UDC 517.942

MATHEMATICS

F. S. ALIEV

ON SOLUTIONS IN GENERALIZED FUNCTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH POLYNOMIAL COEFFICIENTS

(Presented by Academician I. G. Petrovskii, 11 XI 1965)

In the paper \((^1)\) it was proved that an equation of the form

\[ L(y)\equiv \sum_{k=0}^{m} a_k x^{k+r}y^{(k)}(x)+\sum_{q=0}^{n} b_q x^q y^{(q)}(x)=0,\quad m>n,\ r>0, \]

in the space of generalized functions \((S_0^\beta)'\), with some \(\beta>1\), has a fundamental system of \(2m+r\) linearly independent solutions. In the present note we study the more general equation

\[ L(y)\equiv \sum_{p=0}^{n}\sum_{k=0}^{m_p} a_{kp}x^{k+p}y^{(k)}(x)=0, \tag{1} \]

\[ m_n>m_p>0,\quad p=0,1,\ldots,n-1,\quad a_{m_n n}\ne 0,\quad a_{00}\ne 0, \]

in the same space \((S_0^\beta)'\), \(\beta>1\).

Denote by \(\Omega_0\) the totality of all solutions of equation (1) in \((S_0^\beta)'\) concentrated at the point \(x=0\).

Theorem 1. The dimension of the subspace \(\Omega_0\subset (S_0^\beta)'\) is equal to \(n\).

Proof. Let

\[ y(x)=\sum_{q=0}^{\infty} c_q\delta^{(q)}(x). \tag{2} \]

Under the condition of convergence of the series (2), this expression is the general form of a functional in \((S_0^\beta)'\) concentrated at the point \(x=0\) (see \((^2)\)). To prove the theorem it is enough to determine how many free constants \(c_q\) there are in the expansion (2) such that expression (2), with these free constants, is a solution of equation (1). Substituting expression (2) into equation (1) and using the formulas
\(x^k\delta^{(q)}(x)=(-1)^k[q]_k\delta^{(q-k)}(x)\) for \(k<q\) \((=0\) for \(k>q)\), by virtue of the linear independence of \(\delta^{(q)}(x)\) for different \(q\), equating to zero the resulting coefficient of \(\delta^{(q)}(x)\), we obtain

\[ \sum_{p=0}^{n} D_p(q)c_{q+p}=0,\quad q=0,1,\ldots, \tag{3} \]

where

\[ D_p(q)=\sum_{k=0}^{m_p} a_{kp}(-1)^{k+p}[q+k+p]_{k+p},\quad [q]_\rho=q(q-1)\cdots(q-\rho+1). \]

We denote the degree of the polynomial \(D_p(q)\) by \(d_p\). Obviously, \(d_p=m_p+p\), and, by virtue of our assumptions, \(d_n>d_p,\ p=0,1,\ldots,n-1\). It is clear that in equation (3), assigning \(c_0,c_1,\ldots,c_{n-1}\) arbitrarily, it is easy to determine

subsequent \(c_q\) such that

\[ c_{q+n}=-\sum_{p=0}^{n-1}\frac{D_p(q)}{D_n(q)}\,c_{q+p},\qquad q=0,1,2,\ldots \tag{4} \]

It remains for us to verify that the series (2) constructed from the coefficients (4) thus obtained converges in \((S_0^\beta)'\) with some \(\beta>1\). For this it is sufficient that, as \(q\to\infty\), the coefficients \(c_q\) satisfy the estimates

\[ |c_q|\leq \frac{B^q}{q^{q r_0}}\,M_q,\qquad r_0>1,\quad B>1,\quad M_q<c . \tag{5} \]

Indeed, under such an estimate for \(c_q\) we have

\[ \left|\sum_{q=0}^{\infty}(c_q\delta^{(q)}(x),\varphi(x))\right| = \left|\sum_{q=0}^{\infty}c_q(-1)^q\varphi^{(q)}(0)\right| \leq \sum_{q=0}^{\infty}|c_q|\,|\varphi^{(q)}(0)| \leq \]

\[ \leq \sum_{q=0}^{\infty}\frac{B^q B_1^q q^{q\beta}}{q^{q r_0}}\,M_q \leq \sum_{q=0}^{\infty}\left(\frac{\bar B}{q^{r_0-\beta}}\right)^q <\infty \qquad \text{for } \beta<r_0 . \]

Let us prove the estimates (5). Denoting \(M_q=c_q q^{q r_0}/B^q\), from (4) we obtain

\[ M_{q+n} = -\frac{1}{B^{q+n}} \sum_{p=0}^{n-1}\frac{D_p(q)}{D_n(q)} \frac{B^q(q+n)^{(q+n)r_0}}{(q+p)^{(q+p)r_0}} \,M_{q+p}. \]

For

\[ r_0=\min_{0<p<n-1}\left\{1+\frac{m_n-m_p}{n-p}\right\} \]

there exists such a \(q_1\) that for \(q>q_1\)

\[ M_{q+n}\leq \frac{c}{(1+q)^{\varepsilon+1}B} \sum_{p=0}^{n-1}M_{q+p}. \]

Let \(\bar M_{n-1}=\max(M_0,M_1,\ldots,M_{n-1})\); then

\[ M_{q+n}\leq \frac{c_n}{(1+q)^\varepsilon}\frac{1}{B}\,\bar M_{q+n-1}; \]

choosing \(B>cn\), we obtain \(M_{q+n}\leq \bar M_{q+n}\leq \bar M_{q+n-1}\), which means boundedness of \(M_q\) as \(q\to\infty\). This proves Theorem 1 completely.

Put

\[ f^\lambda(x)=|x|^\lambda\Big/\Gamma\!\left(\frac{\lambda+1}{2}\right), \qquad g^\lambda(x)=|x|^\lambda \operatorname{sign}x\Big/\Gamma\!\left(\frac{\lambda+2}{2}\right). \]

As is known ([3], p. 81 and following), by analytic continuation these functions define a functional on \(S'\supset (S_0^\beta)'\) for all values of \(\lambda\); in particular, for \(\lambda=-(2n+1)\) and \(\lambda=-2n\) we obtain

\[ f^{-(2n+1)}(x)=(-1)^n n!\,\delta^{(2n)}(x)/(2n)!, \qquad g^{-2n}(x)=\frac{(-1)^n (n-1)!\,\delta^{(2n-1)}(x)}{(2n-1)!}. \]

The following formulas for differentiating and multiplying by powers of \(x\) the functionals \(f^\lambda(x)\) and \(g^\lambda(x)\) hold:

\[ x^q[f^\lambda(x)]^{(r)} = \begin{cases} 2^r\left[\dfrac{\lambda}{2}\right]_{E(r/2+1)} \left[\dfrac{\lambda+q-r-1}{2}\right]_{E(q/2)} f^{\lambda+q-r}(x), & \text{if } q+r \text{ is even},\\[1.2em] 2^r\left[\dfrac{\lambda}{2}\right]_{E(r/2+1)} \left[\dfrac{\lambda+q-r}{2}\right]_{E(q/2+1)} g^{\lambda+q-r}(x), & \text{if } q+r \text{ is odd}; \end{cases} \tag{6} \]

\[ x^q[g^\lambda(x)]^{(r)} = \begin{cases} 2^r\left[\dfrac{\lambda-1}{2}\right]_{E(r/2)} \left[\dfrac{\lambda+q-r}{2}\right]_{E(q/2+1)} g^{\lambda+q-r}(x), & \text{if } q+r \text{ is even},\\[1.2em] 2^r\left[\dfrac{\lambda-1}{2}\right]_{E(r/2)} \left[\dfrac{\lambda+q-r-1}{2}\right]_{E(q/2)} f^{\lambda+q-r}(x), & \text{if } q+r \text{ is odd}. \end{cases} \tag{7} \]

Here \(E(\lambda)\) is the integer part of \(\lambda\).

Put

\[ R_p(\lambda)=\sum_{k=0}^{m_p} a_{kp}[\lambda-p]_k,\qquad A_p(\lambda)=\left[\frac{\lambda+1}{2}\right]_{E(p/2)} R_p(\lambda), \]

\[ B_p(\lambda)=\left[\frac{\lambda}{2}\right]_{E(p/2+1)} R_p(\lambda). \]

We shall call the polynomial \(A_n(\lambda)\) the characteristic polynomial of equation (1).

Definition. We shall say that the polynomial \(A(\lambda)\) has no generalized multiple roots modulo \(r\), if the arithmetic progressions with difference \(r\) constructed on the roots of the polynomial \(A(\lambda)\) do not intersect. If, however, the arithmetic progression with difference \(r\) constructed over the root \(\lambda_1\) contains \(k\) roots of the polynomial \(A(\lambda)\), then we shall say that \(\lambda_1\) is a generalized multiple root modulo \(r\) of the polynomial \(A(\lambda)\) of order \(k\).

Theorem 2. If the polynomial \(A_n(\lambda)\) has no generalized multiple roots modulo \(n\) for even \(n\) and modulo \(n-1\) for odd \(n\), then equation (1) in the space \((S_0^\beta)'\) with some \(\beta>1\) has \(2m_n+n\) linearly independent solutions of the form

\[ y(x)=\sum_\lambda \xi_\lambda f^\lambda(x)+\sum_\lambda \eta_\lambda g^\lambda(x), \tag{8} \]

where the summation index \(\lambda\) runs through the values from a certain arithmetic progression with difference \(n\) for even \(n\) and with difference \(n-1\) for odd \(n\).

Proof. Substituting the expressions (8) into equation (1) and using formulas (6), (7), in view of the linear independence of \(f^\lambda(x)\) and \(g^\lambda(x)\) for different \(\lambda\), we obtain

\[ \sum_{p=0}^{n}{}'' A_p(\lambda)\xi_{\lambda-p} + \sum_{p=0}^{n}{}' A_p(\lambda)\eta_{\lambda-p}=0, \tag{9} \]

\[ \sum_{p=0}^{n}{}'' B_p(\lambda)\eta_{\lambda-p} + \sum_{p=0}^{n}{}' B_p(\lambda)\xi_{\lambda-p}=0. \tag{10} \]

Here two primes and one prime over \(\sum\) mean that the summation extends over even and odd \(p\), respectively. The pair of equations (3), (9) is equivalent to equation (1). Suppose that \(n\) is even. In this case the degrees \(a_n\) and \(b_n\) of the polynomials \(A_n(\lambda)\) and \(B_n(\lambda)\) are equal to each other, namely \(a_n=b_n=(n/2+m_n)\).

Using the structure of the polynomials \(A_p(\lambda)\) and \(B_p(\lambda)\) and substituting in (9), (10) the values \(\lambda=1,3,5,\ldots,0,2,4,\ldots\), it is not difficult to see that \(\xi_{2k+1}=\eta_{2k}=0\) for the values \(k=0,1,2,\ldots\). Further, assigning \(\xi_{-1},\xi_{-3},\ldots,\xi_{-(n-1)}\) and \(\eta_{-2},\eta_{-4},\ldots,\eta_{-(n-2)}\) arbitrarily, it is easy to determine all values \(\xi_k\) lying on arithmetic progressions with difference \(n\) constructed over the numbers \(\xi_{-1},\xi_{-3},\ldots,\xi_{-(n-1)}\), and all values \(\eta_k\) lying on arithmetic progressions with difference \(n\) constructed over the numbers \(\eta_{-2},\eta_{-4},\ldots,\eta_{-(n-2)}\). The \(\xi_k\) and \(\eta_k\) found in this way depend only on the values \(\xi_{-1},\xi_{-3},\xi_{-5},\ldots,\xi_{-(n-1)};\ \eta_{-2},\eta_{-4},\ldots,\eta_{-(n-2)}\) and determine \(n\) linearly independent solutions of equation (1). These solutions are concentrated at the point \(x=0\).

Let now \(\lambda\) be a root of the polynomial \(R_n(\lambda)\). Assigning \(\xi_{\lambda_1-n},\eta_{\lambda_1-n}\) arbitrarily and \(\xi_{\lambda_1-p-kn}=\eta_{\lambda_1-p-kn}=0\) for the values \(p=0,1,\ldots,n-1\), \(k=0,1,2,\ldots\), we find all values \(\xi_\lambda\) and \(\eta_\lambda\) belonging to arithme-

tic progression constructed above \(\lambda_1\) with difference \(n\). In this case we obtain

\[ \xi_{\lambda_1-kn}\leq \frac{A_0\bigl(\lambda_1-(k-1)n\bigr)} {A_n\bigl(\lambda_1-(k-1)n\bigr)} \,\xi_{\lambda_1-(k-1)n}, \]

whence

\[ \left|\xi_{\lambda_1-kn}\right|\leq \frac{c}{k^{\,k(m_n+n+m_0)}}\,\xi_{\lambda_1-n}. \tag{11} \]

Similarly,

\[ \left|\eta_{\lambda_1-kn}\right|\leq \frac{c}{k^{\,k(m_n+n-m_0)}}\,\eta_{\lambda_1-n}. \tag{12} \]

The values \(\xi_\lambda\) and \(\eta_\lambda\) thus found determine, by formula (8), two solutions of equation (1). Since \(A_n(\lambda)\) has no multiple generalized roots, these solutions are linearly independent. The convergence of the series (8) with these \(\xi_\lambda\) and \(\eta_\lambda\) follows from the estimates (11), (12). Carrying out the indicated procedure for all roots of the polynomial \(R_n(\lambda)\), we obtain another \(2m_n\) linearly independent solutions of equation (1).

Thus, for even \(n\), equation (1) has \(2m_n+n\) linearly independent solutions of the form (8), of which \(n\) are concentrated at the point \(x=0\).

For odd \(n\) the proof is completely analogous, except that in this case
\(a_n=(n-1)/2+m_n\), \(b_n=(n+1)/2+m_n\), and the arithmetic progressions will be constructed not with difference \(n\), but with difference \(n-1\). This completes the proof of Theorem 2.

Theorem 3. The linearly independent solutions found in Theorem 2 form a fundamental system of solutions of equation (1) in \((S_0^\beta)'\), \(\beta>1\).

The proof of Theorem 3 is carried out in the same way as the proof of the analogous assertion (Theorems 2 and 3) in paper \((^1)\).

Example. The equation

\[ x^3y' + \left({}^{3}\!/\!_{2}x^2+x+1\right)y=0 \]

has in \((S_0^{3/2})'\) the fundamental system of solutions

\[ y_1(x)=\delta(x)+\frac{1}{5}\delta''(x)-\frac{4}{7\cdot5\cdot3!}\delta^{\mathrm{IV}}(x)+\cdots \]

\[ y_2(x)=-\delta' + \frac{1}{5}\delta'''(x)-\frac{2}{5\cdot3!}\delta^{\mathrm{V}}(x)+\cdots \]

\[ y_3(x)=\sum_{k=0}^{\infty} \frac{2^k(-1)^k}{k!\prod_{\mu=1}^{k}(4\mu-1)} \,f^{-(1+4k)/2}(x), \]

\[ y_4(x)=\sum_{n=0}^{\infty} \frac{(-1)^k\,2k}{k!\prod_{\mu=0}^{n}(4\mu+1)} \,g^{-(1+4k)/2}(x). \]

The solutions \(y_1\) and \(y_2\) are concentrated at the point \(x=0\).

In conclusion I express my gratitude to Prof. G. E. Shilov for his attention to the present work.

Received
22 X 1965

References

\(^1\) F. S. Aliev, DAN, 167, No. 2 (1966).
\(^2\) B. S. Mityagin, DAN, 138, No. 2, 289 (1961).
\(^3\) I. M. Gelfand, G. E. Shilov, Generalized Functions and Operations on Them, Moscow, 1959.