UDC 517.949.2

EQUATIONS OF NEUTRAL TYPE WITH VARIABLE SMALL RETARDATION

V. I. ROZHKOV

In previous papers [1] and [2] the behavior was studied of the solution of a system of equations of neutral type with small constant retardation \(\Delta t\)

\[ \dot{x}_i(t)=f_i\bigl(t,x_j(t),x_j(t-\Delta t),\dot{x}_j(t-\Delta t)\bigr), \]

\[ x_i(t)=\varphi_i(t)\quad \text{for } 0\leq t\leq \Delta t \quad (i,j=1,2,\ldots,m). \]

It was established that, under certain restrictions imposed on the functions \(f\) and \(\varphi\), the solution \(x(t,\Delta t)\) of this system tends, as \(\Delta t\to 0\), to the solution of the corresponding degenerate system obtained from the original one by setting \(\Delta t=0\). An asymptotic expansion of the solution in powers of \(\Delta t\) was then constructed.

The aim of the present paper is to extend these results to the case of a variable small retardation. To simplify the exposition, a single equation will be considered, although everything proved is also valid for systems of equations.

In the first section the existence of a solution of a neutral-type equation with small variable retardation and its convergence to the degenerate solution as the retardation decreases will be proved. In the following two sections asymptotic expansions of the solution in powers of the small retardation will be established.

§ 1. EXISTENCE OF A SOLUTION AND LIMITING TRANSITION

Consider the equation of neutral type

\[ \dot{x}(t)=f\bigl(t,x(t),x(t-\mu(t,h)),\dot{x}(t-\mu(t,h))\bigr), \tag{1.1} \]

where \(\mu(t,h)>0\) is a small variable retardation, given on the interval \(0\leq t\leq T\), and \(h>0\) is a small parameter.

Introducing the notation \(\dot{x}=u\), \([z]=z(t-\mu(t,h))\), we rewrite the original equation (1.1) in the form of the system

\[ u=f(t,x,[x],[u]),\quad \dot{x}=u. \tag{1.2} \]

We shall consider the solution of this system on the interval \(0\leq t\leq T\) under the following initial conditions:

\[ x(t)=\varphi(t),\quad u(t)=\dot{\varphi}(t)\quad \text{for } t\in E_0=\{t\geq 0:\ t-\mu(t,h)\leq 0\}, \tag{1.3} \]

where \(\varphi(t)\) is a prescribed function.

Assume that on the interval \(0\leq t\leq T\) there exists a continuous solution of the degenerate problem

\[ \bar u=f(t,\bar x,\bar x,\bar u),\qquad \dot{\bar x}=\bar u,\qquad \bar x(0)=\varphi(0), \tag{1.4} \]

and denote by \(Q\) the domain

\[ \left\{ \begin{array}{l} 0\leq t\leq T,\quad |x-\bar x(t)|\leq \delta,\quad |y-\bar x(t)|\leq \delta,\quad |u-\bar u(t)|\leq \delta \quad \text{for}\\[2mm] \delta\leq t\leq T,\quad |u-\bar u(t)|\leq |\dot\varphi(0)-\bar u(0)|+\delta \quad \text{for } 0\leq t\leq \delta, \end{array} \right. \]

where \(\delta>0\) is an arbitrarily small but fixed number, and \((t,x,y,u)\) are the arguments of the function \(f\).

We shall say that conditions A are satisfied if

1) the function \(\mu(t,h)\) can be represented in the form

\[ \mu(t,h)=h\mu_1(t)+\ldots+h^{\,n+1}\mu_{n+1}(t)+h^{\,n+2}\mu_{n+2}(t,h), \tag{1.5} \]

where the functions \(\mu_k\), \(k=1,2,\ldots,n+1\), have \(n+2-k\) continuous derivatives for \(0\leq t\leq T\), and \(\mu_1(0)>0\); the function \(\mu_{n+2}(t,h)\) is continuous;

2) in the domain \(Q\) the function \(f(t,x,y,u)\) satisfies the following stability condition (see [1]):

\[ \left|\frac{\partial f}{\partial u}\right|<a<1; \tag{1.6} \]

3) the function \(f(t,x,y,u)\) in the domain \(Q\), and the function \(\varphi(t)\) for \(0\leq t\leq \delta\), are \(n+2\) times continuously differentiable.

By the first of conditions A, on the whole interval \(0\leq t\leq T\),

\[ \mu(t,h)<C_0h. \]

Let \(r_1(h)\) denote the first zero of the function \(\mu(t,h)\) in order of increasing \(t\), i.e. \(\mu(r_1(h),h)=0\). Since \(\mu_1(0)>0\), for sufficiently small \(h\)

\[ r_1(h)>d_0>0. \]

The number \(n\) occurring in conditions A denotes the order of the asymptotic formula that will be constructed in §§ 2, 3.

The aim of the present section is to establish the following theorem.

Theorem. If conditions A are satisfied for \(n=0\), then for the solution of problem (1.2), (1.3) the following limiting transition is valid:

\[ \begin{gathered} \lim_{h\to 0}x(t,h)=\bar x(t)\quad \text{for } 0\leq t\leq T,\\[2mm] \lim_{h\to 0}u(t,h)=\bar u(t)\quad \text{for } 0<t\leq T. \end{gathered} \tag{1.7} \]

Proof. At any point \(t\in[0,r_1]\) the solution of problem (1.2), (1.3) can be obtained by the method of steps, just as for a constant delay. Therefore, at any point \(t\in[0,r_1]\) the solution exists and, according to [1] and [2], satisfies the inequalities

\[ |x-\bar x|<Ch,\qquad |u-\bar u|<C e^{-\varkappa t/h}+Ch, \tag{1.8} \]

where \(0<\varkappa<-\dfrac{\ln a}{C_0}\), and the constant \(C\) depends neither on \(t\) nor on \(h\).

To continue the solution to the closed interval \([0,r_1]\), we proceed

in the following way. Since \(u(t)\) is known at every point \(t \in [0,r_1)\) and the estimates (1.8) are satisfied, one can find the number \(x_1\):

\[ x_1=\varphi(0)+\int_0^{r_1} u(t)\,dt, \]

where it is obvious that \(|x_1-\bar{x}(r_1)|<Ch\).

Let us now consider the equation with respect to \(u_1\):

\[ u_1=f(r_1,x_1,x_1,u_1). \]

Its solution can be obtained by the method of successive approximations, taking as the zeroth approximation \(u_1^{(0)}=u(r_1)\). We have

\[ u_1^{(0)}=f(r_1,\bar{x}(r_1),\bar{x}(r_1),u_1^{(0)}), \]

\[ u_1^{(1)}=f(r_1,x_1,x_1,u_1^{(0)}), \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ u_1^{(n+1)}=f(r_1,x_1,x_1,u_1^{(n)}), \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

We further have

\[ |u_1^{(1)}-u_1^{(0)}|<Ch,\quad |u_1^{(n+1)}-u_1^{(n)}|<a|u_1^{(n)}-u_1^{(n-1)}|<\ldots<a^nCh. \]

Consequently, the successive approximations converge and

\[ |u_1-\bar{u}(r_1)|<Ch. \]

Let us now show that

\[ \lim_{t\to r_1}u(t)=u_1, \tag{1.9} \]

i.e., for every \(\varepsilon>0\) there is a \(\delta(\varepsilon)>0\) such that from the inequality \(r_1-t<\delta\) it follows that \(|u(t)-u_1|<\varepsilon\).

We have

\[ u(t)-u_1=f(t,x(t),x(t-\mu(t,h)),u(t-\mu(t,h)))- \]

\[ -f(r_1,x_1,x_1,u_1)=f_x^*(x-x_1)+f_y^*[x-x_1]+ \]

\[ +f_u^*[u-u_1]+f_t^*(t-r_1). \]

Denote

\[ b_1=\max_Q |f_x|,\quad b_2=\max_Q |f_y|,\quad b_3=\max_Q |f_t|,\quad d=\max_{[0,T]}|\bar{u}|+1, \]

then it is clear that

\[ |u(t)-u_1|<[(b_1+b_2)d+b_3](r_1-t)+b_2d\,\mu(t,h)+a[|u(t)-u_1|]. \]

Choose \(t'\) so that

\[ [(b_1+b_2)d+b_3](r_1-t')<\frac{\varepsilon(1-a)}{4}, \]

\[ \mu<\frac{\varepsilon(1-a)}{4b_2d}\quad \text{for } t\in[t',r_1]. \]

Then for \(t>t'\) we obtain the following difference inequality:

\[ |u-u_1| \leq \frac{\varepsilon(1-a)}{2}+a[|u-u_1|]. \]

We solve this inequality by the method of steps, taking as the initial set
\(\{t \geq t' : t-\mu(t,h) \leq t'\}\) and using the fact that certainly
\(|u(t)-u_1|<2d\). Then at the \(N\)-th step we obtain

\[ |u-u_1| < \frac{\varepsilon(1-a)}{2}(1+a+a^2+\ldots+a^{N-1})+a^N\cdot 2d < \frac{3}{4}\varepsilon \]

provided

\[ N > \log_a \frac{\varepsilon}{8d}. \]

If we now denote by \(t''\) the extreme right-hand point of the \(N\)-th step, then it is clear that one may take \(\delta:=r_1-t''\). Formula (1.9) is proved.

Let us now put \(x(r_1)=x_1,\ u(r_1)=u_1\). In this case we have

\[ \dot{x}(r_1)=\lim_{\Delta\to+0}\frac{x(r_1)-x(r_1-\Delta)}{\Delta} =\lim_{\Delta\to+0}\int_{r_1-\Delta}^{r_1}u\,dt=u_1. \]

Thus the solution has been continued up to the point \(t=r_1\), and for \(t>r_1\) we have the following initial-value problem:

\[ u=f(t,x,[x],[u]),\quad \dot{x}=u;\quad x(r_1)=x_1,\quad u(r_1)=u_1, \]

where \(u_1=f(r_1,x_1,x_1,u_1)\).

Now the method of steps is no longer applicable, but it turns out that, by virtue of the stability conditions, the solution can be obtained by the method of successive approximations.

Indeed, let us construct the successive approximations as follows:

\[ u_0=f(t,\bar{x},[\bar{x}],[\bar{u}]),\quad \dot{x}_0=u_0,\quad x_0(r_1)=x_1, \]

\[ u_{n+1}=f(t,x_n,[x_n],[u_n]),\quad \dot{x}_{n+1}=u_{n+1},\quad x_{n+1}(r_1)=x_1. \]

It is clear that

\[ |u_0-\bar{u}|<Ch,\quad |u_1-u_0|<Ch,\quad |x_0-\bar{x}|<Ch. \]

From the construction it follows that, for any \(n\), the functions \(u_n(t)\) are continuous. Hence we may write

\[ \max_{[r_1,t]}|u_{n+1}-u_n| = \max_{[r_1,t]}\left| f\left(t,x_1+\int_{r_1}^{t}u_n\,dt,\left[x_1+\int_{r_1}^{t}u_n\,dt\right],[u_n]\right) -\right. \]

\[ \left. -f\left(t,x_1+\int_{r_1}^{t}u_{n-1}\,dt,\left[x_1+\int_{r_1}^{t}u_{n-1}\,dt\right],[u_{n-1}]\right) \right| < \]

\[ <\bigl[(t-r_1)(b_1+b_2)+a\bigr]\max_{[r_1,t]}|u_n-u_{n-1}| <a_1\max_{[r_1,t]}|u_n-u_{n-1}|, \]

provided only that \(t\in[r_1,t_1]\) and \((t_1-r_1)(b_1+b_2)+a<a_1<1\). From the obtained inequality we have

\[ |u_{n+1}-u_n|<a_1^n Ch\quad \text{for } t\in[r_1,t_1]. \]

Thus we obtain that \(u_n\to u,\ x_n\to x\), where
\(u=f(t,x,[x],[u])\); \(\dot{x}=u\); \(x(r_1)=x_1\); \(u(r_1)=u_1\). Moreover, it is clear that
\(|u-\bar{u}|<Ch,\ |x-\bar{x}|<Ch\) for \(t\in[r_1,t_1]\).

The uniqueness of the solution is obtained as usual: suppose that in the domain \(Q\) on the interval \([r_1,t_1]\) there are two continuous solutions \(u_1(t)\) and \(u_2(t)\); then

\[ \max_{[r_1,t_1]} |u_1-u_2| \leq a_1 \max_{[r_1,t_1]} |u_1-u_2| \quad (a_1<1), \]

therefore \(u_1-u_2=0\).

After the point \(r_1\) the solution is found by the method of successive approximations up to the end of the interval \([0,T]\), and we verify that on the whole interval \([0,T]\) the estimates

\[ |u-\bar u| < Ce^{-\varkappa t/h}+Ch,\quad |x-\bar x| < Ch, \]

hold; whence, as \(h\to0\), the assertion of the theorem follows.

§ 2. ASYMPTOTIC EXPANSION OF THE SOLUTION IN A NEIGHBORHOOD OF THE INITIAL POINT

In this section we shall establish asymptotic formulas valid on the interval \(0\leq t\leq -Ah\ln h\), where \(A>0\) is a sufficiently large positive constant depending only on the functions \(f\) and \(\varphi\).

Let us write the original system once more:

\[ u=f(t,x,[x],[u]),\quad \dot x=u, \tag{2.1} \]

\[ x(t)=\varphi(t),\quad u(t)=\psi(t)\quad \text{for } t\in E_0. \tag{2.2} \]

and substitute in equation (2.1), in place of \(x\) and \(u\),

\[ \sum_{k=0}^{\infty} h^k x_k \quad\text{and}\quad \sum_{k=0}^{\infty} h^k u_k, \]

after first replacing \(t\) in the equations by \(\tau h\). Expanding now the function \(f\) in powers of \(h\) and equating the coefficients of like powers of \(h\), we obtain equations for \(x_k\) and \(u_k\). In the final equations, instead of \(\tau h\) we again introduce \(t\). (The expressions \([x_k]\) and \([u_k]\) are not expanded in \(h\).) For example,

\[ u_0=f(0,\varphi(0),\varphi(0),[u_0]),\quad x_0=\varphi(0). \]

Denote \(M_0=(0,\varphi(0),\varphi(0),[u_0])\); then

\[ h u_1=f_t(M_0)t+f_x(M_0)h x_1+f_y(M_0)[h x_1]+f_u(M_0)[h u_1], \]

\[ \frac{d}{dt}(h x_1)=u_0, \]

and so on.

The initial conditions for \(x_k\) and \(u_k\) are determined by the following formulas:

\[ x_k(0)=0,\quad k\geq 1;\qquad h^k u_k=\frac{t^k}{k!}\,\psi^{(k)}(0)\quad \text{for } t\in E_0,\ k\geq 0. \]

The constructed functions, which satisfy estimates \(|h^k z_k|<Ct^k\), will serve as an asymptotic approximation in the neighborhood indicated above, i.e., the following is true.

Theorem. If the functions \(f\), \(\varphi\), and \(\psi\) satisfy conditions A of § 1, then the estimates

\[ \left|x(t,h)-\sum_{k=0}^{n} h^k x_k\right|<Ct^{\,n+1},\qquad \left|u-\sum_{k=0}^{n} h^k u_k\right|<Ct^{\,n+1}, \]

where the constant \(C\) does not depend on \(t\) and \(h\) for \(h\leq t\leq \delta\) and \(h\leq h_0\). Here \(\delta>0\) is small, but fixed as \(h\to 0\).

Proof.

Zero approximation. Denote by \(\widetilde{\Delta}_0\) and \(\widetilde{\delta}_0\) the differences
\(\widetilde{\Delta}_0=u-u_0,\ \widetilde{\delta}_0=x-x_0=x-\varphi(0)\). Then, obviously, we shall have
\[ |\widetilde{\delta}_0|<Ct,\qquad \widetilde{\Delta}_0=f_u^*[\widetilde{\Delta}_0]+O(t),\qquad \widetilde{\Delta}_0=O(h)\quad \text{for } t\in E_0 . \]
Since for small \(\delta\) the function \(\mu\) does not vanish on the interval \([0,\delta]\), we apply the method of steps and, consequently, by Lemma 1 of [2] obtain
\[ |\widetilde{\Delta}_0|<Ct\quad \text{for } h\leq t\leq \delta, \]
as was required.

First approximation. Introducing, by analogy with the zero approximation, the differences
\(\widetilde{\Delta}_1=u-(u_0+hu_1)\) and \(\widetilde{\delta}_1=x-(x_0+hx_1)\), we obtain for them the equations
\[ \widetilde{\Delta}_1=f_x^*\widetilde{\delta}_1+f_y^*[\widetilde{\delta}_1]+f_u^*[\widetilde{\Delta}_1]+O(t^2),\qquad \frac{d}{dt}\widetilde{\delta}_1=\widetilde{\Delta}_0 \]
and the initial conditions \(\widetilde{\Delta}_1=O(h^2)\), \(\widetilde{\delta}_1=O(h^2)\) for \(t\in E_0\). Hence, as above, we easily obtain the estimates \(|\widetilde{\delta}_1|<Ct^2,\ |\widetilde{\Delta}_1|<Ct^2\). Applying the method of induction further, we prove the theorem.

Before proceeding to the construction of asymptotic formulas on the remaining part of the interval, we introduce one additional system of functions \(\{w_k,v_k\}\).

Set \(w_0=\varphi(0)\), and let \(v_0\) be the root of the equation
\(v_0=f(0,\varphi(0),\varphi(0),v_0)\).
On the interval \([0,\delta]\), by the method of steps, we easily obtain that
\[ \Pi_0(u)=u_0-v_0=O\left(e^{-\frac{\varkappa t}{h}}\right), \]
where
\[ 0<\varkappa<-\frac{\ln a}{m},\qquad m=\sup_{\substack{0\leq t\leq \delta\\ h\leq h_0}}\frac{\mu(t,h)}{h}. \]

We now define \(w_1\) as follows:
\[ w_1=v_0t+\int_0^\delta \Pi_0(u)\,dt, \]
and \(v_1\) as the solution of an equation analogous to that for \(hu_1\), in which \(hx_1\) is replaced by \(w_1\), \(u_0\) by \(v_0\), and \(hu_1\) by \(v_1\), i.e.,
\[ v_1=f_t(N_0)t+f_x(N_0)w_1+f_y(N_0)[w_1]+f_u(N_0)[v_1], \]
where \(N_0=(0,w_0,w_0,v_0)\).

The exact initial value for \(v_1\) will be established in the next paragraph; for now we only note that \(v_1=O(h)\) for \(t\in E_0\).

Again applying the method of steps, we easily find that
\[ \Pi_1(u)=hu_1-v_1=O\left(he^{-\frac{\varkappa t}{h}}\right). \]

Suppose that all \(w_i,v_i\) for \(i<k\) are defined and that the functions
\[ \Pi_i(u)=h^i u_i-v_i \]
have order \(O\left(h^i e^{-\frac{\varkappa t}{h}}\right)\). Then set...

\[ w_k=\int_0^t v_{k-1}\,dt+\int_0^\delta \Pi_{k-1}(u)\,dt, \]

and \(v_k\) is obtained as the solution of the equation for \(h^k u_k\), in which \(h^i x_i\) are replaced by \(w_i\) and \(h^i u_i\) are replaced by \(v_i\) \((0\leq i<k)\), with the initial condition

\[ v_k=O(h^k)\quad \text{for } t\in E_0 . \tag{2.3} \]

(The exact form of the initial condition will be found in the next section.) Repeating verbatim the proof of Lemma 2 of [2], we obtain the estimate

\[ \Pi_k(u)=h^k u_k-v_k=O\left(h^k e^{-\varkappa t/h}\right). \tag{2.4} \]

The constructed functions \(\Pi_k(u)\) play the same role as the boundary-layer functions in [2] for obtaining asymptotic formulas outside a neighborhood of the initial point.

§ 3. ASYMPTOTIC EXPANSION OF THE SOLUTION “AWAY” FROM THE INITIAL POINT

To obtain asymptotic formulas for the values of the argument \(t>\delta>0\), we shall proceed exactly as in the case of an equation with constant delay.

Substituting into the original system (1.2), instead of \(x(t)\) and \(u(t)\), the series

\[ \sum_{k=0}^{\infty} h^k \bar{x}_k(t),\qquad \sum_{k=0}^{\infty} h^k \bar{u}_k(t), \]

and instead of \(\mu(t,h)\) the expansion (1.5), we shall have

\[ \sum_{k=0}^{\infty} h^k \bar{u}_k(t) = f\left( t,\sum_{k=0}^{\infty}h^k\bar{x}_k(t), \sum_{k=0}^{\infty}h^k\bar{x}_k(t-h\mu_1(t)-h^2\mu_2(t)-\ldots), \sum_{k=0}^{\infty}h^k\bar{u}_k(t-h\mu_1(t)-h^2\mu_2(t)-\ldots) \right), \]

\[ \frac{d}{dt}\sum_{k=0}^{\infty}h^k\bar{x}_k = \sum_{k=0}^{\infty}h^k\bar{u}_k . \]

Expanding the right-hand side in the upper relation into a formal power series in \(h\) and comparing coefficients of like powers, we obtain equations for the elements \(\bar{x}_k\) and \(\bar{u}_k\). For example,

\[ \bar{u}_0=f(t,\bar{x}_0,\bar{x}_0,\bar{u}_0),\qquad \dot{\bar{x}}_0=\bar{u}_0, \]

\[ \bar{u}_1=f_x\bar{x}_1+f_y(\bar{x}_1-\mu_1(t)\dot{\bar{x}}_0) +f_u(\bar{u}_1-\mu_1(t)\dot{\bar{u}}_0),\qquad \dot{\bar{x}}_1=\bar{u}_1, \]

and so on.

Since for the functions \(\bar{x}_k(t)\) ordinary differential equations have been obtained (for \(\bar{u}_k=\dot{\bar{x}}_k\)), it is still necessary to specify initial conditions. We shall determine these initial conditions by means of the functions \(\Pi_k(u)\) introduced at the end of the preceding section, according to the same formulas as in [2],

\[ \bar{x}_0(0)=\varphi(0),\qquad h^k\bar{x}_k(0)=\int_0^\delta \Pi_{k-1}(u)\,dt,\quad k>1. \tag{3.1} \]

Thanks to estimates (2.4), such a definition is quite correct for arbitrarily small \(h\). Let us also note that the initial value for the function \(\bar{x}_k\) is expressed through the function \(\Pi_{k-1}(u)\), and consequently also through the function \(v_{k-1}\), i.e., with an index one less.

It remains for us to determine the initial value for the functions \(v_k(t)\), \(k\geqslant 1\). We define these initial values inductively. Suppose that the functions \(v_i(t)\) have been found for the indices \(i=0,1,\ldots,k-1\); then the functions \(\Pi_0(u),\ldots,\Pi_{k-1}(u)\) will be defined and, consequently, \(x_i,u_i\) with indices \(i=0,1,\ldots,k\). If we now denote

\[ x_{p,q}=\frac{1}{p!}\frac{d^p}{dt^p}\bar{x}_q(0),\qquad u_{p,q}=\frac{1}{p!}\frac{d^p}{dt^p}\bar{u}_q(0), \tag{3.2} \]

then the initial value for \(v_k\) will be

\[ v_k(t)=\sum_{i=0}^k t^i h^{k-i}u_{i,k-i}\quad \text{for } t\in E_0 \tag{3.3} \]

and, obviously, has the required order (2.3).

Thus, with the aid of formulas (2.4), (3.1), (3.3), all the necessary functions up to any number can be found successively.

The following holds.

Theorem. If the functions \(f,\varphi,\mu\) satisfy conditions \(A\) of § 1 and the functions \(\bar{x}_k(t),\bar{u}_k(t)\) are defined by the method described above, then for \(t\in[-Ah\ln h,T]\) and sufficiently small \(h\) the uniform estimates hold

\[ \left|x(t,h)-\sum_{k=0}^n h^k\bar{x}_k(t)\right|<Ch^{n+1},\qquad \left|u(t,h)-\sum_{k=0}^n h^k\bar{u}_k\right|<Ch^{n+1}. \]

The proof of this theorem is carried out, as in the case of constant small retardation, by means of the following lemma.

Lemma. For values of the argument \(t\in[0,\delta]\) and \(k=0,1,\ldots,n+1\), the differences

\[ \eta_k=\sum_{i=0}^k w_i-\sum_{i=0}^k\sum_{j=0}^i t^j h^{i-j}x_{j,i-j}, \tag{3.4} \]

\[ \xi_k=\sum_{i=0}^k v_i-\sum_{i=0}^k\sum_{j=0}^i t^j h^{i-j}u_{j,i-j} \tag{3.5} \]

satisfy the estimates

\[ \eta_k=O\left(\sum_{i=0}^{k+1} t^i h^{k+1-i}\right),\qquad \xi_k=O\left(\sum_{i=0}^{k+1} t^i h^{k+1-i}\right). \tag{3.6} \]

We prove the lemma by the method of mathematical induction.

For \(k=0\) we have

\[ \eta_0=w_0-x_{00}=\varphi(0)-\varphi(0)=0,\qquad \xi_0=v_0-u_{00}=u_{00}-u_{00}=0, \]

i.e., the estimates (3.6) are fulfilled. Suppose further that the estimates (3.6) hold for \(k=0,1,\ldots,m-1\), and prove them for \(k=m\). We have

\[ {d\over dt}\eta_m=\sum_{i=0}^{m-1}v_i-\sum_{i=0}^{m}\sum_{j=0}^{i}t^j h^{i-j}x_{j,i-j}. \]

But by definition (3.2) we obtain that

\[ j x_{j,i-j}=u_{j-1,i-j}=u_{j',i-1-j'},\qquad j'=j-1. \]

Therefore the final expression for the derivative of \(\eta_m\) will be

\[ {d\over dt}\eta_m=\sum_{i=0}^{m-1}v_i-\sum_{i=0}^{m-1}\sum_{j=0}^{i}t^j h^{i-j}u_{j,i-j}=\xi_{m-1}. \]

Integrating this equation with allowance for the initial condition

\[ \eta_m(0)=\sum_{i=1}^{m}\int_{0}^{\delta}\Pi_{i-1}(u)\,dt-h^i\overline{x_i}(0)=0, \]

we obtain the estimate (3.6) for \(\eta_m(t)\).

Let us now pass to \(\xi_m(t)\). From the equations for the terms that make up \(\xi_m(t)\), it is not difficult to see the validity of the following relations:

\[ \sum_{i=0}^{m}v_i = f\left( t,\sum_{i=0}^{m}w_i,\sum_{i=0}^{m}[w_i],\sum_{i=0}^{m}[v_i] \right) + O\left(\sum_{i=0}^{m+1}t^i h^{m+1-i}\right), \]

\[ \sum_{i=0}^{m}\sum_{j=0}^{i}t^j h^{i-j}u_{j,i-j} = f\left( t,\sum_{i=0}^{m}\sum_{j=0}^{i}t^j h^{i-j}x_{j,i-j}, \right. \]

\[ \left. \sum_{i=0}^{m}\sum_{j=0}^{i}[t^j h^{i-j}x_{j,i-j}], \sum_{i=0}^{m}\sum_{j=0}^{i}[t^j h^{i-j}u_{j,i-j}] \right) + O\left(\sum_{i=0}^{m+1}t^i h^{m+1-i}\right). \]

Taking the difference of these two relations and taking into account the validity of the estimate (3.6) for \(\eta_m(t)\), we obtain the equation for \(\xi_m(t)\):

\[ \dot{\xi}_m=f_u^{*}[\xi_m]+O\left(\sum_{i=0}^{m+1}t^i h^{m+1-i}\right) \]

with the initial condition, according to (3.3), \(\xi_m(t)=0\) for \(t\in E_0\). Solving this equation, as in [2], by the method of steps, we prove the assertion of the lemma.

Remark. For the case of constant retardation we would have

\[ w_i=\sum_{j=0}^{i}t^j h^{i-j}x_{j,i-j},\qquad v_i=\sum_{j=0}^{i}t^j h^{i-j}u_{j,i-j} \]

and, consequently, \(\eta_i\equiv \xi_i\equiv 0\). Now the theorem is proved easily.

By the construction of \(x_k(t)\) and \(u_k(t)\) it is clear that the differences

\[ \overline{\delta}_n=x-\sum_{i=0}^{n} h^i \overline{x}_i,\qquad \overline{\Delta}_n=u-\sum_{i=0}^{n} h^i u_i \]

satisfy the equations

\[ \overline{\Delta}_n=f_x^* \overline{\delta}_n+f_y^*[\overline{\delta}_n]+f_u^*[\overline{\Delta}_n]+O(h^{n+1}),\qquad \frac{d}{dt}\overline{\delta}_n=\overline{\Delta}_n . \tag{3.7} \]

On the interval \(-\dfrac{1}{2}Ah\ln h\leq t\leq -Ah\ln h\) we have

\[ u-\sum_{k=0}^{n} h^k \overline{u}_k = \left(u-\sum_{k=0}^{n+1} h^k u_k\right) + \left(\sum_{k=0}^{n+1} h^k u_k-\sum_{k=0}^{n+1} v_k\right) + \]

\[ + \left(\sum_{k=0}^{n+1} v_k-\sum_{k=0}^{n+1}\sum_{i=0}^{k} t^i h^{k-i}u_{i,k-i}\right) + \left(\sum_{k=0}^{n+1}\sum_{i=0}^{k} t^i h^{k-i}u_{i,k-i} -\sum_{k=0}^{n} h^k \overline{u}_k\right). \]

Taking into account the validity of the estimates near the initial point, the estimates (2.4), (3.6), as well as formulas (3.2) and the fact that on this interval \(O(t^{n+2})=O(h^{n+1})\), we obtain

\[ \overline{\Delta}_n=O(h^{n+1})\quad \text{for}\quad -\frac{1}{2}Ah\ln h\leq t\leq -Ah\ln h . \]

An analogous estimate is satisfied by \(\overline{\delta}_n\) on this interval. Therefore, passing from the differential equation (3.7) to the integral equation (as in [2]) with an initial set lying on the interval indicated above, we prove the validity of the theorem.

Remark. In exactly the same way one constructs asymptotic formulas for the solution of a system of equations of neutral type with variable small retardation under the condition that the characteristic roots of the matrix \(f_u\) do not exceed unity in absolute value. In this case only the technique of proving the estimates changes, in accordance with § 2 of [2].

I express my gratitude to L. N. Zaits for assistance in the work.

References

1. A. B. Vasil’eva, Zhurnal vychisl. matem. i matem. fiz. 2, No. 5, 768—786, 1962.
2. V. I. Rozhkov, In the collection Numerical Methods for Solving Differential and Integral Equations and Quadrature Formulas, Nauka Publishing House, 1964, pp. 162—176.

Received by the editors
July 19, 1965

Peoples’ Friendship University
named after P. Lumumba