Full Text
ON A CERTAIN CASE OF A VOLTERRA INTEGRAL EQUATION
E. I. GRUDO
Consider the integral equation
\[ xu(x)=\int_0^x H(x,t,u(t))\,dt, \tag{1} \]
where \(H(x,t,u)\) is a function holomorphic in a neighborhood of \(x=t=u=0\) and vanishing at \(x=t=u=0\). Equation (1) can be written in the form
\[ xu(x)=f(x)+\lambda\int_0^x u(t)\,dt+\int_0^x K(x,t,u(t))\,dt, \tag{2} \]
where \(f(x)\) and \(K(x,t,u)\) are functions holomorphic in a neighborhood of \(x=t=u=0\),
\[ f(0)=f'(0)=0,\quad K(x,t,0)\equiv 0,\quad \frac{\partial k(0,0,0)}{\partial u}=0. \]
It is proved in [1] that if there exists a formal solution
\[ u=c_1x+c_2x^2+\cdots \tag{3} \]
of equation (2), then this series converges in a neighborhood of \(x=0\) and represents a solution of (2). A formal solution (3) can fail to exist only in the case \(\lambda-1=m\), where \(m\) is a positive integer. It is further shown that if \(\lambda-1\) is neither zero, nor a positive integer, nor a negative rational number, then equation (2) with \(f(x)\equiv 0\) has a unique formal solution in powers of \(x\) and \(Cx^{\lambda-1}\),
\[ u=\sum_{j,k=0}^{\infty} p_{jk}x^j(Cx^{\lambda-1})^{k+1} \tag{4} \]
(\(p_{00}=1,\ C\) is an arbitrary constant).
Series (4) converges and represents a solution of equation (2) with \(f(x)\equiv 0\) for sufficiently small \(|x|\) and \(|Cx^{\lambda-1}|\), provided that \(\lambda-1\) is neither zero nor a negative number; in this case the path of integration in (2) is chosen in the appropriate way.
However, apparently, the nonlinear equation (2) was first considered by Horn [3] in the cases where \(\lambda-1\) is not equal to a positive integer and \(\operatorname{Re}(\lambda-1)>0\), although [1] does not refer to [3]. Horn’s results in these cases coincide with the results obtained in [1].
We note that in the calculation of the coefficients of series (4) in [1] an error was made, and therefore the assertion that in the case of a positive integer ...
of \(\lambda-1\) for equation (2) with \(f(x)\equiv 0\), the series (4) may fail to exist, is not true. A unique series of the form (4) always exists and in this case converges and represents a solution of equation (2). Indeed, to compute the coefficients \(p_{jk}\), substituting (4) into (2), we obtain the formulas
\[ \frac{j+(\lambda-1)k}{1+j+(\lambda-1)(k+1)}\,p_{jk} = B_{jk}(p_{lm})\quad (j+k>0,\ p_{00}=1), \]
which allow one to find successively all \(p_{jk}\). Here \(B_{jk}(p_{lm})\) is a polynomial in the \(p_{lm}\) \((l+m<j+k)\). The convergence of the series (4), in the case of its existence for \(\lambda-1\) equal to a positive integer, was proved in [1].
In the present note we consider equation (2) in the case where \(\lambda-1\) is a positive integer, when for this equation there is no series (3) (in this case, of course, \(f(x)\not\equiv 0\)). We shall show that there exists an infinite set of solutions holomorphic with respect to \(x\) and \(x^{\lambda-1}\ln x\).
Make in equation (2) the transformation
\[ u=p_1x+p_2x^2+\cdots+p_{\lambda-2}x^{\lambda-2}+w, \]
so that in the transformed equation the Maclaurin expansion of the free term begins with degree \(\lambda\). To determine \(p_i\) \((i=1,2,\ldots,\lambda-2)\) we obviously obtain the equations
\[ p_i=f_{i+1}+\frac{\lambda p_i}{i+1}+Q_i(p_1,\ldots,p_{i-1}),\quad Q_1\equiv 0, \]
where \(f_{i+1}\) are the coefficients of the Maclaurin expansion of the function \(f(x)\), and \(Q_i\) are polynomials in their arguments. These equations make it possible to find \(p\) uniquely.
For \(w\) we obtain the equation
\[ xw(x)=g(x)+\lambda\int_0^x w(t)\,dt+\int_0^x K_2(x,t,w(t))\,dt, \tag{5} \]
where
\[ g(0)=g'(0)=\cdots=g^{(\lambda-1)}(0)=0, \]
\[ g^{(\lambda)}(0)\ne 0,\quad K_2(x,t,0)\equiv 0,\quad \frac{\partial K_2(0,0,0)}{\partial u}=0. \]
Let
\[ g(x)=\sum_{i=\lambda}^{\infty} g_i x^i,\quad K_2(x,t,w)=\sum_{i,j,k} b_{ijk}x^i t^j w^k. \]
We shall seek a solution of equation (5) in the form of a series
\[ w=\sum_{i,j} p_{ij}x^i(-ax^{\lambda-1}\ln Cx)^j \quad (i=0,\ j\ge 1;\ i=0,\ j\ge \lambda-1,\ i+j>0), \tag{6} \]
where \(C\) is an arbitrary constant; \(a\) is a constant which we shall determine shortly.
For what follows we shall need the formula
\[ \int_0^x t^i(-at^{\lambda-1}\ln Ct)^j\,dt = \frac{x^{i+1}(-ax^{\lambda-1}\ln Cx)^j}{i+j(\lambda-1)+1} + \]
\[ +\sum_{m=1}^{j}\frac{a^m j(j-1)\cdots(j-m+1)}{[i+j(\lambda-1)+1]^{m+1}}x^{i+m(\lambda-1)+1}(-a x^{\lambda-1}\ln Cx)^{j-m}. \tag{7} \]
We now substitute the series (6) into equation (5) and compare the coefficients at like powers of \(x\) and \(-a x^{\lambda-1}\ln Cx\) on the left and on the right, using formula (7). For \(i=0,\ j=1\) we obtain the identity
\[ p_{01}=p_{01}, \]
i.e. the coefficient \(p_{01}\) may be taken arbitrarily. Since \(a\) has not yet been determined, we may assume that \(p_{01}=1\). For \(i=\lambda-1,\ j=0\) we obtain
\[ p_{\lambda-1,0}=g_\lambda+ap_{01}+p_{\lambda-1,0}. \]
From this equation we find
\[ a=-g_\lambda, \]
and \(p_{\lambda-1,0}\) may be taken arbitrarily. We take \(p_{\lambda-1,0}=0\). To determine \(p_{k1}\) \((k=1,2,\ldots,\lambda-2)\), we have the equations
\[ p_{k1}=\frac{\lambda}{k+\lambda}p_{k1}+B_{k1}(p_{01},p_{11},\ldots,p_{k-1,1}), \]
where \(B_{k1}\) are polynomials in their arguments. From these equations we successively find \(p_{k1}\) \((k=1,2,\ldots,\lambda-2)\). After this we find \(p_{k0}\) \((k=\lambda,\ldots,2\lambda-3)\). To determine them we have the equations
\[ p_{k0}=g_{k+1}+\frac{\lambda}{k+1}p_{k0} +B_{k0}(p_{11},p_{21},\ldots,p_{k-\lambda+1,1},p_{\lambda0},\ldots,p_{k-1,0}), \]
where \(B_{k0}\) are polynomials in their arguments. From these equations we successively find \(p_{k0}\) \((k=\lambda,\ldots,2\lambda-3)\).
We shall call the number \(i+j(\lambda-1)\) the dimension of the coefficient \(p_{ij}\). If all coefficients \(p_{ij}\) of dimensions less than \(m(\lambda-1)\) have already been determined, then the coefficients of dimensions less than \((m+1)(\lambda-1)\) can be found in the following sequence. First we determine the coefficient \(p_{0m}\). For it we have the equation
\[ p_{0m}=\frac{\lambda}{m(\lambda-1)+1}\,p_{0m}+B_{0m}(p_{ij}), \]
where \(B_{0m}(p_{ij})\) is a polynomial in the \(p_{ij}\) with dimensions less than \(m(\lambda-1)\). Then we determine the coefficients \(p_{km}\) \((k=1,2,\ldots,\lambda-2)\). For them we have the equations
\[ p_{km}=\frac{\lambda p_{km}}{k+m(\lambda-1)+1}+B_{km}(p_{1m},\ldots,p_{k-1,m},p_{ij}), \]
where \(B_{km}\) is a polynomial in \(p_{1m},\ldots,p_{k-1,m}\) and \(p_{ij}\) with dimensions less than \(m(\lambda-1)\). Hence we successively find \(p_{km}\) \((k=1,2,\ldots,\lambda-2)\). Next, in an analogous way, we find \(p_{\lambda-1,m-1},\ldots,p_{2\lambda-3,m-1}\), and so on. Penultimately we find the coefficients \(p_{(m-1)(\lambda-1),1},\ldots,p_{m(\lambda-1)-1,1}\), and then \(p_{m(\lambda-1),0},\ldots,p_{(m+1)(\lambda-1)-1,0}\). After this we proceed to determine the coefficients \(p_{ij}\) with dimensions less than \((m+2)(\lambda-1)\).
Thus, we have shown that in our case for equation (5) there exists the series (6), which is a formal solution of (5).
We shall now prove the convergence of the series (6). Let
\[ \frac{M_1 w}{ \left(1-\dfrac{x}{r}\right) \left(1-\dfrac{t}{r}\right) \left(1-\dfrac{w}{\rho}\right) }, \]
where \(r\) and \(\rho\) are the radii of convergence of \(K_2(x,t,w)\), respectively in \(x,t\), and \(w\), be a majorant of the function \(K_2(x,t,w)\), and
\[ \frac{M x^\lambda}{1-\dfrac{x}{r}} \]
be a majorant of the function \(g(x)\). We assume that \(g(x)\) and \(K_2(x,t,w)\) have one and the same radius of convergence in \(x\). Consider the auxiliary integral equation
\[ xW(x)=-Mx^\lambda+\frac{M}{r}\frac{x^{\lambda+1}}{1-\dfrac{x}{r}} +\alpha\int_0^x W(t)\,dt+ \]
\[ +\int_0^x \frac{M_1 W(t)\,dt}{ \left(1-\dfrac{x}{r}\right) \left(1-\dfrac{t}{r}\right) \left(1-\dfrac{W(t)}{\rho}\right) }, \tag{8} \]
where the coefficient \(\alpha\) is chosen so that
\[ \alpha+M_1=\lambda. \]
It is clear that equation (8) also has a formal solution in powers of \(x\) and \(-ax^{\lambda-1}\ln Cx\) (here \(a=M\)):
\[ W=\sum_{i,j} P_{ij}x^i(-Mx^{\lambda-1}\ln Cx)^j \quad (i=0,\ j\geqslant 1;\ j=0,\ i\geqslant \lambda-1;\ i+j>0), \tag{9} \]
where all coefficients \(P_{ij}\) are positive. (As in the series (6), we put here \(P_{01}=1,\ P_{\lambda-1,0}=0\).) Further, it is clear that
\[ |p_{ij}|\leqslant P_{ij}. \]
Therefore, to prove the convergence of the series (6), it is sufficient to prove the convergence of the series (9).
Equation (8) can be rewritten in the form
\[ x\left(1-\frac{x}{r}\right)W(x) = -Mx^\lambda+\frac{2M}{r}x^{\lambda+1} +\alpha\left(1-\frac{x}{r}\right)\int_0^x W(t)\,dt+ \]
\[ +\int_0^x \frac{M_1 W(t)\,dt}{ \left(1-\dfrac{t}{r}\right) \left(1-\dfrac{W(t)}{\rho}\right) }. \]
Differentiating this equation twice with respect to \(x\), we obtain
\[ x\left(1-\frac{x}{r}\right)W''' +\left[ 2-\alpha-\frac{4-\alpha}{r}x -\frac{M_1}{\left(1-\frac{x}{r}\right)\left(1-\frac{W}{\rho}\right)} -\frac{M_1W}{\rho\left(1-\frac{x}{r}\right)\left(1-\frac{W}{\rho}\right)^2} \right]W'' \]
\[ +\left[ \frac{2(\alpha-1)}{r} -\frac{M_1}{r}\, \frac{1}{\left(1-\frac{x}{r}\right)^2\left(1-\frac{W}{\rho}\right)} \right]W' = -M(\lambda-1)\lambda x^{\lambda-2} +\frac{2M(\lambda+1)\lambda}{r}x^{\lambda-1}. \tag{10} \]
It is clear that equation (10) has the formal solution (9). Let us form the characteristic equation of equation (10):
\[ s(s-1)+(2-\alpha-M_1)s=0. \]
It has roots \(s=0\) and \(s=\lambda-1\). To the root \(\lambda-1\) there corresponds the formal solution (9) of equation (10). By Horn’s theorem [2], equation (10) has a solution, holomorphic in \(x\) and \(-x^{\lambda-1}\ln x\), depending on one arbitrary constant \(C_1\). The expansion of this solution in a series in powers of \(x\) and \(-x^{\lambda-1}\ln x\) will coincide with the series obtained from (9), if we expand it in powers of \(x\) and \(-x^{\lambda-1}\ln x\), replacing \(-Mx^{\lambda-1}\ln Cx\) by \(-Mx^{\lambda-1}\ln x+C_1x^{\lambda-1}\) \((C_1=-M\ln C)\). Hence the convergence of the series (9), and therefore of the series (6), follows for sufficiently small \(|x|\) and \(|-ax^{\lambda-1}\ln Cx|\).
Remark. In determining the coefficients of the series (6) we assumed \(p_{\lambda-1,0}=0\). If we had set \(p_{\lambda-1,0}\) equal to an arbitrary number \(C_2\), this would be equivalent to changing the constant \(C\) to \(C_3\) by means of
\[ -a\ln C+C_2=-a\ln C_3. \]
From all the preceding we obtain the theorem.
Theorem. If in equation (2) \(\lambda-1\) is equal to a positive integer, then this equation has a solution holomorphic in powers of \(x\) and \(-ax^{\lambda-1}\ln x+Cx^{\lambda-1}\),
\[ u=\sum_{i,j}p_{ij}x^i\left(-ax^{\lambda-1}\ln x+Cx^{\lambda-1}\right)^j, \tag{11} \]
where the constant \(a\) depends on a finite number of coefficients in the Maclaurin expansions of the functions \(f(x)\) and \(K(x,t,u)\). If \(a=0\), then the solution (11) is holomorphic in \(x\) and \(Cx^{\lambda-1}\).
References
- Sato T. Journal of the Mathematical Society of Japan, 5, no. 2, 1953.
- Horn J. Journal für Mathematik, bd. CXVI, heft 4, 1896, 305–306.
- Horn J. Über eine nicht lineare volterrasche Integralgleichung, Jahresbericht d. D. Math.-Ver., 23, 1914.
Received by the editors
October 5, 1964
Institute of Mathematics
Academy of Sciences of the BSSR