UDC 517.948.34

ON SOLUTIONS OF VOLTERRA’S MULTIDIMENSIONAL INTEGRAL EQUATION IN A NEIGHBORHOOD OF A SINGULAR POINT

E. I. GRUDO

The equation

\[ xu(x)=\varphi(x)+\int_0^x K[x,t,u(t)]\,dt, \tag{1} \]

where \(\varphi(x)\), \(K(x,t,u)\) are holomorphic functions in a neighborhood of the zero values of their arguments; \(\varphi(0)=\varphi'(0)=0\); \(K(x,t,0)\equiv 0\), was studied in [1—4].

The equation

\[ \begin{aligned} x_1x_2\ldots x_n u(x_1,x_2,\ldots,x_n) &=x_1x_2\ldots x_n \varphi(x_1,x_2,\ldots,x_n)+{}\\ &\quad+\int_0^{x_1}\int_0^{x_2}\ldots\int_0^{x_n} K[x_1,x_2,\ldots,x_n;\ t_1,t_2,\ldots,t_n;\\ &\qquad\qquad u(t_1,t_2,\ldots,t_n)]\,dt_1dt_2\ldots dt_n, \end{aligned} \tag{2} \]

where \(\varphi(x_1,x_2,\ldots,x_n)\), \(K(x_1,x_2,\ldots,x_n;\ t_1,t_2,\ldots,t_n;\ u)\) are holomorphic functions in a neighborhood of the zero values of their arguments; \(\varphi(0,0,\ldots,0)=0\), \(K(x_1,x_2,x_n;\ t_1,t_2,\ldots,t_n;\ 0)\equiv 0\), is a natural generalization of equation (1) to the multidimensional case.

In the present note we shall attempt to carry over to equation (2) certain results obtained for equation (1).

Denote

\[ \frac{\partial K(0,0,\ldots,0)}{\partial u}=\lambda \]

and seek the solution of equation (2) in the form of the series

\[ u(x_1,x_2,\ldots,x_n)= \sum_{i_1+\cdots+i_n=1}^{\infty} \alpha_{i_1i_2\ldots i_n}x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n}. \tag{3} \]

To determine the coefficients \(\alpha_{i_1\ldots i_n}\) we have the equations:

\[ \alpha_{i_1\ldots i_n} = \frac{\lambda\alpha_{i_1\ldots i_n}} {(i_1+1)(i_2+1)\ldots(i_n+1)} + \varphi_{i_1i_2\ldots i_n} + R_{i_1\ldots i_n}(\alpha_{i'_1i'_2\ldots i'_n}), \tag{4} \]

where

\[ \varphi_{i_1\ldots i_n} = \frac{1}{i_1!i_2!\ldots i_n!} \frac{\partial^{i_1+i_2+\cdots+i_n}\varphi(0,0,\ldots,0)} {\partial x_1^{i_1}\partial x_2^{i_2}\ldots\partial x_n^{i_n}}; \]

\(R_{i_1\ldots i_n}\) are polynomials with respect to those \(\alpha_{i'_1\ldots i'_n}\) for which \(i'_1+\ldots+i'_n<i_1+\ldots+i_n\),

\(i'_1 \leq i_1, \ldots, i'_n \leq i_n\). These polynomials are the coefficients of \(x_1^{i_1}\cdots x_n^{i_n}\) in the series obtained from

\[ \int_0^{x_1}\int_0^{x_2}\cdots\int_0^{x_n} \{K[x_1,\ldots,x_n;t_1,\ldots,t_n; \]

\[ u(t_1,\ldots,t_n)]-\lambda u(t_1,\ldots,t_n)\}\,dt_1\cdots dt_n \]

by substituting the series (3) for \(u\) in the last expression.

If

\[ (i_1+1)(i_2+1)\cdots(i_n+1)-\lambda\ne 0 \tag{5} \]

for all nonnegative integers \(i_1,\ldots,i_n\), \(i_1+\cdots+i_n\geq 1\), then the recurrence formulas (4) make it possible to find uniquely all coefficients \(a_{i_1\ldots i_n}\) of the series (3):

\[ a_{i_1\ldots i_n} = \frac{(i_1+1)\cdots(i_n+1)} {(i_1+1)\cdots(i_n+1)-\lambda} \,[R_{i_1\ldots i_n}(a_{i'_1\ldots i'_n})+\varphi_{i_1\ldots i_n}]. \tag{6} \]

Thus, if \(\lambda\) is not a positive integer, then equation (2) has a unique formal solution (3).

Let us prove the convergence of the series (3) for sufficiently small \(|x_1|,\ldots,|x_n|\). To this end, note that there exists a positive number \(B\) such that we have

\[ \left| \frac{(i_1+1)\cdots(i_n+1)} {(i_1+1)\cdots(i_n+1)-\lambda} \right| = \]

\[ = \left| \frac{1} {1-\dfrac{\lambda}{(i_1+1)\cdots(i_n+1)}} \right|<B \tag{7} \]

for \(i_1+\cdots+i_n\geq 1\), and \(B\) may be chosen so that, along with (7),

\[ \left| \frac{1}{(i_1+1)\cdots(i_n+1)} \right|<B \tag{8} \]

for \(i_1+\cdots+i_n\geq 1\).

Let \(\Phi(x_1,\ldots,x_n)\), \(K_1(x_1,\ldots,x_n;t_1,\ldots,t_n;u)\) be majorants respectively of the series \(\varphi(x_1,\ldots,x_n)\), \(K(x_1,\ldots,x_n;t_1,\ldots,t_n;u)-\lambda u\) in a neighborhood of the zero values of the arguments, \(\Phi(0,0,\ldots,0)=0\),

\[ K_1(x_1,\ldots,x_n;t_1,\ldots,t_n;0)\equiv 0, \qquad \frac{\partial K_1(0,\ldots,0)}{\partial u}=0. \]

Consider the equation

\[ U=B\Phi(x_1,\ldots,x_n)+B^2K_1(x_1,\ldots,x_n;x_1,\ldots,x_n;U). \]

In a neighborhood of \(x_1=\cdots=x_n=0\) this equation has a holomorphic solution

\[ U(x_1,\ldots,x_n) = \sum_{i_1+\cdots+i_n=1}^{\infty} A_{i_1\ldots i_n}x_1^{i_1}\cdots x_n^{i_n}, \tag{9} \]

where all \(A_{i_1\ldots i_n}\) are positive. The coefficients \(A_{i_1\ldots i_n}\) are formed by formulas analogous to (6):

\[ A_{i_1,\ldots,i_n} = B\Phi_{i_1\ldots i_n} + B^2\widehat R_{i_1\ldots i_n}(A_{i'_1\ldots i'_n}), \tag{10} \]

where

\[ \Phi_{i_1\ldots i_n}= \frac{1}{i_1!\cdots i_n!}\, \frac{\partial^{i_1+\cdots+i_n}\Phi(0,\ldots,0)} {\partial x_1^{i_1}\cdots \partial x_n^{i_n}}, \qquad \widetilde R_{i_1\ldots i_n} \]

are polynomials in those \(A_{i'_1\ldots i'_n}\) for which \(i'_1+\cdots+i'_n<i_1+\cdots+i_n\), \(i'_1\leq i_1,\ldots,i'_n\leq i_n\). These polynomials are the coefficients of \(x_1^{i_1}\cdots x_n^{i_n}\) in the series obtained from the series \(K_1(x_1,\ldots,x_n;\,x_1,\ldots,x_n;\,U)\) by substituting into it the series (9). On the basis of (6), (7), (8), (10) it is not difficult to see that

\[ |\alpha_{i_1\ldots i_n}|\leq A_{i_1\ldots i_n} \]

for \(i_1+\cdots+i_n\geq 1\), which implies the convergence of the series (3) for sufficiently small \(|x_1|,\ldots,|x_n|\).

Thus we have

Theorem 1. If \(\dfrac{\partial K(0,\ldots,0)}{\partial u}\) is not equal to a positive integer, then equation (2) has a unique holomorphic solution (3) in a neighborhood of \(x_1=\cdots=x_n=0\).

Let now \(\lambda\) be equal to a positive integer. Let \((i_1^{(k)},\ldots,i_n^{(k)})\) \((k=1,\ldots,m)\) be all distinct systems of nonnegative integers for which

\[ (i_1^{(k)}+1)\cdots(i_n^{(k)}+1)=\lambda \qquad (k=1,\ldots,m). \]

If

\[ \varphi_{i_1^{(k)}\ldots i_n^{(k)}}+ R_{i_1^{(k)}\ldots i_n^{(k)}}=0 \qquad (k=1,\ldots,m), \tag{11} \]

then \(\alpha_{i_1^{(k)}\ldots i_n^{(k)}}=C_k\) \((k=1,\ldots,m)\) remain arbitrary. It is not difficult to see that the series (3) will converge in this case.

Consequently, if \(\dfrac{\partial K(0,\ldots,0)}{\partial u}\) is equal to a positive integer and the conditions (11) are satisfied, then equation (2) has a family of holomorphic solutions (3), depending on \(m\) arbitrary constants \(C_1,\ldots,C_m\).

If at least one of the relations (11) is not satisfied, then equation (2) has no solutions of the form (3).

Suppose now that in equation (2) \(\varphi(x_1,\ldots,x_n)\equiv0\), i.e., consider the equation

\[ x_1\cdots x_n u(x_1,\ldots,x_n) = \int_0^{x_1}\int_0^{x_2}\cdots\int_0^{x_n} K[x_1,\ldots,x_n; \]

\[ t_1,\ldots,t_n;\ u(t_1,\ldots,t_n)]\,dt_1\cdots dt_n . \tag{12} \]

Henceforth we shall assume that \(\lambda\ne0\).

We shall seek a solution of this equation in the form of the series

\[ u= \sum_{i_1+\cdots+i_n+j=1}^{\infty} a_{i_1\ldots i_n j}\, x_1^{i_1}\cdots x_n^{i_n} \left(Cx_1^{\alpha_1}\cdots x_n^{\alpha_n}\right)^j, \tag{13} \]

where \(C\) is an arbitrary constant and \(\alpha_1,\ldots,\alpha_n\) are certain constants, not all equal to zero. To construct a formal solution (13) of equation (12), the nonzero constants \(\alpha_k\) and the corresponding paths of integration in equation (12) should be regarded as connected in such a way that \(x_k^{\alpha_k}\to0\) as \(x_k\to0\).

along the paths of integration in the planes \(x_k\). Obviously, none of the \(\alpha_k\) must be equal to a negative real number. Therefore all the expressions

\[ i_1+j\alpha_1+1,\ldots,i_n+j\alpha_n+1 \]

for \(i_1\geqslant 0,\ldots,i_n\geqslant 0,\quad j\geqslant 1\) will be, in modulus, greater than some positive number.

Substituting now the series (13) into equation (12), to determine the coefficients \(a_{i_1\ldots i_n j}\) we obtain the equations

\[ a_{i_1\ldots i_n j} = \frac{\lambda a_{i_1\ldots i_n j}^{\prime}} {(i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)} + R_{i_1\ldots i_n j}(a_{i_1\ldots i_n j}^{\prime}), \tag{14} \]

where \(R_{i_1\ldots i_n j}\) are polynomials with respect to those \(a_{i'_1\ldots i'_n j'}\) for which
\(i'_1+\cdots+i'_n+j'<i_1+\cdots+i_n+j,\quad i'_1\leqslant i_1,\ldots,\quad i'_n\leqslant i_n,\quad j'\leqslant j\).

For \(i_1=0,\ldots,i_n=0,\quad j=1\), formulas (14) give

\[ a_{00\ldots 01} = \frac{\lambda a_{00\ldots 01}} {(\alpha_1+1)\ldots(\alpha_n+1)}. \]

Thus one may put \(a_{00\ldots 01}=1\), while the constants \(\alpha_1,\ldots,\alpha_n\) must also be subject to the condition

\[ (\alpha_1+1)\ldots(\alpha_n+1)=\lambda. \tag{15} \]

It is clear that the constants \(\alpha_1,\ldots,\alpha_n\) and the paths of integration in equation (12) can always be chosen, and with a large degree of arbitrariness, so that equality (15) is satisfied and so that \(x_k^{\alpha_k}\to 0\) as \(x_k\to 0\) for \(\alpha_k\) different from zero along the paths of integration in the planes \(x_k\). The paths of integration corresponding to \(\alpha_k\) equal to zero may be chosen arbitrarily. We note that if \(\lambda-1\) is equal to a real nonpositive number, then among \(\alpha_1,\ldots,\alpha_n\) at least two must be different from zero.

As is seen from the recurrence formulas (14), if

\[ (i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)-\lambda\ne 0 \tag{16} \]

for \(i_1+\cdots+i_n+j>1,\quad j>0\), then all the coefficients \(a_{i_1\ldots i_n j}\) of the series (13) are determined successively in a unique way:

\[ a_{i_1\ldots i_n j} = \frac{(i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)} {(i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)-\lambda} \times \]

\[ {}\times R_{i_1\ldots i_n j}(a_{i'_1\ldots i'_n j'}). \tag{17} \]

Since none of \(\alpha_1,\ldots,\alpha_n\) is equal to a negative real number, the equality

\[ (i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)-\lambda=0 \]

can hold only for a finite number of systems \((i_1,\ldots,i_n,j)\) satisfying the conditions
\(i_1+\cdots+i_n+j>1,\quad j>0\). As \(i_1+\cdots+i_n+j\to\infty\),
\[ \left|(i_1+j\alpha_1+1)\ldots(i_n+j\alpha_n+1)-\lambda\right|\to\infty. \]
Therefore, by changing, if necessary, the nonzero \(\alpha_1,\ldots,\alpha_n\) arbitrarily little, but so that condition (15) is fulfilled, we shall ensure that inequality (16) holds for all
\(i_1+\cdots+i_n+j>1,\quad j>0\). In doing so, if necessary, we change, in accordance with the change of \(\alpha_1,\ldots,\alpha_n\), the paths of integration, so that for nonzero \(\alpha_k\), \(x_k^{\alpha_k}\to 0\) as \(x_k\to 0\) along the paths of integration.

We shall now prove the convergence of the series (13) for sufficiently small \(|x_1|,\ldots,|x_n|\), \(|C x_1^{\alpha_1}\cdots x_n^{\alpha_n}|\), assuming that inequality (16) holds.

Obviously, by virtue of our conditions on \(\alpha_1,\ldots,\alpha_n\), there exists a positive number \(B\) such that

\[ \left| \frac{(i_1+j\alpha_1+1)\cdots(i_n+j\alpha_n+1)} {(i_1+j\alpha_1+1)\cdots(i_n+j\alpha_n+1)-\lambda} \right|<B \tag{18} \]

for \(i_1+\cdots+i_n+j>1,\quad j>0\); moreover, \(B\) can be chosen so that, along with (18), the inequalities

\[ \left| \frac{1}{(i_1+j\alpha_1+1)\cdots(i_n+j\alpha_n+1)} \right|<B \tag{19} \]

are satisfied for \(i_1\geq 0,\ldots,i_n\geq 0,\quad j\geq 1\).

Consider the equation

\[ U=T_1+B^2 K_1(x_1,\ldots,x_n;\ x_1,\ldots,x_n;\ U). \]

This equation has a holomorphic solution in a neighborhood of \(x_1=\cdots=x_n=T_1=0\),

\[ U=\sum_{i_1+\cdots+i_n+j=1}^{\infty} A_{i_1\ldots i_n j}x_1^{i_1}\cdots x_n^{i_n}T_1^j,\qquad A_{0\ldots 01}=1. \tag{20} \]

The coefficients \(A_{i_1\ldots i_n j}\) are determined by the recurrence formulas

\[ A_{i_1\ldots i_n j}=B^2\widetilde R_{i_1\ldots i_n j}(A_{i'_1\ldots i'_n j'}) \tag{21} \]

\[ (i_1+\cdots+i_n+j>1,\ j>0), \]

where \(\widetilde R_{i_1\ldots i_n j}\) are polynomials in those \(A_{i'_1\ldots i'_n j'}\) for which \(i'_1+\cdots+i'_n+j'<i_1+\cdots+i_n+j,\ i'_1\leq i_1,\ldots,i'_n\leq i_n,\ j'\leq j\). These polynomials are obtained as the coefficients of \(x_1^{i_1}\cdots x_n^{i_n}T_1^j\) in the series obtained from \(K_1(x_1,\ldots,x_n;\ x_1,\ldots,x_n;\ U)\) after substituting into it the series (20). On the basis of (17), (18), (19), (21) it is easy to see that

\[ |a_{i_1\ldots i_n j}|\leq A_{i_1\ldots i_n j} \]

for \(i_1+\cdots+i_n+j>0,\quad j>0\), whence follows the convergence of the series (13) for sufficiently small \(|x_1|,\ldots,|x_n|\), \(|C x_1^{\alpha_1}\cdots x_n^{\alpha_n}|\).

Thus, the following can be formulated.

Theorem 2. Let the constants \(\alpha_1,\ldots,\alpha_n\), not all equal to zero, satisfy the following conditions:

1)

\[ (\alpha_1+1)\cdots(\alpha_n+1)=\lambda\ne 0, \qquad \text{where }\lambda=\frac{\partial K(0,\ldots,0)}{\partial u}; \]

2)

\[ (i_1+j\alpha_1+1)\cdots(i_n+j\alpha_n+1)\ne\lambda \]

for nonnegative integers \(i_1,\ldots,i_n,j\) subject to the conditions
\[ i_1+\cdots+i_n+j>1,\qquad j>0; \]

3) for nonzero \(a_1,\ldots,a_n\), \(x_k^{\alpha_k}\to 0\) as \(x_k\to 0\) along the corresponding paths of integration of equation (12).

Then equation (12) has a solution, representable for sufficiently small \(|x_1|,\ldots,|x_n|\), \(|C x_1^{\alpha_1}\cdots x_n^{\alpha_n}|\), by the convergent series (13).

Theorem 2 can be generalized. We shall seek a solution of equation (12) in the form of the series

\[ U= \sum_{i_1+\cdots+i_n+j_1+\cdots+j_m=1}^{\infty} a_{i_1\ldots i_n j_1\ldots j_m} x_1^{i_1}\cdots x_n^{i_n}\times \]

\[ \times \left(C_1x_1^{\alpha_1^{(1)}}\cdots x_n^{\alpha_n^{(1)}}\right)^{j_1} \cdots \left(C_mx_1^{\alpha_1^{(m)}}\cdots x_n^{\alpha_n^{(m)}}\right)^{j_m}, \tag{22} \]

where \(m\) is an arbitrary finite positive integer; \(C_1,\ldots,C_m\) are arbitrary constants; \(\alpha_1^{(k)},\ldots,\alpha_n^{(k)}\) \((k=1,\ldots,m)\) are certain constants. For each \(k\), among \(\alpha_1^{(k)},\ldots,\alpha_n^{(k)}\), some, but not all, may be equal to zero. We shall regard all nonzero constants \(\alpha_q^{(k)}\) \((k=1,\ldots,m;\ 1\le q\le n)\) and their corresponding path of integration in equation (12) in the plane \(x_q\) as being related in such a way that \(x_q^{\alpha_q^{(k)}}\to 0\) as \(x_q\to 0\) along the path of integration. If all \(\alpha_q^{(k)}\) \((k=1,\ldots,m)\) are equal to zero, then the path of integration in the plane \(x_q\) for equation (12) may be arbitrary. Clearly, under our conditions on \(\alpha_q^{(k)}\), the expressions

\[ i_q+j_1\alpha_q^{(1)}+\cdots+j_m\alpha_q^{(m)}+1 \qquad (q=1,\ldots,n) \]

for all finite \(i_1,\ldots,i_n,\ j_1,\ldots,j_m\) satisfying the conditions \(i_1\ge 0,\ldots,i_n\ge 0,\ j_1+\cdots+j_m\ge 1\), will be nonzero. Further, under these same conditions, as follows from [5], the nonzero constants \(\alpha_q^{(k)}\) \((k=1,\ldots,m;\ 1\le q\le n)\) and \(1\) will lie in the plane \(x_q\) on one side of some straight line passing through the origin. Then, on the basis of [6], the moduli of the preceding expressions tend to infinity as \(i_q+j_1+\cdots+j_m\to\infty\).

Substituting now the series (22) into equation (12), for determining the coefficients \(a_{i_1\ldots i_n j_1\ldots j_m}\) we obtain the recurrence formulas

\[ a_{i_1\ldots i_n j_1\ldots j_m}= \]

\[ = \frac{\lambda a_{i_1\ldots i_n j_1\ldots j_m}} {(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1)} + \]

\[ +R_{i_1\ldots i_n j_1\ldots j_m} \bigl(a_{i_1'\ldots i_n'j_1'\ldots j_m'}\bigr), \tag{23} \]

where \(R_{i_1\ldots i_n j_1\ldots j_m}\) are polynomials in those \(a_{i_1'\ldots i_n'j_1'\ldots j_m'}\) for which

\[ i_1'+\cdots+i_n'+j_1'+\cdots+j_m' < i_1+\cdots+i_n+j_1+\cdots+j_m, \]

\[ i_1'\le i_1,\ldots,i_n'\le i_n,\qquad j_1'\le j_1,\ldots,j_m'\le j_m. \]

From formulas (23), for \(i_1=0,\ldots,i_n=0,\ j_1+\cdots+j_m=1\), we obtain that

\[ a_{00\ldots 0j_1\ldots j_m}=1 \qquad (j_1+\cdots+j_m=1), \]

and \(\alpha_q^{(k)}\) \((k=1,\ldots,m;\ q=1,\ldots,n)\) must also satisfy the relations

\[ (\alpha_1^{(k)}+1)\cdots(\alpha_n^{(k)}+1)=\lambda \qquad (k=1,\ldots,m). \tag{24} \]

It is clear that one can always choose, and moreover with a large degree of arbitrariness, the constants \(\alpha_q^{(k)}\) \((k=1,\ldots,m;\ q=1,\ldots,n)\) and the paths of integration for equation (12) so that for nonzero \(\alpha_q^{(k)}\) \((k=1,\ldots,m;\ 1\le q\le n)\) one has \(x_q^{\alpha_q^{(k)}}\to 0\) as \(x_q\to 0\) along the path of integration and so that the equalities (24) are satisfied.

Since

\[ \left|i_q+j_1\alpha_q^{(1)}+\cdots+j_m\alpha_q^{(m)}+1\right|\to\infty \]

as \(i_q+j_1+\cdots+j_m\to\infty\), it is clear that the expression

\[ (i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+ \]

\[ +\cdots+j_m\alpha_n^{(m)}+1)-\lambda \tag{25} \]

can vanish, for \(i_1+\cdots+i_n+j_1+\cdots+j_m>1,\ j_1+\cdots+j_m\ge 1\), only for a finite number of systems \((i_1,\ldots,i_n,j_1,\ldots,j_m)\).

Therefore, if necessary, we can change the nonzero \(\alpha_q^{(k)}\) arbitrarily little so that the equalities (24) hold and so that expression (25) does not vanish for all \(i_1+\cdots+i_n+j_1+\cdots+j_m>1,\ j_1+\cdots+j_m>1\). In doing so, if necessary, we change the paths of integration in equation (12) so that the relation indicated above is realized between \(\alpha_q^{(k)}\) and the paths of integration.

Then the coefficients \(a_{i_1\ldots i_n j_1\ldots j_m}\) \((i_1+\cdots+i_n+j_1+\cdots+j_m>1,\ j_1+\cdots+j_m>0)\) of the series (22) are determined successively in a unique way:

\[ a_{i_1\ldots i_n j_1\ldots j_m}= \]

\[ = \frac{(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1)} {(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1)-\lambda} \times \]

\[ \times R_{i_1\ldots i_n j_1\ldots j_m}. \]

The proof of convergence of the series (22) for \(|x_1|,\ldots,|x_n|\), \(\left|C_1x_1^{\alpha_1^{(1)}}\cdots x_n^{\alpha_n^{(1)}}\right|,\ldots,\left|C_mx_1^{\alpha_1^{(m)}}\cdots x_n^{\alpha_n^{(m)}}\right|\) sufficiently small is carried out analogously to the proof of convergence of the series (13).

By virtue of the conditions imposed on \(\alpha_q^{(k)}\), there exists a positive number \(B\) such that we have

\[ \left| \frac{(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_1^{(n)}+\cdots+j_m\alpha_n^{(m)}+1)} {(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1)-\lambda} \right|<B \]

for \(i_1+\cdots+i_n+j_1+\cdots+j_m>1,\ j_1+\cdots+j_m>0\). This number \(B\) can be chosen so that the inequalities

\[ \left| \frac{1} {(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1)\cdots (i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1)} \right|<B \]

also hold for \(i_1\ge0,\ldots,i_n\ge0,\ j_1+\cdots+j_n\ge1\).

Considering now the equation

\[ U=T_1+\cdots+T_m+B^2K_1(x_1;\ldots,x_n;\ x_n,\ldots,x_n,\ U), \]

which has a solution holomorphic in a neighborhood of \(x_1=\cdots=x_n=T_1=\cdots=T_m=0\),

\[ U=\sum_{i_1+\cdots+i_n+j_1+\cdots+j_m=1}^{\infty} A_{i_1\ldots i_n j_1\ldots j_m} x_1^{i_1}\cdots x_n^{i_n}T_1^{j_1}\cdots T_m^{j_m}, \]

it is not hard to see that

\[ |a_{i_1\ldots i_n j_1\ldots j_m}|<A_{i_1\ldots i_n j_1\ldots j_m}, \]

whence follows the convergence, for \(|x_1|,\ldots,|x_n|\), \(\left|C_1x_1^{\alpha_1^{(1)}}\cdots x_n^{\alpha_n^{(1)}}\right|,\ldots,\left|C_mx_1^{\alpha_1^{(m)}}\cdots x_n^{\alpha_n^{(m)}}\right|\) sufficiently small, of the series (22).

Thus, the following can be formulated.

Theorem 3. Let the constants \(\alpha_1^{(k)},\ldots,\alpha_n^{(k)}\) \((k=1,\ldots,m)\), not all equal to zero for each \(k\), satisfy the following conditions:

\[ 1)\quad (\alpha_1^{(k)}+1)\cdots(\alpha_n^{(k)}+1)=\lambda \quad (k=1,\ldots,m),\qquad \lambda=\frac{\partial K(0,\ldots,0)}{\partial u}\ne0; \]

2)
\[ \left(i_1+j_1\alpha_1^{(1)}+\cdots+j_m\alpha_1^{(m)}+1\right)\cdots \left(i_n+j_1\alpha_n^{(1)}+\cdots+j_m\alpha_n^{(m)}+1\right)\ne\lambda \]
for nonnegative integers \(i_1,\ldots,i_n,j_1,\ldots,j_m\) subject to the conditions
\(i_1+\cdots+i_n+j_1+\cdots+j_m>1,\ j_1+\cdots+j_m>0\);

3) for \(\alpha_q^{(k)}\ne0\) \((k=1,\ldots,m;\ 1\le q\le n)\),
\(x_q^{\alpha_q^{(k)}}\to0\) as \(x_q\to0\) along the path of integration of equation (12) in the \(x_q\)-plane.

Then equation (12) has a solution representable, for
\[ |x_1|,\ldots,|x_n|,\quad \left|C_1x_1^{\alpha_1^{(1)}}\cdots x_n^{\alpha_n^{(1)}}\right|,\ldots, \left|C_mx_1^{\alpha_1^{(m)}}\cdots x_n^{\alpha_n^{(m)}}\right| \]
sufficiently small, by the convergent series (22).

References

1. Horn J. Jahresbericht d. D. Math.-Ver., 23, 1914.
2. Sato T. Journal of the Math. Soc. of Japan, 5, N. 2, 1953.
3. Grudo E. I. Differential Equations, 1, No. 2, 214—218, 1965.
4. Grudo E. I. Differential Equations, 1, No. 4, 535—544, 1965.
5. Grudo E. I. Differential Equations, 3, No. 5, 1967.
6. Picard E., Traité d’Analyse, t. 3, 1908, 6—7.

Received by the editors
February 3, 1966

Institute of Mathematics, Academy of Sciences of the BSSR