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).