly convergent sequence of polynomials in I,. Therefore, given E > 0, there exists a polynomial P , ( T ~ ..., T , ; t ) such that
K , ( T ~..., , rn;t )
= Pn(rl,
...,
7,;
t)
+ Q , ( T ~ ..., ,
t),
7 ;,
(4.7.2)
where Qn(rl, ..., T , ; t ) is a continuous function defined in I , and
I Q n ( r 1 , ..-,rn; in I ,
. Clearly
t) I
(4.7.3)
each polynomial P,(T~, ..., 1% ; t ) is separable; i.e.,
Pn(r,, ..., r n ; t)
= i+i,+.
2
..+i,=O
aii,,,.i,t irli i ... r2
,
(4.7.4)
aiil.. being constants. Assuming that each K , ( T ~..., , T , ; t ) , n = 2, 3, ..., is written in the form (4.7.2), consider the nonlinear functionals
and
128
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
Since the solutions x ( t ) = x ( t , to ,4, A) and Z ( t ) = Z(t, to , 4, A), with the usual notation, are defined and continuous in I [ t o ,j?], there exists a positive number A such that
I x(t) I
< A,
I “ q t )I
(4.7.9)
for t E I [ t o , is]. Let L be the Lipschitz constant of Eq. (4.7.7) for t € I [ t O j,?] and 1 x I A. Then, we have
<
I w t , 4 . )) - W t , C( . )) I
< L 144 - q t ) I
(4.7.10)
and
1 W ( t ,2( . ))
- @ ( t , 2(
.)) I < c A ( t - to)[eAA(t-to)
-
I]
(4.7.11)
for t E ] [ t o , is]. Since
I W t , 4 . )) - @P> a( . 1) I
< I W(t,4 . 1)
-
w, q )) I ’
+ I w, q . )) - w, q . )) I,
the inequalities (4.7.10) and (4.7.1 1) yield
I W ( t ,x( . 1)
-
@ ( t , a( . 1) 1
< L 1 x ( t ) - ~ ( tI )+ c ~ ( t
- to)[eAA‘t-to) -
11
(4.7.12)
for t d [ t o j,?]. We now integrate both sides of Eqs. (4.7.7) and (4.7.8) with respect to t, from to to t . Putting u(t) = I x ( t ) - q
and using the inequality (4.7.12), we find u(t)
<(c/~)(t
- t,,)eAA(t-to)
(4.7.13)
t)I
+L
st
(4.7.14)
u(s) ds
to
for t E I [ t o ,is]. Therefore, by Gronwall’s lemma, the function u ( t ) , i.e., 1 x ( t ) - Z ( t ) l , is an infinitesimal with E for every finite interval I [ t o ,j?]. Thus, the solutions x(t, to , 4, A ) of Eq. (4.7.8) with separable kernels approximate well the solutions x ( t , to ,4, A) of the original equation (4.7.7). 4.8. Extension
In this section we shall deal with an equation of the form n(t) = V(t, X( . ))
+ @ ’ ( ~ ( t ) ,~
(-t hj), ..., x(t
-
hm), t ) ,
(4.8.1)
4.8. Extension
129
where p is a parameter,
and F(u, , u2 , ..., urn+,, t ) is an analytic function of u1 , ..., urn+,, t in a compact set D of the ( m 1)-dimensional u space and for t € I [ t o ,PI, where ,B may be arbitrarily large. We suppose that the parameter h is fixed, and the functions A,(t), v = 0, 1, ..., m , f ( t ) ,K n ( ~,l..., 7, ; t ) , n = 1,2, ..., are continuous for t E I [ t o ,PI. Since Eq. (4.8.1) depends analytically upon the parameter p, by Theorem 2.12, the solution x ( t ) = x ( t , t o , 4, p), with the usual notation, is also an analytic function of p. T h e method of Section 4.3 may be easily modified for the present equation. For simplicity, we shall work on the equation
+
k(t) = f(t)
t A(t)x(t)t B(t)x(t - W)
+ f KI(T; t)x(T) $0
+ f (" + W t ) ,x(t
&(TI
to
where
w
, 72;
&4T,)x(Tz)
d71d72
to
-
w),
0,
is a positive constant. Let
with the initial conditions
(4.8.3)
130
I V . NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
Hence, zo(t)can be computed by the method of Section 4.3. Therefore, we may assume that zo(t)is known. Differentiating Eq. (4.8.3) with respect to p and letting p = 0, we find
which is linear with respect to z,(t). Since z,(t) = 0 for t E I[to ,/3], we can calculate zl(t) by the method of Chapter 111. Differentiating twice Eq. (4.8.3) with respect to p and putting p = 0, we find %(t)
=
A(t)z,(t)
+ B(t)z,(t
-
w)
+f
&(7;
t ) 4 4 d7
t0
+ A J"; o
It
KZ(T1 Y 72;
t"O(T1)~2(72)
+ 2~1(7,)~1(~2) + %(71)X0(72)1d71d72
*n
+ 2[I;,(zo(t), zo(t
-
w),
t)z,(t)
+ Fz(zo(t), zo(t
-
w),
t)zl(t
-
w)I,
(4.8.8)
which is linear with respect to zz(t).I n (4.8.8) F, and F, denote the partial derivatives of the function F(u, , u 2 , t ) with respect to u1 and u2 respectively. Since z2(t) = 0 for t E I[t, , /3], we may easily find the function zz(t)for t E ][to, 81. Following this process we obtain all zn(t)as solutions of linear equations satisfying the initial conditions (4.8.5). Having then constructed the series (4.8.9)
we can show, making use of the analyticity of F(u, , ug , t ) and the continuity of A(t),B(t),K1(7; t ) , and K z ( r l ,r 2 ; t ) ,that the series (4.8.9) converges absolutely and uniformly for t E I [ t o ,81. Hence the series (4.8.9) represents the solution x ( t , t o ,4, p). T h e above method may be applied to the general equation (4.8.1) without any difficulty.
4.9. Boundedness and Stability
131
4.9. Boundedness and Stability
I n Sections 2.19-2.22, we have investigated stability and boundedness of delay-differential equations in general. I n this section we shall briefly discuss, with a different approach, some stability and boundedness properties of some nonlinear equations. We shall consider in particular equations of the form
(4.9.1)
where X is a constant parameter and the functions f ( t ) , AY(t),Y = 0, 1, 2, ..., m,K,(T;t ) , and K , ( T , ,T~ ; t ) are supposed to be continuous for T , T , , T ~ t ,3 t o . Equations of the form (4.4.1) can be investigated similarly. Let M(o, t ) and N(o, t ) be the kernel functions of the first and second kind of the linear equation m
(4.9.2) “=O
Consider the solution x(t, t o , 6,A) of Eq. (4.9.1) corresponding to the initial condition x(t, t o , 4, 4 = + ( t ) for t E I[%t o ] , (4.9.3) where $(t) E C(l[a,to],D), D being a closed bounded set on the real line. I n addition to the above continuity hypotheses, we assume that
s:
I M(o, t ) I do
s’ I
No,
t ) I do
for
t
for
t >, t o ,
for
t >, t o ,
for t >, t o ,
t0
where a,8, y , 6, , and 6, are positive constants.
E I[a,to],
132
IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS
I n Section 4.3 we have shown that (4.9.5) where
the function zo(t)is given by
and the functions zJt), n
=
1, 2, ..., satisfy the recurrence formula
for t 2 t o . Using the inequalities (4.9.4) and the formula (4.9.7) we find (4.9.9)
I zo(t) I G $0 for 1
2 t o , where 9.0
= 61B
+ 6,Y.
(4.9.10)
Making use of the general recurrence formula (4.9.8),we obtain by induction (4.9.11) I zn(t) 1 < 8 n for t 3 t o , where $n -__ n2n-l$n+lan
y n,
n = l , 2 ,....
(4.9.12)
Therefore, the series (4.9.5)is majorated by the series (4.9.13) Thus: Theorem 4.2. Let Eq. (4.9.1) be given and assume that the inequalities (4.9.4)are satisfied, all functions involved in (4.9.1)being continuous f o r
133
4.9. Boundedness and Stability
t 2 t o . Then, the solution x(t, t o , 4, A) of Eq. (4.9.1) is bounded by B, y , 4 f o r t >, to ,
4 6 0 , %
I x(t,
to
9
474 I < 4 8 0 , B, Y , 4,
(4.9.14)
where A(9, , a, p, y, A) is the sum of the series (4.9.13);i.e.,
+ 8oayX e~p(28~ayA)l.
a, B, y, A) = $.,[I
(4.9.15)
Remembering that zo(t) = x ( t , tn 3 $7
O),
the above argument also shows that: Under the hypotheses of Theorem 4.2, we have
Theorem 4.3.
I x(t, to 4, A) - x(t, t o , A 0 ) I < A(80, a, B, y , A) 9
- 80
(4.9.16)
for t 2 t o . We now compare the solutions x(t, t o , 4, A) and 2(t, t o , A) of the same equation (4.9.1) corresponding to the initial functions + ( t ) and +(t) in the same initial interval ][a,to]. Let
4,
(4.9.18)
and
lp, = max I zn(t)I ,
= 0,
I , 2,
t>to
... .
(4.9.19)
Since Zo(t) is the solution of the linear equation (4.9.2) with the initial function &t), we may write Zo(t) =
J
t0
M(u, t)&u) du
+
(4.9.20)
Thus, by virtue of Eqs. (4.9.7) and (4.9.20), we have
I for t
- %(t)
!
<
(4.9.21)
t o . Therefore, f o ( t ) = zo(t)
70 1
$0
+
= 81)
3
(4.9.22)
134
IV. NONLINEAR DELAY-DIFFERENTIAL
EQUATIONS
where (4.9.23) Since
and
making use of the equality (4.9.22) and the inequalities (4.9.4) and (4.9.23), we find
I zi(t) - zi(t) I
< a2+Y8n + 8,)
(4.9.24)
for t 3 t o . Therefore, %(t> = 4
4
+-
71 >
where
I 71 !, I q 1 I Similarly, from rt
and
we obtain
"U
"(1
$1
< a"Y9.0
= 9.1
+
+
$0).
7j1,
(4.9.25) (4.9.26)
4.9. Boundedness and Stability
135 (4.9.29)
By induction, we can easily show that (4.9.30)
(4.9.31) (4.9.32) Consider now the function
(4.9.27),and (4.9.30)yield T h e inequalities (4.9.21),(4.9.24), 9.3 ) Since the right-hand side of (4.9.33)is infinitesimal with 6 , we may state the following: Theorem 4.4. Let the inequalities (4.9.4) be satisfied. If all the functions involved in Eq. (4.9.1)are continuous and if
I B ( t ) - 4(t) I < 6,
(4.9.34)
where 6<
41
+ 4 a o + $,)A
c
exp(2v(80
+ 80)A)l’
(4.9.35)
then
I Z ( t , t o , 6,A)
- x ( t , t o , $ , A)
I
(4.9.36)
for t 3 t o .
It should be noticed that Theorem 4.4expresses the uniform stability of the system (4.9.1) under small variation of the initial functions. Similar results can be established if to and h also vary. T h e method described above can also be extended to the systems of the form (4.4.1).