MATHEMATICS

M. A. EVGRAFOV

ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENCE EQUATIONS

(Presented by Academician M. V. Keldysh, February 10, 1958)

For a difference equation of the form

[
y(n+k)+\sum_{m=1}^{k} a_m(n)y(n+k-m)=0,
\tag{1}
]

whose coefficients satisfy the conditions

[
\lim_{n\to\infty} a_m(n)=a_m,\qquad m=1,2,\ldots,k;
]

[
\lambda^k+a_1\lambda^{k-1}+\cdots+a_k
=(\lambda-\lambda_1)\cdots(\lambda-\lambda_k),
\qquad |\lambda_i|\ne|\lambda_j|,\quad i\ne j,
]

Perron’s theorem (({}^{1-4})) is known, asserting that every solution of equation (1) has the form

[
y(n)=C_1y_1(n)+\cdots+C_ky_k(n),
\tag{2}
]

where

[
\lim_{n\to\infty}\frac{y_m(n)}{y_m(n+1)}=\lambda_m.
]

The assertions of Perron’s theorem concerning the asymptotic behavior of the solutions (y_m(n)) can be substantially refined if one makes certain additional assumptions on the regularity with which the coefficients (a_m(n)) approach their limiting values.

Theorem 1. Let the coefficients of equation (1) satisfy the conditions

[
a_k(n)\ne0,\qquad n\ge1;\qquad
\lim_{n\to\infty}a_m(n)=a_m,\qquad
\sum^\infty |a_m(n+1)-a_m(n)|<\infty
]

[
(m=1,2,\ldots,k);
]

[
\lambda^k+a_1\lambda^{k-1}+\cdots+a_k
=(\lambda-\lambda_1)\cdots(\lambda-\lambda_k);
\qquad
\lambda_i\ne\lambda_j,\quad i=j,\quad \lambda_i\ne0
]

[
(i,j=1,2,\ldots,k).
]

Put

[
P_n(\lambda)=\lambda^k+a_1(n)\lambda^{k-1}+\cdots+a_k(n)
=(\lambda-\lambda_1(n))\cdots(\lambda-\lambda_k(n)),
]

[
\lim_{n\to\infty}\lambda_m(n)=\lambda_m.
]

Every solution of equation (1) has the form (2), where

[
y_m(n)\sim \lambda_m^{-1}(1)\cdots\lambda_m^{-1}(n),
\qquad n\to\infty.
]

An analogous result is also easily obtained for systems of difference equations.

Let a system be given by

[
y(n+1)=A(n)y(n),
\tag{3}
]

where

[
y(n)={y_1(n),\ldots,y_k(n)},\qquad A(n)=(a_{ij}(n))_1^k.
]

Denote by (\lambda_m(n)), (m=1,2,\ldots,k), the proper values of the matrix (A(n)) (for simplicity we shall assume that the (\lambda_m(n)) are distinct for all (n \geq 1)), and by (t_m(n)={t_{m1}(n),\ldots,t_{mk}(n)}) the corresponding proper vectors, and, finally, (T(n)=(t_{ij}(n))_1^k).

Theorem 2. Suppose that

[
\sum_{n=1}^{\infty} |a_{ij}(n+1)-a_{ij}(n)|<\infty,\qquad
i,j=1,2,\ldots,k;\qquad
\lim_{n\to\infty} A(n)=A .
]

Then, as is not hard to see, there exist limits (under a corresponding choice of the indices (\lambda_m(n)))

[
\lambda_m=\lim_{n\to\infty}\lambda_m(n),\qquad
T=\lim_{n\to\infty}T(n).
]

If (\lambda_m\ne0), (m=1,2,\ldots,k); (\lambda_m(n)\ne0), (n\geq1), (m=1,2,\ldots,k); (\lambda_i\ne\lambda_j), (i\ne j), then every solution of system (3) has the form (2), where

[
y_m(n)\sim \lambda_m^{-1}(1)\cdots \lambda_m^{-1}(n)\,T e_m(n);
]

[
e_m(n)={e_{m1}(n),\ldots,e_{mk}(n)},\qquad
\lim_{n\to\infty} e_{ij}(n)=\delta_{ij}.
]

The assertions of Theorems 1 and 2 can be generalized to the case when (\lambda_m(n)) may increase without bound or tend to zero. We shall give the formulation of such a generalization of Theorem 1.

Put, as above:

[
P_n(\lambda)=\lambda^k+a_1(n)\lambda^{k-1}+\cdots+a_k(n)
=(\lambda-\lambda_1(n))\cdots(\lambda-\lambda_k(n)),
]

[
P_{n,m}(\lambda)=\frac{P_n(\lambda)}{\lambda-\lambda_m(n)}.
]

Theorem 3. If the coefficients of equation (1) satisfy the conditions

[
a_k(n)\ne0,\qquad n\geq1;\qquad
\sum_{n=1}^{\infty}
\left|
\frac{\lambda_i(n)P_{n+1,j}(\lambda_i(n))}
{\lambda_j(n)P_{n+1,j}(\lambda_j(n))}
\right|<\infty,\quad
i\ne j,\ i,j=1,2,\ldots,k,
\tag{4}
]

then every solution of equation (1) has the form (2), where

[
y_m(n)\sim \mu_m(1)\cdots \mu_m(n),\qquad
\mu_m(n)=
\frac{P_{n+1,m}(\lambda_m(n+1))}
{\lambda_m(n)P_{n+1,m}(\lambda_m(n))}.
]

Let us pass to difference equations of infinite order. Consider the recurrence relation

[
y_n+\sum_{m=1}^{n} a_{m,n}y_{n-m}=0,\qquad y_0=1.
\tag{5}
]

Denote

[
P_n(\lambda)=1+\sum_{m=1}^{n} a_{m,n}\lambda^m
]

and suppose that there exists a function (\varphi(n)) possessing the following properties:

1. For (|\lambda|<\varphi(n)) the function (P_n(\lambda)) has exactly (k) simple zeros, say (\lambda_1(n),\ldots,\lambda_k(n)).

2.

[
\lim_{n\to\infty}\max_{|z|=1}
\left|
\frac{
P_{n+1}\left(\dfrac{z}{\varphi(n+1)}\right)
}{
P_n\left(\dfrac{z}{\varphi(n)}\right)
}
\right|=1.
]

Theorem 4. Put

[
P_{n,m}(\lambda)=\frac{P_n(\lambda)}{\lambda-\lambda_m(n)}.
]

If

[
\lim_{n\to\infty}\frac{P_{n+1,i}(\lambda_i(n))}{P_{n+1,i}(\lambda_i(n+1))}=1,\quad
\sum_{n=1}^{\infty}\left|
\frac{\lambda_i(n)P_{n+1,j}(\lambda_i(n))}
{\lambda_j(n)P_{n+1,j}(\lambda_j(n))}
\right|<\infty,\quad i\ne j,\quad i,j=1,2,\ldots,k,
\tag{6}
]

then the relation holds ((\varepsilon>0) arbitrarily small)

[
y_n=\sum_{m=1}^{k} C_m y_n^{(m)}+
O\bigl((1+\varepsilon)^n\varphi(1)\cdots\varphi(n)\bigr),
]

[
y_n^{(m)}\sim \mu_m(1)\cdots\mu_m(n),\qquad
\mu_m(n)=
\frac{P_{n+1,m}(\lambda_m(n+1))}
{\lambda_m(n)P_{n+1,m}(\lambda_m(n))}.
]

Let us also consider an infinite system of linear equations of the form

[
y_n+\sum_{m=1}^{\infty} a_{m,n}y_{m+n}=0,\qquad n=1,2,\ldots,
\tag{7}
]

and suppose that there exists a function (\varphi(n)) such that the series

[
P_n(\lambda)=1+\sum_{m=1}^{\infty} a_{m,n}\lambda^m
]

converge for (|\lambda|\leq \varphi(n)) and conditions 1 and 2 are satisfied.

Theorem 5. If (6) holds, then every solution of equation (7) satisfying the condition

[
y_n=O\bigl((1-\varepsilon)^n\varphi(1)\cdots\varphi(n)\bigr),\qquad \varepsilon>0,
]

has the form (y_n=C_1y_n^{(1)}+\cdots+C_ky_n^{(k)}), where (y_n^{(m)}) are the same as in Theorem 4.

Application of Theorems 4 and 5 gives interesting refinements of the results obtained in (\left({}^{5}\right)) and partly in (\left({}^{6}\right)).

Let us now consider the differential equation

[
y^{(k)}+\sum_{m=1}^{k} a_m(x)y^{(k-m)}=0,\qquad 0\leq x<\infty,
\tag{8}
]

whose coefficients are twice continuously differentiable functions. Introduce the notation

[
P(x;\lambda)=\lambda^k+a_1(x)\lambda^{k-1}+\cdots+a_k(x)
=(\lambda-\lambda_1(x))\cdots(\lambda-\lambda_k(x)),
]

[
P_m(x;\lambda)=\frac{P(x;\lambda)}{\lambda-\lambda_m(x)},\qquad
P'_m(x;\lambda)=\frac{\partial}{\partial\lambda}P_m(x;\lambda).
]

Theorem 6. If

[
\int_{0}^{\infty}\left|
\frac{d}{dx}\frac{\lambda_i'(x)}{\lambda_j(x)-\lambda_i(x)}
\frac{P'_j(x;\lambda_i(x))}{P_j(x;\lambda_j(x))}
\right|\,dx<\infty,\qquad
i,j=1,2,\ldots,k,\quad i\ne j,
]

then every solution of equation (8) has the form (y=C_1y_1+\cdots+C_ky_k), where

[
y_m(x)\sim \exp\left{\int_{0}^{x}\mu_m(t)\,dt\right},\qquad
\mu_m(x)=\lambda_m(x)-\lambda'_m(x)
\frac{P'_m(x;\lambda_m(x))}{P_m(x;\lambda_m(x))}.
]

Consider the integral equation of the form

[
y(x)+\int_{0}^{x} a(x,x-t)y(t)\,dt=\delta(x),\qquad 0\leq x<\infty,
\tag{9}
]

where (\delta(x)) is the delta function, and (a(x,t)) has two continuous derivatives

with respect to (x) (with respect to (t) the function (a(x,t)) may be a generalized function). Denote

[
P(x;\lambda)=1+\int_0^x a(x,t)e^{-\lambda t}\,dt
]

and suppose that there exists a function (\varphi(x)) satisfying the conditions:

(1^*). In the half-plane (\operatorname{Re}\lambda>\varphi(x)) the function (P(x;\lambda)) has exactly (k) simple zeros, say (\lambda_m(x)), (m=1,2,\ldots,k).

(2^*). There is an (\alpha) such that (\lambda^\alpha P(x;\lambda)\to\infty) as (|\operatorname{Im}\lambda|\to\infty), uniformly in (\operatorname{Re}\lambda), (\varphi(x)\leqslant \operatorname{Re}\lambda<\infty).

(3^*). Uniformly in (\lambda) on any finite segment of the line (\operatorname{Re}\lambda=1),

[
\lim_{n\to\infty}\frac{d}{dx}\ln P!\left(x;\frac{\lambda}{\varphi(x)}\right)=0.
]

Theorem 7. Put

[
P_m(x;\lambda)=\frac{P(x;\lambda)}{\lambda-\lambda_m(x)},\qquad
P_m'(x;\lambda)=\frac{\partial}{\partial\lambda}P_m(x;\lambda).
]

If

[
\lim_{x\to\infty}\lambda_i'(x)\frac{P_i'(x;\lambda_i(x))}{P_i(x;\lambda_i(x))}=0,\qquad
\int_0^\infty\left|\frac{d}{dx}\frac{\lambda_i'(x)P_j'(x;\lambda_i(x))}{\lambda_j(x)-\lambda_i(x)\,P_j(x;\lambda_j(x))}\right|dx<\infty,
\tag{10}
]

[
i,\ j=1,2,\ldots,k,\quad i\ne j,
]

then for a solution of equation (9) we have ((\varepsilon>0) arbitrarily small)

[
y(x)=\sum_{m=1}^k C_m y_m(x)+O!\left(\exp\left{(1+\varepsilon)\int_0^x\varphi(t)\,dt\right}\right),
]

where

[
y_m(x)\sim \exp\left{\int_0^x \mu_m(t)\,dt\right},\qquad
\mu_m(x)=\lambda_m(x)-\lambda_m'(x)\frac{P_m'(x;\lambda_m(x))}{P_m(x;\lambda_m(x))}.
\tag{11}
]

Finally, consider an integral equation of the form

[
y(x)+\int_x^\infty a(x,t-x)y(t)\,dt=0,\qquad 0\leqslant x<\infty,
\tag{12}
]

where (a(x,t)), as before, is assumed to have two continuous derivatives with respect to (x). Suppose that there exists a (\varphi(x)) such that the integral entering the formula

[
P(x;\lambda)=1+\int_0^\infty a(x,t)e^{-\lambda t}\,dt,
]

converges uniformly in the half-plane (\operatorname{Re}\lambda\geqslant\varphi(x)) and the conditions (1^, 2^, 3^*) are fulfilled.

Theorem 8. If (10) holds, then the assertion of Theorem 6 is valid.

Received
8 II 1958

CITED LITERATURE

(^{1}) A. O. Gelfond, Calculus of Finite Differences, Moscow, 1952.
(^{2}) A. O. Gelfond, I. M. Kubenskaya, Izv. AN SSSR, Ser. Mat., 17, No. 2, 83 (1953).
(^{3}) M. A. Evgrafov, Izv. AN SSSR, Ser. Mat., 17, No. 2, 77 (1953).
(^{4}) G. A. Freiman, Uspekhi Mat. Nauk, 12, issue 3, 241 (1957).
(^{5}) A. D. Solov’ev, DAN, 113, No. 5 (1957).
(^{6}) M. A. Evgrafov, A. D. Solov’ev, DAN, 113, No. 3 (1957).