UDC 517.948.34

ON THE EXISTENCE OF BOUNDED AND PERIODIC SOLUTIONS OF INTEGRO-DIFFERENTIAL EQUATIONS WITH RETARDED ARGUMENT

G. F. YAKOVLEVA

Consider the system of equations

\[ \frac{dx}{dt}=A(t)x(t)+B(t)x(t-\tau)+f(t)+\varepsilon F[t,x(t),x(t-\tau),\varepsilon]+ \]

\[ +\varepsilon\int_{t_0}^{\infty}K[t,z,x(z),x'(z),x(z-\tau),\varepsilon]\,dz, \tag{1} \]

where \(A(t), B(t)\) are \(n\)-square matrix functions; \(x(t), f(t)\) are \(n\)-dimensional vectors; \(F(t,u,u_1,\varepsilon)\) and \(K(t,z,u,v,u_1,\varepsilon)\) are \(n\)-dimensional vector functions of the numerical arguments \(t\) and \(z\) and of \(n\)-dimensional vectors; \(\varepsilon\) is a numerical parameter; \(\tau>0\) is a constant delay.

We investigate the question of the existence of bounded and periodic solutions of equation (1).

This question for systems of linear and quasilinear differential-difference equations has been studied, for example, by Wan Chou-Hua [1], S. N. Shimanov [2], V. P. Rubanik and V. I. Fodchuk [3], A. Halanay [4], [5], and for systems of integro-differential-difference equations of a somewhat different type and with constant coefficients, by V. P. Misnik [6].

Suppose that the following conditions are satisfied:

a) the functions \(A(t), B(t), f(t)\) are continuous and bounded for \(t\geq t_0\), and

\[ |A(t)|\leq A_0,\quad |B(t)|\leq B_0,\quad |f(t)|\leq f_0,\quad A_0+B_0<1; \]

b) the vector functions \(F(t,u,u_1,\varepsilon)\) and \(K(t,z,u,v,u_1,\varepsilon)\) are continuous jointly in their arguments in the domain

\[ D_r\{\,t\geq t_0;\ z\geq t_0;\ |\varepsilon|\leq \varepsilon^0;\ |u|,\ |u_1|,\ |v|\leq r\,\}; \]

c)

\[ |F(t,0,0,\varepsilon)|\leq M_1;\quad |K(t,z,0,0,0,\varepsilon)|\leq M_2|\psi(z)| \]

for all \(t,z\geq t_0\), \(|\varepsilon|\leq \varepsilon^0\), where \(\psi(z)\) is a scalar function for which

\[ \int_{t_0}^{\infty}|\psi(z)|\,dz=p \]

converges;

d) in the domain \(D_r\),

\[ |F(t,u,u_1,\varepsilon)-F(t,u',u_1',\varepsilon)| \leq N_1(|u-u'|+|u_1-u_1'|), \]

\[ |K(t,z,u,v,u_1,\varepsilon)-K(t,z,u',v',u_1',\varepsilon)|\leq \]

\[ \leq N_2|\psi(z)|\bigl(|u-u'|+|v-v'|+|u_1-u_1'|\bigr), \]

where \(N_1, N_2\) are constants.

Conditions c) and d) ensure the convergence, uniform with respect to \(\varepsilon\) and \(t\), of the integral

\[ \int_{t_0}^{\infty} K[t, z, x(z), x'(z), x(z-\tau), \varepsilon]\,dz \]

in the domain \(D_r\).

First we prove the existence of a unique bounded solution of equation (1) for \(t \geqslant t_0\) under the initial condition

\[ x(t)=\varphi(t)\quad \text{for } \quad t_0-\tau \leqslant t \leqslant t_0 . \tag{2} \]

For this purpose we introduce a matrix function \(Y(s,t)\), which for all \(t>t_0\) satisfies the equations

\[ \frac{\partial}{\partial s}Y(s,t)+Y(s,t)A(s)=0 \quad \text{for } \quad t-\tau \leqslant s \leqslant t,\quad Y(t,t)=E, \]

\[ \frac{\partial}{\partial s}Y(s,t)+Y(s,t)A(s)+Y(s+\tau,t)B(s+\tau)=0 \]

\[ \text{for } \quad t_0 \leqslant s \leqslant t-\tau . \]

Since the matrix functions \(A(s)\) and \(B(s)\) are continuous for \(s \geqslant t_0\), using the results of Bellman and Cooke [7], we conclude that \(Y(s,t)\) is determined uniquely and is continuous for \(t>t_0,\ t_0 \leqslant s \leqslant t\).

Let \(z(t,t_0,\varphi)\) be a solution of the linear homogeneous system

\[ \frac{dx}{dt}=A(t)x(t)+B(t)x(t-\tau), \tag{3} \]

satisfying the initial condition (2).

Lemma. Every solution of equation (1) satisfying the initial condition (2) is a solution of the equation

\[ x(t)=z(t,t_0,\varphi)+\int_{t_0}^{t}Y(s,t)\{f(s)+\varepsilon F[s,x(s),x(s-\tau),\varepsilon]+ \]

\[ +\varepsilon \int_{t_0}^{\infty}K[s,z,x(z),x'(z),x(z-\tau),\varepsilon]\,dz\}\,ds \tag{4} \]

with the initial condition (2), and conversely.

The validity of the lemma is easy to prove using the results of Bellman and Cooke [7].

Next consider the space \(C_1(r)\) of vector functions \(x(t,\varepsilon)\), continuously differentiable for all \(t \geqslant t_0,\ |\varepsilon|\leqslant \varepsilon^0\), satisfying the inequalities \(|x(t,\varepsilon)|\leqslant r,\ |x'(t,\varepsilon)|\leqslant r\), and reducing to \(\varphi(t)\) for \(t_0-\tau \leqslant t \leqslant t_0\).

Defining in the space \(C_1(r)\) a metric by the rule

\[ \rho(x_1,x_2)= \max\left\{ \sup_{t\geqslant t_0}|x_1(t,\varepsilon)-x_2(t,\varepsilon)|;\ \sup_{t\geqslant t_0}|x_1'(t,\varepsilon)-x_2'(t,\varepsilon)| \right\}, \]

we obtain a complete metric space.

In this space introduce the operator

\[ I(x)\equiv z(t,t_0,\varphi)+\int_{t_0}^{t}Y(s,t)\left\{f(s)+\varepsilon F[s,x(s),x(s-\tau),\varepsilon]+\right. \]

\[ \left.+\varepsilon\int_{t_0}^{\infty}K[s,z,x(z),x'(z),x(z-\tau),\varepsilon]\,dz\right\}\,ds . \tag{5} \]

Theorem 1. Suppose that for system (1): 1) conditions a)—c) are satisfied;

\[ 2)\quad \int_{t_0}^{t}|Y(s,t)|\,ds\leq \Gamma;\qquad 3)\quad |z(t,t_0,\varphi)|\leq \gamma<r;\qquad 4)\quad |\varepsilon|<\bar\varepsilon^{0}, \]

where

\[ \bar\varepsilon^{0}=\min\{L_1,L_2,L_3,L_4,\varepsilon^{0}\},\qquad L_1=\frac{r-\gamma-\Gamma f_0}{\Gamma[M_1+pM_2+r(2N_1+3pN_2)]}, \]

\[ L_2=\frac{r-f_0-(A_0+B_0)r}{M_1+pM_2+r(2N_1+3pN_2)},\qquad L_3=\frac{1}{\Gamma(2N_1+3pN_2)},\qquad L_4=\frac{1-A_0-B_0}{2N_1+3pN_2}; \]

\[ 5)\quad f_0<\mu,\quad \mu=\min\left\{\frac{r-\gamma}{\Gamma};\, r(1-A_0-B_0)\right\}, \]

then system (1) has a unique solution \(x=\Phi(t,\varepsilon)\), continuously differentiable for all \(t\geq t_0\), \(|\varepsilon|<\bar\varepsilon^{0}\), satisfying the initial condition (2) and the inequalities

\[ |\Phi(t,\varepsilon)|\leq r,\qquad |\Phi'(t,\varepsilon)|\leq r . \]

Proof. The validity of this theorem follows from the contraction mapping principle, which the operator (5) satisfies, and from the lemma. First we prove that the operator (5) maps elements of the space \(C_1(r)\) into elements of the same space. By virtue of the conditions of the theorem,

\[ |I(x)|\leq |z(t,t_0,\varphi)|+\int_{t_0}^{t}|Y(s,t)|\times \]

\[ \times\left\{|f(s)|+|\varepsilon|\,|F[s,x(s),x(s-\tau),\varepsilon]|+\right. \]

\[ \left.+|\varepsilon|\int_{t_0}^{\infty}|K[s,z,x(z),x'(z),x(z-\tau),\varepsilon]|\,dz\right\}\,ds< \]

\[ <\gamma+\Gamma\left\{f_0+\bar\varepsilon^{0}[M_1+pM_2+r(2N_1+3pN_2)]\right\}\leq \]

\[ \leq \gamma+\Gamma f_0+L_1\Gamma[M_1+pM_2+r(2N_1+3pN_2)]=r, \]

i.e. \(|I(x)|<r\).

Similarly,

\[ \left|\frac{dI}{dt}\right|\leq (A_0+B_0)r+f_0+|\varepsilon|[M_1+pM_2+r(2N_1+3pN_2)]< \]

\[ <(A_0+B_0)r+f_0+\bar\varepsilon^{0}[M_1+pM_2+r(2N_1+3pN_2)]\leq \]

\[ \leq (A_0+B_0)r+f_0+L_2[M_1+pM_2+r(2N_1+3pN_2)]=r, \]

i.e.

\[ \left|\frac{dI}{dt}\right|<r . \]

Moreover, from conditions a) and b) it follows that \(I(x)\) and \(\dfrac{dI}{dt}\) are continuous in the domain \(D_r\).

We now show that the mapping effected by the operator (5) is a contraction:

\[ \begin{aligned} \sup_{t\ge t_0}|I(x_1)-I(x_2)| &\le |\varepsilon|\Gamma N_1\left(\sup_{s\ge t_0}|x_1(s)-x_2(s)|+\right.\\ &\quad \left.+\sup_{s\ge t_0}|x_1(s-\tau)-x_2(s-\tau)|\right)\\ &\quad +|\varepsilon|\Gamma pN_2\left(\sup_{z\ge t_0}|x_1(z)-x_2(z)|+\right.\\ &\quad \left.+\sup_{z\ge t_0}|x'_1(z)-x'_2(z)|+\sup_{z\ge t_0}|x_1(z-\tau)-x_2(z-\tau)|\right)\\ &\le |\varepsilon|\Gamma(2N_1+3pN_2)\rho(x_1,x_2) = \alpha_1\rho(x_1,x_2), \end{aligned} \]

where \(0<\alpha_1=|\varepsilon|\Gamma(2N_1+3pN_2)<1\) by condition 4) of Theorem 1;

\[ \begin{aligned} \sup_{t\ge t_0}\left|\frac{dI(x_1)}{dt}-\frac{dI(x_2)}{dt}\right| &\le A_0\sup_{t\ge t_0}|x_1(t)-x_2(t)|+\\ &\quad +B_0\sup_{t\ge t_0}|x_1(t-\tau)-x_2(t-\tau)|\\ &\quad +|\varepsilon|N_1\left(\sup_{t\ge t_0}|x_1(t)-x_2(t)|+\right.\\ &\quad \left.+\sup_{t\ge t_0}|x_1(t-\tau)-x_2(t-\tau)|\right)\\ &\quad +|\varepsilon|pN_2\left(\sup_{z\ge t_0}|x_1(z)-x_2(z)|+\right.\\ &\quad \left.+\sup_{z\ge t_0}|x'_1(z)-x'_2(z)|+\sup_{z\ge t_0}|x_1(z-\tau)-x_2(z-\tau)|\right)\\ &\le [A_0+B_0+|\varepsilon|(2N_1+3pN_2)]\rho(x_1,x_2) = \alpha_2\rho(x_1,x_2), \end{aligned} \]

where \(0<\alpha_2=A_0+B_0+|\varepsilon|(2N_1+3pN_2)<1\) by condition 4) of Theorem 1. Let \(\alpha=\max(\alpha_1,\alpha_2)\). Then

\[ \rho[I(x_1),I(x_2)]\le \alpha\rho(x_1,x_2),\quad \text{where } 0<\alpha<1. \]

Thus the conditions of the contraction mapping principle are fulfilled. Therefore the initial-value problem (4), (2), and hence also (1)—(2), has a unique solution \(x=\Phi(t,\varepsilon)\), continuously differentiable for all \(t\ge t_0\), \(|\varepsilon|<\bar\varepsilon^{\,0}\), for which the estimates

\[ |\Phi(t,\varepsilon)|\le r,\qquad |\Phi'(t,\varepsilon)|\le r \]

are valid.

We note some special cases.

A) If in equation (1) the function \(K\) does not contain \(x'(z)\), then instead of the space \(C_1(r)\) one may take the ordinary space \(C(r)\) with the Chebyshev metric. In this case Theorem 1 (without requiring boundedness of \(\Phi'(t,\varepsilon)\)) will hold under weaker restrictions, namely:

1) the condition \(A_0+B_0<1\) is removed;

2) condition 4) is replaced by the condition \(|\varepsilon|<\bar\varepsilon^{\,0}\), where
\[ \bar\varepsilon^{\,0}=\min\{L_1,L_3,\varepsilon^0\}; \]

3) condition 5) is replaced by the condition
\[ f_0<\frac{r-\gamma}{\Gamma}. \]

B) If one proves only the existence and uniqueness of a solution of equation (1), without requiring boundedness of the solution and of its derivative, then

1) conditions 3) and 5) of the theorem are removed;

2) condition 4) is replaced by the condition \(|\varepsilon|<\bar\varepsilon^{\,0}\), where
\[ \bar\varepsilon^{\,0}=\min\{L_3,L_4,\varepsilon^0\}. \]

If, in addition, in this case \(K\) does not contain \(x'\), then instead of the space \(C_1(r)\) one must take the space \(C(r)\). Then the condition \(A_0+B_0<1\) is also removed, and
\[ \bar\varepsilon^{\,0}=\min\{L_3,\varepsilon^0\}. \]

C) Let in equation (1) \(F \equiv 0\) and \(K \equiv 0\) (or \(\varepsilon = 0\)). Then the system of integro-differential equations is transformed into a system of differential equations with retarded argument

\[ \frac{dx}{dt}=A(t)x(t)+B(t)x(t-\tau)+f(t). \tag{6} \]

The solution of this system satisfying the initial condition (2) is obtained from formula (4), if in it one sets \(F \equiv 0\), \(K \equiv 0\) (or \(\varepsilon=0\)):

\[ x(t)=z(t,t_0,\varphi)+\int_{t_0}^{t}Y(s,t)f(s)\,ds, \tag{7} \]

and, as follows from [8], this solution is unique.

In this special case, from the conditions of Theorem 1 there automatically drop out conditions 1)—b), c), d) and 4). The remaining conditions ensure the validity of the assertion of the theorem.

Indeed, from (7), taking into account conditions 2), 3) and 5) of the theorem, we obtain

\[ |x(t)|\leq \gamma+f_0\Gamma<\gamma+\mu\Gamma\leq \gamma+\frac{r-\gamma}{\Gamma}\,\Gamma=r, \tag{8} \]

i.e. \(|x(t)|<r\).

From (6), by virtue of conditions 1)—a) and 5) of the theorem and estimate (8), we find

\[ \left|\frac{dx}{dt}\right| < (A_0+B_0)r+f_0<(A_0+B_0)r+ \]

\[ +\mu\leq (A_0+B_0)r+r(1-A_0-B_0)=r. \]

Consequently,

\[ \left|\frac{dx}{dt}\right|<r. \]

Theorem 2. Let conditions a)—e) be fulfilled, with the exception of \(A_0+B_0<1\). If, moreover, the functions \(A(t)\), \(B(t)\), \(f(t)\), \(F\) and \(K\) are periodic in \(t\) with period \(\omega>\tau\), and system (3) has no periodic solutions except the trivial one, then system (1), for sufficiently small values of the parameter \(\varepsilon\), has a unique periodic solution of period \(\omega\).

Proof. This theorem is easily proved by the method of successive approximations. Here one uses a result of A. Halanay [4], which consists in the fact that the system

\[ \frac{dx}{dt}=A(t)x(t)+B(t)x(t-\tau)+\Psi(t) \tag{9} \]

has a unique periodic solution of period \(\omega\), if the functions \(A(t)\), \(B(t)\) and \(\Psi(t)\) are periodic with period \(\omega\) and system (3) has no periodic solutions except the trivial one; moreover, this periodic solution admits an estimate of the form

\[ |x(t)|\leq R\sup|\Psi(t)|, \]

where the constant \(R\) does not depend on the function \(\Psi(t)\).

As the zeroth approximation we take an arbitrary continuously differentiable periodic function \(x_0(t)\), of period \(\omega\), defined for all \(t\geq t_0-\tau\). Let

\[ |x_0(t)|\leq T,\quad |x_0'(t)|\leq T. \]

The subsequent approximations are defined by the rule

\[ \frac{d x_n}{dt}=A(t)x_n(t)+B(t)x_n(t-\tau)+f(t)+ \]

\[ +\varepsilon F[t,\ x_{n-1}(t),\ x_{n-1}(t-\tau),\ \varepsilon]+ \tag{10} \]

\[ +\varepsilon\int_{t_0}^{\infty}K[t,\ z,\ x_{n-1}(z),\ x'_{n-1}(z),\ x_{n-1}(z-\tau),\ \varepsilon]\,dz \]

for \(t\geq t_0\),

\[ x_n(t)=\varphi_n(t)\quad \text{for } t_0-\tau\leq t\leq t_0\quad (n=1,2,\ldots), \]

where \(\varphi_n(t)\) \((n=1,2,\ldots)\) are initial functions, which are determined in each approximation uniquely from the condition of periodicity of the solution (see [4]).

Considering the systems (10) successively, in each approximation we shall have systems of type (9). Consequently, for any \(n=1,2,\ldots\), the system (10) has a unique periodic solution of period \(\omega\), for which the estimate

\[ |x_n(t)|\leq R\sup |f_n(t)|, \tag{11} \]

is valid, where

\[ f_n(t)=f(t)+\varepsilon F[t,\ x_{n-1}(t),\ x_{n-1}(t-\tau),\ \varepsilon]+ \]

\[ +\varepsilon\int_{t_0}^{\infty}K[t,\ z,\ x_{n-1}(z),\ x'_{n-1}(z),\ x_{n-1}(z-\tau),\ \varepsilon]\,dz. \]

Using estimate (11) and conditions a)—г), we shall prove the convergence of the sequences

\[ x_0(t),\ x_1(t),\ x_2(t),\ \ldots,\ x_n(t),\ \ldots \tag{12} \]

\[ x'_0(t),\ x'_1(t),\ x'_2(t),\ \ldots,\ x'_n(t),\ \ldots \tag{12'} \]

For this purpose we form the series

\[ x_0(t)+[x_1(t)-x_0(t)]+[x_2(t)-x_1(t)]+\ldots \tag{13} \]

By \((13')\) we denote the series obtained from (13) by termwise differentiation with respect to \(t\). Let us estimate the terms of the series (13) and \((13')\):

\[ |x_1(t)-x_0(t)|\leq |x_1(t)|+|x_0(t)|\leq R\sup |f_1(t)|+T\leq RG+T, \]

where

\[ G=f_0+|\varepsilon|[M_1+pM_2+(2N_1+3pN_2)T], \]

\[ |x'_1(t)-x'_0(t)|\leq |x'_1(t)|+|x'_0(t)|\leq [1+R(A_0+B_0)]G+T. \]

Let

\[ R_1=\max\{R;\ [1+R(A_0+B_0)]\}. \]

Then

\[ |x^{(k)}_1(t)-x^{(k)}_0(t)|\leq S\quad (k=0,1), \]

where

\[ S=R_1G+T, \]

\[ |x_2(t)-x_1(t)|\leq R\sup |f_2(t)-f_1(t)|\leq R|\varepsilon|(2N_1+3pN_2)S; \]

\[ |x'_2(t)-x'_1(t)|\leq [1+R(A_0+B_0)]|\varepsilon|(2N_1+3pN_2)S. \]

Therefore,

\[ \left|x_2^{(k)}(t)-x_1^{(k)}(t)\right|\leq R_1|\varepsilon|\,(2N_1+3pN_2)S \quad (k=0,1). \]

Further,

\[ \left|x_3(t)-x_2(t)\right|\leq R\sup |f_3(t)-f_2(t)| \leq RR_1|\varepsilon|^2(2N_1+3pN_2)^2S, \]

\[ \left|x'_3(t)-x'_2(t)\right|\leq [1+R(A_0+B_0)]|\varepsilon|^2(2N_1+3pN_2)^2R_1S. \]

Hence,

\[ \left|x_3^{(k)}(t)-x_2^{(k)}(t)\right| \leq [|\varepsilon|R_1(2N_1+3pN_2)]^2S \quad (k=0,1). \]

Similarly,

\[ \left|x_n^{(k)}(t)-x_{n-1}^{(k)}(t)\right| \leq [|\varepsilon|R_1(2N_1+3pN_2)]^{\,n-1}S \quad (k=0,1). \]

It follows from this that the series (13) and (13′) are majorized by the series

\[ T+S\sum_{n=0}^{\infty}[|\varepsilon|R_1(2N_1+3pN_2)]^n. \tag{14} \]

The series (14) converges when

\[ |\varepsilon|<\frac{1}{R_1(2N_1+3pN_2)}. \tag{15} \]

Consequently, the series (13) and (13′), and therefore also the sequences (12) and (12′), converge uniformly for all \(t\geq t_0\).

The limit of the sequence (12) gives the unique periodic solution of equation (1) for \(\varepsilon\) satisfying condition (15).

References

1. Wang Rou Hwai. Sci. Record. N. S., 2, 1, 23—26, 1958.
2. Shimanov S. N. PMM, 23, 5, 836—844, 1959.
3. Rubanik V. P., Fodchuk V. I. Ukr. Math. Journal, vol. XIV, No. 1, 87—92, 1962.
4. Halanay A. Compt. Rend. Acad. Sci. Paris, 249, 2708—2709, 1959.
5. Halanay A. Compt. Rend. Acad. Sci. Paris, 250, 797—798, 1960.
6. Misnik V. P. Studies on integro-differential equations in Kirgizia, iss. 2, 233—237, 1962.
7. Bellman R. and Cooke K. L. Trans. Amer. Math. Soc., 92, 470—500, 1959.
8. Halanay A. Studii și cercetări mat. Acad. RPR, 12, No. 2, 367—391, 1961.

Received by the editors
December 30, 1965

Krasnodar Polytechnic
Institute