Reports of the Academy of Sciences of the USSR
1968. Volume 182, No. 3

UDC 517.948.32

MATHEMATICS

V. B. UVAROV

ASYMPTOTIC PROPERTIES OF SOLUTIONS OF SYSTEMS OF INTEGRAL EQUATIONS ON THE HALF-AXIS

(Presented by Academician A. N. Tikhonov on February 9, 1968)

Introduction. Consider the system of integral equations

\[ y(x)=f(x)+\int_{0}^{\infty} k(x,s)y(s)\,ds . \tag{1} \]

Here \(y(x)=\{y_1(x),\,y_2(x),\ldots\}\) and \(f(x)=\{f_1(x),\,f_2(x),\ldots\}\) are vectors; \(k(x,s)\) is the matrix \(k_{ij}(x,s)\), with \(f(x)\equiv 0\) for \(x\ge x_0\), \(k(x,s)=0\) for \(x-s>a\).

In the paper it is shown that the solutions of system (1), for a rather general class of kernels \(k(x,s)\), have a common asymptotic behavior as \(x\to\infty\) (up to a constant factor) for various functions \(f(x)\). This fact is a generalization of the well-known property of solutions of integral equations with kernels depending on the difference of the arguments, when an explicit form of the solution \(y(x)\) can be found.

Many equations of mathematical physics, in particular problems connected with the transport equation (Milne’s problem, etc.), lead to equations of the form (1), but with \(f(x)\to 0\) as \(x\to\infty\) and with kernels that do not vanish for \(x-s>a\), but decrease rapidly as \(x-s\to+\infty\).

§ 1. Auxiliary formulas. Everywhere in what follows we shall assume that the solution of equation (1) can be obtained by the method of successive approximations. In addition, we shall assume that \(k(x,s)\ge 0\) (we shall write \(a\ge b\) when \(a_{ij}\ge b_{ij}\)) and \(y(x)>0\) for \(x\ge x_0\), if \(f(x)>0\). Introduce the notation: \(x_m=x_0+ma\), \(u_m=[x_m,x_{m+1}]\), \(G_m(x,s)\) is the resolvent of the equation

\[ y(x)=f(x)+\int_{x_m}^{\infty} k(x,s)y(s)\,ds, \]

considered for \(x\ge x_m\). Writing equation (1) for \(x\ge x_m\) in the form

\[ y(x)=f_m(x)+\int_{x_m}^{\infty} k(x,s)y(s)\,ds, \]

\[ f_m(x)= \begin{cases} \displaystyle \int_{x-a}^{x_m} k(x,s)y(s)\,ds, & \text{for } x\in u_m,\\[1.2ex] 0, & \text{for } x>x_{m+1}, \end{cases} \]

we obtain, for \(x\in u_m\),

\[ y(x)=\int_{x_{m-1}}^{x_m} \psi^{(m)}(x,s)y(s)\,ds, \tag{2} \]

where

\[ \psi^{(m)}(x,s)=k(x,s)+\int_{x_m}^{x_{m+1}} G_m(x,t)k(t,s)\,dt. \tag{3} \]

Let \(\bar y(x)\) be a solution of equation (1) with \(f(x)\) replaced by \(\underline f(x)\), and let \(\bar f(x)>0\) for \(x\le x_0\). The function \(z(x)\) with components \(z_i(x)=y_i(x)/\bar y_i(x)\) is a solution of the equation

\[ z(x)=\int_{x_{m-1}}^{x_m}\varphi^{(m)}(x,s)z(s)\,ds. \tag{4} \]

Here \(\varphi^{(m)}(x,s)\) is the matrix with components

\[ \varphi_{ij}^{(m)}(x,s)=\psi_{ij}^{(m)}(x,s)\bar y_j(s)/\bar y_i(x). \tag{5} \]

In what follows we shall need the obvious equality

\[ \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m}\varphi_{ij}^{(m)}(x,s)\,ds=1. \tag{6} \]

§ 2. Asymptotic properties of solutions of integral equations

1. We shall show that, under fairly general conditions, \(z_i(x)\to c\) as \(x\to\infty\), where \(c\) is some constant independent of \(i\).

Theorem 1. Let

\[ P_m=\max_i\sup_{x\in u_m} z_i(x) \quad\text{and}\quad p_m=\min_i\inf_{x\in u_m} z_i(x). \]

Suppose there exist finite limits

\[ P=\lim_{m\to\infty}P_m \quad\text{and}\quad p=\lim_{m\to\infty}p_m. \]

If, moreover,

\[ \prod_{k=1}^{\infty}(1-\delta_k)=0, \]

where

\[ \delta_m=\min_{i_1,i_2,\;x',x'',\;E_{mj}}\inf \sum_{j=1}^{n} \left[ \int_{E_{mj}}\varphi_{i_1j}^{(m)}(x',s)\,ds+ \int_{\bar E_{mj}}\varphi_{i_2j}^{(m)}(x'',s)\,ds \right] \]

(\(E_{mj}\) are arbitrary measurable sets, \(\bar E_{mj}\) is the complement of \(E_{mj}\) to \(u_{m-1}\)), then \(P=p\), i.e., there exists a common finite limit

\[ z(\infty)=\lim_{x\to\infty} z_i(x), \]

and moreover

\[ |z_i(x)-z(\infty)|\le (P_0-p_0)\prod_{k=1}^{m}(1-\delta_k), \quad \text{for } x\in u_m. \]

Proof. We have

\[ P_m=\max_i\sup_{x\in u_m} \left[ \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m}\varphi_{ij}^{(m)}(x,s)z_j(s)\,ds \right]\le \]

\[ \le P_{m-1}\max_i\sup_{x\in u_m} \left[ \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m}\varphi_{ij}^{(m)}(x,s)\,ds \right] =P_{m-1} \]

by virtue of (6). Similarly, we obtain \(p_m\ge p_{m-1}\). Thus the sequences \(\{P_m\}\) and \(\{p_m\}\) are monotone and bounded, whence the first part of the theorem follows.

Next,

\[ P_m-p_m=\max_{i_1,i_2}\sup_{x',x''\in u_m} \left[z_{i_1}(x')-z_{i_2}(x'')\right]. \]

But

\[ z_{i_1}(x')-z_{i_2}(x'') = \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m} \left[ \varphi_{i_1j}^{(m)}(x',s)-\varphi_{i_2j}^{(m)}(x'',s) \right]ds. \]

Let \(E_{mj}\) be the set on which \(\varphi_{i_1j}^{(m)}(x',s)\ge \varphi_{i_2j}^{(m)}(x'',s)\). Splitting the integral over \(u_{m-1}\) into integrals over \(E_{mj}\) and \(\bar E_{mj}\) (\(E_{mj}+\bar E_{mj}=u_{m-1}\)), we note

them in the first integral \(z_j(s)\) by \(P_{m-1}\), and in the second integral by \(p_{m-1}\). Then, by virtue of the obvious relation:

\[ \sum_{j=1}^{n}\int_{E_{mj}} \left[\varphi_{i_1j}^{(m)}(x',s)-\varphi_{i_2j}^{(m)}(x'',s)\right]\,ds = \]

\[ = \sum_{j=1}^{n}\int_{E_{mj}} \left[\varphi_{i_1j}^{(m)}(x'',s)-\varphi_{i_2j}^{(m)}(x',s)\right]\,ds \leqslant 1-\delta_m \]

we obtain the inequality \(P_m-p_m\leqslant (1-\delta_m)(P_{m-1}-p_{m-1})\). From this inequality the second part of the theorem follows.

1. The expression for \(\delta_m\) in Theorem 1 is rather complicated. Therefore we shall give cruder, but simpler and obvious estimates for \(\delta_m\):

\[ \delta_m\geqslant \min_{i_1,i_2} \left\{ \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m} \min\left[\alpha_{i_1j}^{(m)}(s),\alpha_{i_2j}^{(m)}(s)\right]\,ds \right\}, \tag{7} \]

where

\[ \alpha_{ij}^{(m)}(s)=\inf_{x\in u_m}\varphi_{ij}^{(m)}(x,s). \]

If \(\varphi^{(m)}(x,s)>0\), then the estimates can be simplified still further:

\[ \delta_m\geqslant \sum_{j=1}^{n}\int_{x_{m-1}}^{x_m} \left[\min_i \alpha_{ij}^{(m)}(s)\right]\,ds. \tag{8} \]

If the solution \(\overline y(x)\) is not known explicitly, then it is necessary to obtain estimates that do not contain \(\overline y(x)\). For simplicity we shall obtain such estimates under \(\psi^{(m)}(x,s)>0\) for \(s>x_{m-1}\). We have

\[ \alpha_{ij}(s)\geqslant \frac{ \displaystyle \inf_x \psi_{ij}^{(m)}(x,s)\cdot \sum_{l=1}^{n}\int_{x_{m-2}}^{x_{m-1}} \overline y_l(t)\psi_{jl}^{(m-1)}(s,t)\,dt }{ \displaystyle \sup_x\left\{ \sum_{l=1}^{n}\int_{x_{m-2}}^{x_{m-1}} \overline y_l(t)\,dt\cdot \sum_{p=1}^{n}\int_{x_{m-1}}^{x_m} \psi_{ip}^{(m)}(x,z)\psi_{pl}^{(m-1)}(z,t)\,dz \right\} } \geqslant \]

\[ \geqslant \frac{ \displaystyle \left[\min_i \inf_x \psi_{ij}^{(m)}(x,s)\right] \left[\min_l \inf_t p(t)\psi_{jl}^{(m-1)}(s,t)\right] }{ \displaystyle \left[\max_{ij}\sup_{xs}\psi_{ij}^{(m)}(x,s)\right] \left[\max_{ij}\sup_t p(t)\cdot \int_{x_{m-1}}^{x_m}\psi_{ij}^{(m-1)}(z,t)\,dz\right] }. \tag{9} \]

Here \(p(t)\) is an arbitrary nonnegative function. We cannot put \(p(t)=1\), since

\[ \inf_t \psi_{jl}^{(m-1)}(s,t)=\psi_{jl}^{(m-1)}(s,x_{m-2})=0. \]

Taking this consideration into account, one may put \(p(t)=1/(t-x_{m-2})\). Substituting (9) into (8), we obtain an estimate for \(\delta_m\) that does not contain \(\overline y(x)\).

1. Thus, we obtain that the estimate for the quantity \(\delta_m\) reduces to estimating the functions \(\psi^{(m)}(x,s)\) from above and below. If the functions \(\psi^{(m)}(x,s)\) are known explicitly, then such estimates present no difficulty. If, however, the functions \(\psi^{(m)}(x,s)\) are not known explicitly, then one may use the obvious consideration: if \(\underline{k}(x,s)\leqslant k(x,s)\leqslant \widetilde{k}(x,s)\), then \(\underline{\psi}^{(m)}(x,s)\leqslant \psi^{(m)}(x,s)\leqslant \widetilde{\psi}^{(m)}(x,s)\) (the functions \(\underline{\psi}^{(m)}(x,s)\) and \(\widetilde{\psi}^{(m)}(x,s)\) are obtained with the help of the kernels \(\underline{k}(x,s)\) and \(\widetilde{k}(x,s)\) in the same way as the function \(\psi^{(m)}(x,s)\) is obtained with the help of \(k(x,s)\)). In this case, as the kernels \(\underline{k}(x,s)\) and \(\widetilde{k}(x,s)\) it is convenient to take such kernels for which the functions \(\underline{\psi}^{(m)}(x,s)\) and \(\widetilde{\psi}^{(m)}(x,s)\) can be computed explicitly. If, in particular, \(k(x,s)\geqslant \overline c_m\)

for \(x-a\leq s\leq x,\ x\in u_m\) (\(\bar c_m\) is a constant matrix), then \(\psi^{(m)}(x,s)\geq \bar\psi^{(m)}(x,s)\), where

\[ \bar\psi^{(m)}(x,s)= \begin{cases} \bar c_m e^{\bar c_m(x-x_m)}, & \text{for } s>x-a,\\ \bar c_m\left[e^{\bar c_m(x-x_m)}-e^{\bar c_m(x-a-s)}\right], & \text{for } s\leq x-a. \end{cases} \]

If we consider Volterra equations, then to estimate the function \(\psi^{(m)}(x,s)\) from above one may use a similar estimate: if \(k(x,s)\leq \tilde c_m\) for \(x-a\leq s\leq x\) (\(x\in u_m\)), then \(\psi^{(m)}(x,s)\leq \tilde\psi^{(m)}(x,s)\) (the expression for \(\tilde\psi^{(m)}(x,s)\) is obtained from the expression for \(\psi^{(m)}(x,s)\) by replacing \(\bar c_m\) by \(\tilde c_m\)).

It is easy to obtain for the function \(\psi^{(m)}(x,s)\) an upper estimate also in the case of Fredholm equations with kernels satisfying the condition

\[ \sum_{j=1}^{n}\int_{0}^{\infty} k_{ij}(x,s)\,ds\leq \alpha<1. \]

In this case, to estimate the resolvent \(G_m(x,s)\), it is convenient to use the obvious estimates:

\[ G_m(x,s)\leq G(x,s),\qquad \sum_{j=1}^{n}\int_{0}^{\infty} G_{ij}(x,s)\,ds\leq \frac{\alpha}{1-\alpha}. \]

From the functional relation for the resolvent \(G_m(x,s)\)

\[ G_m(x,s)=k(x,s)+\int_{x_m}^{s+a} G_m(x,t)k(t,s)\,dt \]

for \(x\in u_m,\ s\in u_m\), we have

\[ \max_{ij}[G_m(x,s)]_{ij}\leq \frac{1}{1-\alpha} \left[\max_{i,j}\ \sup_{s\in u_m,\ x_m\leq x\leq s+a} k_{ij}(x,s)\right] =\frac{q_m}{1-\alpha}. \]

Hence

\[ \psi_{ij}^{(m)}(x,s)\leq q_{m-1}+\frac{q_{m-1}q_m(s-x_{m-1})}{1-\alpha} \quad \text{for } s>x-a, \]

\[ \psi_{ij}^{(m)}\leq \frac{q_{m-1}q_m(s-x_{m-1})}{1-\alpha} \quad \text{for } s\leq x-a. \]

Substituting these estimates into the expression for \(\delta_m\), one can obtain conditions for the existence of a single limit
\[ z(\infty)=\lim_{x\to\infty} z_i(x). \]
In view of the cumbersomeness of the estimates \(\delta_m\) for the cases considered above, we shall not present them. We note only that, when the inequalities
\[ 0<\bar c\leq k_{ij}(x,s)\leq \tilde c \]
(\(\bar c,\tilde c\) are certain constants) hold in the domain \(2a\geq x-s\geq a\), for the cases considered above \(\delta_m\geq \delta>0\), and, consequently, there exists a single limit
\[ z(\infty)=\lim_{x\to\infty} z_i(x). \]

§ 3. All the results obtained are easily generalized to the continuum case, when instead of equation (1) one considers equations of the form

\[ y(x,\lambda)=f(x,\lambda)+\int_{\Omega} d\mu \int_{0}^{\infty} k(x,\lambda;s,\mu)y(s,\mu)\,ds, \]

where \(f(x,\lambda)\equiv 0\) for \(x\geq x_0\), \(k(x,\lambda;s,\mu)=0\) for \(x-s>a\), and \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_p)\) belongs to some closed domain \(\Omega\). In view of the obviousness of such a generalization, we shall not consider it in detail.

The author expresses gratitude to I. V. Fryazinov and I. M. Sobol’ for discussing a number of questions touched upon in this work.

Received
19 XII 1967