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ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A LINEAR DIFFERENTIAL EQUATION OF THE \(n\)-TH ORDER WITH RETARDED ARGUMENT
V. I. LOGUNOV
We consider the differential equation
\[ x^{(n)}(t)=\sum_{i=0}^{n-1}\sum_{j=1}^{m} a_{ij}(t)\,x^{(i)}(t-g_{ij}(t)). \tag{1} \]
Let \(E_{t_0}\) denote the initial set, i.e., the set consisting of the point \(t_0\) and the values \(t-g_{ij}(t)\leq t_0\) for \(t\in[t_0,\infty)\), \(i=0,1,\ldots,n-1\), and \(j=1,2,\ldots,m\). The coefficients of equation (1) and the retardations \(g_{ij}(t)\) are assumed to be continuous functions. The theorem below is a generalization, to equations with retarded argument, of a result known in the theory of ordinary differential equations (see, for example, [1, 2]).
Theorem. Suppose
\[ \int_{t_0}^{\infty}\sum_{i=0}^{n-1}\sum_{j=1}^{m} \left|a_{ij}(s)(s-g_{ij}(s))^{\,n-1-i}\right|\,ds<\infty, \]
\[ \lim_{t\to\infty} t-g_{ij}(t)=\infty,\quad i=0,1,\ldots,n-1;\ j=1,2,\ldots,m. \]
Suppose further that for a solution \(x(t)\) of equation (1) we have \(x^{(n-1)}(t)=O(1)\) for \(t\in E_{t_0}\). Then
\[ x^{(k)}(t)=\left[\frac{d}{(n-1-k)!}+O(1)\right]t^{\,n-1-k} \quad \text{as } t\to\infty \]
\[ (k=0,1,\ldots,n-1;\ d=\mathrm{const}). \]
Remark. The condition \(x^{(n-1)}(t)=O(1)\) for \(t\in E_{t_0}\) is satisfied for every solution of equation (1) if, for example, \(\inf E_{t_0}>-\infty\).
Proof. Setting
\[ f_k(t)=t^{k+1-n}\sum_{i=k}^{n-1}\frac{(t-t_0)^{i-k}}{(i-k)!}\,x^{(i)}(t_0), \]
\[ y_k(t)=x^{(k)}(t)t^{k+1-n}\quad (k=0,1,\ldots,n-1,\ t\geq t_0), \]
we obtain the system of integral equations
\[ y_k(t)=f_k(t)+\int_{t_0}^{t} \frac{(t-s)^{n-1-k}}{(n-1-k)!\,t^{\,n-1-k}} \sum_{i=0}^{n-1}\sum_{j=1}^{m} a_{ij}(s)(s-g_{ij}(s))^{\,n-1-i}\times \]
\[ \times\, y_i(s-g_{ij}(s))\,ds \quad (k=0,1,\ldots,n-1;\ t\geq t_0). \]
Changing, if necessary, \(t_0\), we may assume that
\[ \int_{t_0}^{\infty}\sum_{i=0}^{n-1}\sum_{j=1}^{m}|a_{ij}(s)(s-g_{ij}(s))^{n-1-i}|\,ds<\frac12. \]
Let \(C=\mathrm{const}\) be such that
\[ C\ge \sup_{t>t_0}|f_k(t)|;\qquad C\ge \sup_{t\in E_{t_0}}|y_k(t)|,\quad k=0,1,\ldots,n-1. \]
Then
\[ 2C\ge C+ \int_{t_0}^{t}\frac{(t-s)^{n-1-k}}{(n-1-k)!\,t^{\,n-1-k}} \sum_{i=0}^{n-1}\sum_{j=1}^{m}|a_{ij}(s)(s-g_{ij}(s))^{n-1-i}|\,2C\,ds \]
\[ (k=0,1,\ldots,n-1,\ t\ge t_0). \]
Since, further,
\[ |y_k(t)|\le C+ \int_{t_0}^{t}\frac{(t-s)^{n-1-k}}{(n-1-k)!\,t^{\,n-1-k}} \sum_{i=0}^{n-1}\sum_{j=1}^{m}|a_{ij}(s)(s-g_{ij}(s))^{n-1-i}| \]
\[ {}\times |y_i(s-g_{ij}(s))|\,ds, \]
it follows, by virtue of the theorem on integral inequalities (see, for example, [3–5]), that
\[ |y_k(t)|\le 2C,\quad k=0,1,\ldots,n-1,\quad t\in[t_0,\infty). \]
Hence
\[ \int_{t_0}^{\infty}\sum_{i=0}^{n-1}\sum_{j=1}^{m}a_{ij}(s)(s-g_{ij}(s))^{n-1-i}y_i(s-g_{ij}(s))\,ds \]
converges. Denote this integral by \(M(t_0)\). Applying L’Hospital’s rule, we obtain
\[ \lim_{t\to\infty}y_k(t)=\lim_{t\to\infty}f_k(t)+ \lim_{t\to\infty}\int_{t_0}^{t}\frac{1}{(n-1-k)!} \sum_{i=0}^{n-1}\sum_{j=1}^{m}a_{ij}(s)(s-g_{ij}(s))^{n-1-i} \]
\[ {}\times y_i(s-g_{ij}(s))\,ds = M(t_0)+x^{(n-1)}(t_0)\frac{1}{(n-1-k)!}=d_k,\quad k=0,1,\ldots,n-1. \]
Since \(d_k\) does not depend on \(t_0\) and \(M(t_0)\to0\) as \(t\to\infty\), it follows that
\[ d_k=\frac{d}{(n-1-k)!},\quad k=0,1,\ldots,n-1, \]
where \(d=\lim_{t\to\infty}x^{(n-1)}(t)\). The theorem is proved.
The author thanks the members of the Izhevsk Mathematical Seminar for their attention to the present work.
References
- I. M. Sobol, Dokl. Akad. Nauk SSSR, 61, 1948.
- E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations. IL, 1958.
- A. I. Guseinov, Ya. D. Mamedov, Investigation of the solution of integral equations with retarded argument. Scientific Notes of Azerbaijan University, No. 3, 1960.
- A. I. Logunov, Dokl. Akad. Nauk SSSR, 150, No. 2, 1963.
- Z. B. Seidov, Dissertation, Azerbaijan State University, 1963.
Received by the editors
9 November 1964
Udmurt State
Pedagogical Institute