Full Text
INVESTIGATION OF THE SOLUTIONS OF A CERTAIN CLASS OF INTEGRO-DIFFERENTIAL EQUATIONS
E. I. GRUDO
The integral equation
\[ xu(x)=g(x)+\lambda\int_0^x u(t)\,dt+\int_0^x K_1(x,t,u(t))\,dt, \tag{1} \]
where \(g(x)\), \(K_1(x,t,u)\) are functions holomorphic in a neighborhood of \(x=t=u=0\),
\[ g(0)=g'(0)=0,\qquad K_1(x,t,0)\equiv 0,\qquad \frac{\partial K_1(0,0,0)}{\partial u}=0, \]
and \(\lambda\) is a constant number, was considered under various assumptions on \(\lambda\) in [1–3]. However, the case \(\lambda=1\) in [1, 2] was not considered completely. It was shown only that equation (1) in this case has a holomorphic solution \(u=\varphi(x)\) in a neighborhood of \(x=0\), \(\varphi(0)=0\). Whether equation (1), in the case \(\lambda=1\), has other solutions tending to zero as \(x\to 0\) remained an open question. In the present note we shall deal with this question.
Making in equation (1), for \(\lambda=1\), the substitution
\[ u=\varphi(x)+v, \]
we obtain
\[ xv(x)=\int_0^x v(t)\,dt+\int_0^x K(x,t,v(t))\,dt, \tag{2} \]
where \(K(x,t,v)\) is a function holomorphic in a neighborhood of \(x=t=v=0\),
\[ K(x,t,0)\equiv 0,\qquad \frac{\partial K(0,0,0)}{\partial u}=0. \]
Differentiating equation (2) with respect to \(x\), we obtain
\[ x\frac{dv}{dx}=K(x,x,v)+\int_0^x \frac{\partial K(x,t,v(t))}{\partial x}\,dt. \tag{3} \]
1. Instead of equation (3), let us consider the more general integro-differential equation
\[ x\frac{dv}{dx}=M(x,v)+\int_0^x N(x,t,v(t))\,dt, \tag{4} \]
where the path of integration does not pass through the point \(t=0\), and \(M(x,v)\), \(N(x,t,v)\) are functions holomorphic in a neighborhood of \(x=t=v=0\),
\[ M(x,0)\equiv N(x,t,0)\equiv 0, \]
\(M(0,v)\not\equiv 0,\ \dfrac{\partial M(0,0)}{\partial v}=0.\) Obviously, the expansions of the functions \(M(x,v)\), \(N(x,t,v)\) in series can be written as
\[ M(x,v)=\sum_{j=0}^{\infty} m_j(v)x^j,\quad m_0(v)=\sum_{j=r}^{\infty} m_{0j}v^j,\quad m_{0r}\ne 0,\quad r\ge 2,\quad m_j(0)=0; \]
\[ N(x,t,v)=\sum_{j+k=0}^{\infty} n_{jk}(v)x^j t^k,\quad n_{jk}(0)=0. \]
For what follows we shall need the following two lemmas.
Lemma 1. The equation
\[ \frac{b_0+b(\psi)}{\psi^n}+a\ln\psi = \frac{b_0+\gamma(z)}{z^n}+a\ln z, \tag{5} \]
where \(b(\psi)\) and \(\gamma(z)\) are functions holomorphic, respectively, in neighborhoods of \(\psi=0\) and \(z=0\); \(b_0,a\) are constants; \(n\) is a positive integer, determines \(n\) holomorphic functions \(\psi_j=\psi_j(z)\), \(\psi_j(0)=0\), \(\psi'_j(0)=k_j\) \((j=1,2,\ldots,n)\); \(k_j\) is an \(n\)-th root of \(1\).
First let \(a=0\). Then equation (5) can be written in the form
\[ \psi=z\left(\frac{b_0+b(\psi)}{b_0+\gamma(z)}\right)^{\frac{1}{n}}. \]
From this equation the assertion of the lemma in the case \(a=0\) is obvious.
Now let \(a\ne 0\). Then we have
\[ a\ln\frac{\psi}{z} = \frac{b_0+\gamma(z)}{z^n} - \frac{b_0+b(\psi)}{\psi^n}. \]
Making in the last equation the substitution
\[ \psi=\bar{k}z+z\bar{\psi}_1, \]
we obtain
\[ az^n(\bar{k}+\bar{\psi}_1)^n\ln(\bar{k}+\bar{\psi}_1) = [b_0+\gamma(z)](\bar{k}+\bar{\psi}_1)^n-b_0-b(\bar{k}z+z\bar{\psi}_1). \]
From the last equation it is immediately seen that, if \(\bar{k}\) is equal to an \(n\)-th root of \(1\), then this equation admits a holomorphic solution \(\bar{\psi}_1(z)\), with \(\bar{\psi}_1(0)=0\). Hence the assertion of the lemma follows in the case \(a\ne 0\).
It is clear from the proof of the lemma that equation (5) cannot determine more than \(n\) functions tending to zero as \(z\to 0\).
Lemma 2. Let the function \(f(\bar{\tau}(x,C))\) be asymptotically expandable in a power series in \(\bar{\tau}(x,C)\) in a sufficiently small neighborhood of \(x=0\), holomorphic for \(|x|\ne 0\) sufficiently small and tending to zero as \(x\to 0\), where \(\bar{\tau}(x,C)\) is the general solution of the equation
\[ x\frac{d\bar{\tau}}{dx}=\bar{\tau}^q h(\bar{\tau}), \tag{6} \]
where \(q\ge 2\) is a positive integer, \(h(\bar{\tau})\) is a function holomorphic in a neighborhood of \(\bar{\tau}=0\), \(h(0)\ne 0\). Then
\[ \int_{0}^{x} t^k f(\bar{\tau}(t,C))\,dt = x^{k+1}F(\bar{\tau}(x,C)), \tag{7} \]
where the path of integration does not pass through the point \(t=0\) and \(k\) is a nonnegative integer, while the function \(F(\bar{\tau}(x,C))\) has the same properties as the function \(f(\bar{\tau}(x,C))\).
Indeed, differentiating (7) with respect to \(x\), taking (6) into account, we have
\[ \bar{\tau}^{q}h(\bar{\tau})\frac{dF}{d\tau}=-(k+1)F+f(\bar{\tau}). \]
From the last equation the assertion of the lemma is obvious.
We shall now seek a solution of the integro-differential equation (4) in the form of the series
\[ v=\sum_{j=0}^{\infty}\varphi_j(\tau(x,C))x^j, \tag{8} \]
where \(\tau(x,C)\) is the general solution of the equation
\[ x\frac{d\tau}{dx}=\alpha(\tau), \tag{9} \]
and the function \(\alpha(\tau)\) will be determined shortly.
To determine \(\varphi_0(\tau(x,C))\) and \(\alpha(\tau)\) we obtain the equation
\[ \alpha(\tau)\frac{d\varphi_0}{d\tau}=m_0(\varphi_0). \tag{10} \]
We define \(\alpha(\tau)\) so that equation (10) has a holomorphic solution in a neighborhood of \(\tau=0\) that vanishes for \(\tau=0\). Let the series
\[ \sum_{j=0}^{\infty}\alpha_j\varphi_0^{-r+j},\qquad \alpha_0\ne0 \]
be the Laurent series for
\[ \frac{1}{m_0(\varphi_0)} \]
in a neighborhood of \(\varphi_0=0\). On the basis of Lemma 1, in order that equation (10) have a holomorphic solution in a neighborhood of \(\tau=0\), it is sufficient to take \(\alpha(\tau)\) in the form
\[ \alpha(\tau)=\frac{\tau^r}{\alpha_0+\alpha_{r-1}\tau^{r-1}}. \]
Clearly, equations (9) and (10) do not change if we replace \(\tau\) by \(k_j\tau\), where \(k_j\) is an \((r-1)\)-st root of unity; and since in the series (8) we take as \(\tau(x,C)\) the general solution of equation (9), in determining \(\varphi_0(\tau(x,C))\) on the basis of Lemma 1 we may assume that \(k_j=1\). Thus, the function \(\varphi_0(\tau(x,C))\) in the series (8) is determined uniquely; it is holomorphic in a neighborhood of \(\tau=0\), \(\varphi_0(0)=0\), \(\varphi_0'(0)=1\).
For \(\tau(x,C)\) we have the equation
\[ x\frac{d\tau}{dx}=\frac{\tau^r}{\alpha_0+\alpha_{r-1}\tau^{r-1}}. \tag{11} \]
If \(\alpha_{r-1}=0\), then the general solution of equation (11) is written as
\[ \tau(x,C)=\left[\frac{\alpha_0}{(1-r)(\ln x+C)}\right]^{\frac{1}{r-1}}. \tag{12} \]
Suppose now that \(\alpha_{r-1} \ne 0\). We shall show that in this case \(\tau(x,C)\), for sufficiently small \(|x|\), is a holomorphic function of \(\tau_1(x,C)\) and \(\tau_1^{r-1}(x,C)\ln \tau_1(x,C)\), where \(\tau_1(x,C)\) is equal to the right-hand side of (12).
Indeed, between \(\tau(x,C)\) and \(\tau_1(x,C)\) there is the relation
\[ \frac{\alpha_0}{(1-r)\tau^{r-1}}+\alpha_{r-1}\ln \tau = \frac{\alpha_0}{(1-r)\tau_1^{r-1}} . \]
Making in the last equation the substitution
\[ \tau=\tau_1+\tau_1\xi, \]
to determine \(\xi\) as a function of \(\tau_1\), we obtain the equation
\[ \xi=\Phi(\xi)-\frac{\alpha_{r-1}}{\alpha_0}\tau_1^{r-1}\ln \tau_1 -\frac{\alpha_{r-1}}{\alpha_0}\tau_1^{r-1}\ln(1+\xi), \]
where \(\Phi(\xi)\) is a holomorphic function in a neighborhood of \(\xi=0\), \(\Phi(\xi)=0\), \(\Phi'(0)=0\). The last equation has the solution
\[ \xi=\xi\left(\tau_1^{r-1},\ \tau_1^{r-1}\ln \tau_1\right), \]
where \(\xi\left(\tau_1^{r-1},\tau_1^{r-1}\ln \tau_1\right)\) is a holomorphic function of its arguments in a neighborhood of \((0,0)\).
Thus, for sufficiently small \(|x|\),
\[ \tau(x,C)=\tau_1(x,C)+\tau_1(x,C)\xi\left(\tau_1^{r-1}(x,C),\ \tau_1^{r-1}(x,C)\ln \tau_1(x,C)\right). \]
Let us now return to the construction of the series (8). To determine \(\varphi_1(\tau(x,C))\), we have the equation
\[ x\alpha(\tau)\frac{d\varphi_1}{d\tau} + x\varphi_1 = xm_0'(\varphi_0)\varphi_1 + \int_0^x n_{00}(\varphi_0)\,dt . \]
Applying Lemma 2, we can rewrite this equation in the form
\[ \alpha(\tau)\frac{d\varphi_1}{d\tau} + \varphi_1 = m_0'(\varphi_0)\varphi_1+n_1(\tau), \]
where \(n_1(\tau(x,C))\) is asymptotically expandable in a power series in \(\tau(x,C)\) in a sufficiently small neighborhood of \(x=0\), is holomorphic for sufficiently small \(|x|\ne 0\), and tends to zero as \(x\to 0\). From the last equation it follows that the function \(\varphi_1(\tau(x,C))\) in the series (8) is determined as a function having the same properties as the function \(n_1(\tau(x,C))\), and moreover it is determined uniquely.
Suppose now that we have uniquely determined the functions \(\varphi_1(\tau(x,C))\), \(\varphi_2(\tau(x,C))\), \(\ldots\), \(\varphi_j(\tau(x,C))\) as functions asymptotically expandable in power series in \(\tau(x,C)\) in a sufficiently small neighborhood of \(x=0\), holomorphic for sufficiently small \(|x|\ne 0\), and tending to zero as \(x\to 0\). We shall show that then the function \(\varphi_{j+1}(\tau(x,C))\) in the series (8) is also determined uniquely and has the same properties as the functions \(\varphi_1(\tau(x,C))\), \(\varphi_2(\tau(x,C))\), \(\ldots\), \(\varphi_j(\tau(x,C))\). Indeed, for \(\varphi_{j+1}(\tau(x,C))\), applying Lemma 2, we obtain the equation
\[ \alpha(\tau)\frac{d\varphi_{j+1}}{d\tau} + (j+1)\varphi_{j+1} = m_0'(\varphi_0)\varphi_{j+1}+n_{j+1}(\tau), \]
where \(\eta_{j+1}(\tau(x,C))\) has the same properties as the function \(\eta_1(\tau(x,C))\). From the last equation our assertion is obvious.
Thus, we have shown that there exists a series (8) formally satisfying equation (4); the functions \(\varphi_j(\tau(x,C))\) are determined uniquely; \(\varphi_0(\tau(x,C))\) is holomorphic in a neighborhood of \(\tau=0\); \(\varphi_j(\tau(x,C))\) \((j>0)\) are asymptotically expandable into power series in \(\tau(x,C)\) in a sufficiently small neighborhood of \(x=0\), are holomorphic for sufficiently small \(|x|\ne 0\), and tend to zero as \(x\to 0\).
We now prove the convergence of the series (8). To this end consider the equation
\[ x_1\frac{dv}{dx_1}=M(\beta x_1,v)+\int_0^{\beta x_1}N(\beta x_1,t,v(t))\,dt, \]
obtained from equation (4) by means of the substitution \(x=\beta x_1\), where \(\beta\) will be regarded as a parameter. The last equation becomes
\[ x_1\frac{dv}{dx_1}=M(\beta x_1,v)+\beta\int_0^{x_1}N(\beta x_1,\beta t_1,v(t_1))\,dt_1, \tag{13} \]
if we make the substitution \(t=\beta t_1\). Obviously, equation (13) has the formal solution
\[ v=\sum_{j=0}^{\infty}\varphi_j(\tau_2(x_1,C_1))\beta^j x_1^j, \tag{14} \]
where \(\tau_2(x_1,C_1)\) is the general solution of the equation
\[ x_1\frac{d\tau_2}{dx_1}=\frac{\tau_2^r}{\alpha_0+\alpha_{r-1}\tau_2^{r-1}}, \]
and the functions \(\varphi_j(\tau_2(x_1,C_1))\) are obtained from the functions \(\varphi_j(\tau(x,C))\) in (8) by replacing \(\tau(x,C)\) in the latter by \(\tau_2(x_1,C_1)\). Therefore the functions \(\varphi_j(\tau_2(x_1,C_1))\) in (14) have the same properties as the functions \(\varphi_j(\tau(x,C))\) in (8).
Make in (13) the substitution
\[ v=\varphi_0(\tau_2(x_1,C_1))+\varphi_1(\tau_2(x_1,C_1))\beta x_1+\beta x_1w. \]
For \(w\) we obtain the equation
\[ x_1\frac{dw}{dx_1}=\sum_{j=0}^{\infty}\mu_j(x_1,\beta)w^j+\frac{1}{x_1}\int_0^{x_1}\sum_{j=0}^{\infty}\nu_j(x_1,t_1,\beta)w^j(t_1)\,dt_1, \tag{15} \]
where the path of integration does not pass through the point \(t_1=0\), and the functions \(\mu_j(x_1,\beta)\), \(\nu_j(x_1,t_1,\beta)\) \((j=0,1,2,\ldots)\) are representable in the following form:
\[ \mu_j(x_1,\beta)=\sum_{n=1}^{\infty}\mu_{jn}(x_1)\beta^n \qquad (j=0,2,3,\ldots), \]
\[ \mu_1(x_1,\beta)=-1+m_0'(\varphi_0)+\sum_{n=1}^{\infty}\mu_{1n}(x_1)\beta^n, \tag{16} \]
\[ \nu_j(x_1,t_1,\beta)=\sum_{n=1}^{\infty}\nu_{jn}(x_1,t_1)\beta^n \qquad (j=0,1,2,\ldots). \]
where all the functions \(\mu_{jn}(x_1)\) and \(\nu_{jn}(x_1,t_1)\) tend to zero, respectively, as \(x_1\to 0\) and as \(x_1\to 0,\ t_1\to 0\). The series (16) converge absolutely and uniformly for sufficiently small \(|x_1|,\ |t_1|,\ |\tau_2(x_1,C_1)|\), and \(|\beta|\). The series on the right-hand side of (15) converge absolutely and uniformly for sufficiently small \(|x_1|,\ |\tau_2(x_1,C_1)|,\ |t_1|,\ |\beta|\), and \(|w|\), and we shall assume that \(|\beta|\leqslant 1\).
Equation (15) has the formal solution
\[ w=\sum_{j=1}^{\infty}\overline{\varphi}_j(x_1)\beta^j,\qquad \overline{\varphi}_j(x_1)=\varphi_{j+1}(\tau_2(x_1,C_1))x_1^j . \tag{17} \]
Obviously, \(\overline{\varphi}_j(x_1)\) tends to zero as \(x_1\to 0\).
Substituting the series (17) into equation (15) and comparing the coefficients of equal powers of \(\beta\) on the right and on the left, we obtain that the functions \(\overline{\varphi}_j(x_1)\) satisfy the equations
\[ x_1\frac{d\overline{\varphi}_j}{dx_1} = [-1+m_0'(\varphi_0)]\overline{\varphi}_j + \Phi_j(\overline{\varphi}_1,\ldots,\overline{\varphi}_{j-1},x_1) + \frac{1}{x_1}\int_0^{x_1} \Psi_j(\overline{\varphi}_1(t_1),\ldots,\overline{\varphi}_{j-1}(t_1),x_1,t_1)\,dt_1 \quad (j=1,2,\ldots), \]
where \(\Phi_j\) and \(\Psi_j\) are the coefficients of \(\beta^j\) in the series obtained from the series
\[ \mu_0(x_1,\beta)+\sum_{j=1}^{\infty}\mu_{1j}(x_1)\beta^j w + \sum_{j=2}^{\infty}\mu_j(x_1,\beta)w^j, \tag{18} \]
and
\[ \sum_{j=0}^{\infty}\nu_j(x_1,t_1,\beta)w^j(t_1) \]
respectively, after substituting in them, instead of \(w\), the series (17).
Then it is clear that \(\overline{\varphi}_j(x_1)\) can be represented in the form
\[ \begin{aligned} \overline{\varphi}_j(x_1) &= \frac{1}{x_1} e^{\int_{x_{10}}^{x_1}\frac{m_0'(\varphi_0)}{x_1}\,dx_1} \int_0^{x_1} \Bigl[ \Phi_j(\overline{\varphi}_1,\ldots,\overline{\varphi}_{j-1},x_1) \\ &\qquad\qquad\qquad +\frac{1}{x_1}\int_0^{x_1} \Psi_j(\overline{\varphi}_1(t_1),\ldots,\overline{\varphi}_{j-1}(t_1),x_1,t_1)\,dt_1 \Bigr] e^{-\int_{x_{10}}^{x_1}\frac{m_0'(\varphi_0)}{x_1}\,dx_1} \,dx_1 \quad (j=1,2,\ldots), \end{aligned} \tag{19} \]
where the path of integration in the outer integral from \(0\) to \(x_1\) is taken along the straight line joining the points \(0\) and \(x_1\), and the point \(x_{10}\) is chosen so that \(\tau_2(x_{10},C_1)\) lies in the disk of holomorphy of \(m_0'(\varphi_0)\) with respect to \(\tau_2\).
It is easy to see that
\[ \int \frac{m_0'(\varphi_0)}{x_1}\,dx_1 = \int \frac{dm_0(\varphi_0)}{d\varphi_0}\frac{d\tau_2}{\alpha(\tau_2)} = \int \frac{dm_0(\varphi_0)}{d\varphi_0}\frac{d\varphi_0}{m_0(\varphi_0)} = \]
\[ =\ln m_0(\varphi_0(\tau_2(x_1,C_1))). \tag{20} \]
Then
\[ \overline{\varphi}_j(x_1) = \frac{1}{x_1} m_0(\varphi_0(\tau_2(x_1,C_1))) \int_0^{x_1} \Bigl[ \Phi_i(\overline{\varphi}_1,\ldots,\overline{\varphi}_{j-1},x_1) + \]
\[ +\frac{1}{x_1}\int_0^{x_1}\Psi_j\bigl(\overline{\varphi}_1(t_1),\ldots,\overline{\varphi}_{j-1}(t_1),x_1,t_1\bigr)\,dt_1 \frac{dx_1}{m_0\bigl(\varphi_0(\tau_2(x_1,C_1))\bigr)} \quad (j=1,2,\ldots). \]
Let the series
\[ \sum_{j=0}^{\infty} Q_j(\beta) W^j, \tag{21} \]
where \(Q_j(\beta)\) are functions holomorphic in a neighborhood of \(\beta=0\), \(Q_j(0)=0\), be a majorant of the series (18) in the domain of absolute and uniform convergence of the latter, and let the series
\[ \sum_{j=0}^{\infty} P_j(\beta) W^j, \tag{22} \]
where \(P_j(\beta)\) are functions holomorphic in a neighborhood of \(\beta=0\), \(P_j(0)=0\), be a majorant of the series \(\sum_{j=0}^{\infty} \nu_j(x_1,t_1,\beta) w^j\) in the domain of its absolute and uniform convergence.
Denote the domain of absolute and uniform convergence of the series (18) with respect to \(x_1\) by \(R_1\), and that of the series \(\sum_{j=0}^{\infty}\nu_j(x_1,t_1,\beta)w^j\) by \(R_2\).
Further, as follows from the definition of the function \(\xi=\xi(\tau_1^{r-1},\tau_1^{r-1}\ln\tau_1)\), this function does not vanish in the domain of its holomorphy with respect to the arguments at \(-1\). From (10) (replacing \(\tau\) by \(\tau_2\)) and (20) it is easy to see that neither \(\varphi_0(\tau_2)\) nor \(m_0(\varphi_0(\tau_2))\), in the domain of their holomorphy with respect to \(\tau_2\), vanish except at the point \(\tau_2=0\). Therefore, in the intersection \(R_3\) of the domains of absolute and uniform convergence with respect to \(x_1\) of the functions \(m_0(\varphi_0(\tau_2))\) and \(1+\xi(\tau_3^{r-1},\tau_3^{r-1}\ln\tau_3)\), where \(\tau_3(x_1,C_1)\) is obtained from formula (12) by replacing \(x\) by \(x_1\) and \(C\) by \(C_1\), we shall have:
\[ \left| \frac{m_0\bigl(\varphi_0(\tau_2(x_1,C_1))\bigr)}{\tau_2^r(x_1,C_1)} \bigl[1+\xi(\tau_3^{r-1}(x_1,C_1),\tau_3^{r-1}(x_1,C_1)\ln\tau_3(x_1,C_1))\bigr]^r \right|\leq M_1, \]
\[ \left| \frac{\tau_2^r(x_1,C_1)}{m_0\bigl(\varphi_0(\tau_2(x_1,C_1))\bigr)} \bigl[1+\xi(\tau_3^{r-1}(x_1,C_1),\tau_3^{r-1}(x_1,C_1)\ln\tau_3(x_1,C_1))\bigr]^{-r} \right|\leq M_2, \]
where \(M_1\) and \(M_2\) are positive numbers.
Suppose that in the intersection of the domains \(R_1\), \(R_2\), and \(R_3\), which we denote by \(R_4\), the inequalities
\[ |\overline{\varphi}_n(x_1)|\leq \overline{\Phi}_n \quad (n=1,2,\ldots,j-1), \tag{23} \]
hold, where \(\overline{\Phi}_n\) are certain positive constant numbers. We shall show that then in \(R_4\) the inequality
\[ |\overline{\varphi}_j(x_1)|\leq \overline{\Phi}_j, \]
holds, where \(\overline{\Phi}_j\) is a certain positive constant.
Indeed, putting
\[ x_1=\rho_1 e^{i\theta}, \]
we have, in \(R_4\),
\[ |\bar{\varphi}_j(x_1)|\leq \frac{M_1M_2}{\rho_1} \left|\frac{1}{\ln\rho_1+i\theta+C_1}\right|^{\frac{r}{r-1}} \int_0^{\rho_1} \left[ |\bar{Q}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})| + \right. \]
\[ \left. + |\bar{P}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})| \right] |\ln\rho_1+i\theta+C_1|^{\frac{r}{r-1}}\,d\rho_1 . \tag{24} \]
In deriving the last inequality we used the fact that, in formulas (19), the path of integration in the inner integrals does not go around the point \(t_1=0\). In this case, on the basis of the properties of the functions \(\Psi_j\), it is easy to see that the integrals along the straight line from \(0\) to \(x_1\) of the functions \(\Psi_j\) are equal to the integrals of the corresponding functions \(\Psi_j\) along any curve not going around the point \(t_1=0\) and joining \(0\) and \(x_1\). Therefore, in deriving the last inequality, in the inner integrals in formulas (19) we took the path of integration along the straight line from \(0\) to \(x_1\).
In (24), \(\bar{Q}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})\) and \(\bar{P}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})\) are the coefficients of \(\beta^j\) in the series obtained respectively from the series (21) and (22) by substituting into them, in place of \(W\), the series
\[ W=\sum_{j=1}^{\infty}\bar{\Phi}_j\beta^j . \tag{25} \]
Consider the expression
\[ S(\rho_1,\theta)= \frac{1}{\rho_1} \left|\frac{1}{\ln\rho_1+i\theta+C_1}\right|^{\frac{r}{r-1}} \int_0^{\rho_1} |\ln\rho_1+i\theta+C_1|^{\frac{r}{r-1}}\,d\rho_1 . \]
It is easy to show that for \(0<\rho_1<|e^{-C_1}|\), \(-\infty<\theta<+\infty\),
\[ S(\rho_1,\theta)\leq 1- \frac{2(\ln\rho_1+\bar{C}_1)-2} {(\ln\rho_1+\bar{C}_1)^2+(\theta+\bar{C}_2)^2}, \]
where \(\bar{C}_1=\operatorname{Re}(C_1)\), \(\bar{C}_2=\operatorname{Im}(C_1)\).
Obviously, in the domain \(R_4\), \(|x_1|\leq r_1<e^{-\bar{C}_1}\). Further, it is clear that there exists a positive number \(S_1>1\) such that for \(0\leq\rho_1\leq r_1\), \(|\theta|\leq\infty\),
\[ S(\rho_1,\theta)<S_1 . \]
Therefore, by formula (24) we have
\[ |\bar{\varphi}_j(x_1)| \leq M_1M_2S_1 \left[ |\bar{Q}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})| + |\bar{P}_j(\bar{\Phi}_1,\ldots,\bar{\Phi}_{j-1})| \right] =\bar{\Phi}_j \]
in the domain \(R_4\).
Since \(|\bar{\varphi}_1(x_1)|\) in the domain \(R_4\) does not exceed a certain positive number \(\bar{\Phi}_1\), it follows that the inequalities (23) hold for all \(n=1,2,\ldots\) in the domain \(R_4\). Obviously, the constants \(\bar{\Phi}_j\) coincide with the coefficients of the expansion in a series in a neighborhood of \(\beta=0\) of the holomorphic solution \(W(\beta)\) of the equation
\[ W=M_1M_2S_1\sum_{j=0}^{\infty}[Q_j(\beta)+P_j(\beta)]W^j . \]
Thus, the series (25) converges in a neighborhood of \(\beta=0\), and since we have the inequalities (23) for all \(n=1,2,\ldots\) in the domain \(R_4\), the series (17) converges absolutely and uniformly for \(x_1\in R_4\), \(|\beta|\) sufficiently small, and consequently—
Consequently, the series (14) also converges for the same \(x_1\) and \(\beta\). Fixing now \(\beta\) in the circle of convergence of the series (25) and then passing in the series (14) and in equation (13) to the variable \(x\) by the formula \(x=\beta x_1\), we obtain that the series (8) converges absolutely and uniformly for \(|x|\), \(|\tau(x,C)|\) sufficiently small.
Thus, we have obtained
Theorem. The integro-differential equation (4) has a family of solutions depending on one parameter and representable in the form of the series
\[ v=\sum_{j=0}^{\infty}\varphi_j(\tau(x,C))x^j, \]
which converges absolutely and uniformly for \(|x|\), \(|\tau(x,C)|\) sufficiently small, where \(\tau(x,C)\) is the general solution of equation (11)
\[ x\frac{d\tau}{dx}=\frac{\tau^r}{\alpha_0+\alpha_{r-1}\tau^{r-1}}. \]
The function \(\varphi_0(\tau)\) is holomorphic in a neighborhood of \(\tau=0\), \(\varphi_0(0)=0\), \(\varphi_0'(0)=1\); the functions \(\varphi_j(\tau(x,C))\) \((j=1,2,\ldots)\) are asymptotically expandable in power series in powers of \(\tau(x,C)\) in a sufficiently small neighborhood of \(x=0\), tend to zero as \(x\to0\), and are holomorphic for sufficiently small \(|x|\ne0\).
- The preceding theorem gives a representation of the general solution of equation (4) in a neighborhood of \(x=v=0\) in the case \(M(0,v)\not\equiv0\). However, in the case \(M(0,v)\equiv0\) this theorem is not applicable. We shall show that in this case equation (4) has no other solutions tending to zero as \(x\to0\), except \(v\equiv0\). To this end, we first give a representation of the general solution of equation (4) in the case under consideration in a neighborhood of \(x=v=0\).
We shall seek this general solution in the form
\[ v=C+\sum_{j=1}^{\infty}\delta_j x^j, \tag{26} \]
where \(C\) is an arbitrary constant. To determine the coefficients \(\delta_j\) we shall have the equations
\[ j\delta_j=L_j(C,\delta_1,\ldots,\delta_{j-1})+G_j(C,\delta_1,\ldots,\delta_{j-1}) \quad (j=1,2,\ldots), \]
where \(L_j\) and \(G_j\) are the coefficients of the powers \(x^j\) in the series obtained from the series \(M(x,v)\) and
\[ \int_0^x N(x,t,v(t))\,dt \]
when the series (26) is substituted for \(v\). The last relations make it possible to find successively and uniquely \(\delta_j\) in terms of \(C\) and the coefficients of the series \(M(x,v)\) and \(N(x,t,v)\). We now prove the convergence of the series (26) for \(|C|\) and \(|x|\) sufficiently small.
Let \(xB(x,V)\) and \(D(x,t,V)\) be majorants, respectively, of the series \(M(x,v)\) and \(N(x,t,v)\) in the domain of their holomorphy in a neighborhood of \(x=t=v=0\). Consider the equation
\[ \frac{dV}{dx}=B(x,V)+D(x,x,V). \]
This equation has a solution holomorphic in a neighborhood of \(x=0\),
\[ V=|C|+\sum_{j=1}^{\infty}\Delta_j x^j, \]
if \(|C|\) is sufficiently small.
Obviously,
\[ |\delta_j| \leq \Delta_j \quad (j=1, 2, \ldots). \]
Consequently, the series (26) converges for sufficiently small \(|x|\) and \(|C|\). Hence it easily follows that there are no other solutions of equation (4), in the case \(M(0, v) \equiv 0\), tending to zero as \(x \to 0\), except for \(v \equiv 0\).
From the preceding it follows
Theorem. The general solution of equation (4) in the case \(M(0, v) \equiv 0\) in a neighborhood of \(x = v = 0\) is representable in the form of the series
\[ v = C + \sum_{j=1}^{\infty} \delta_j(C)x^j, \]
which converges absolutely and uniformly for sufficiently small \(|x|\) and \(|C|\). Here \(C\) is an arbitrary constant, and \(\delta_j(C)\) are holomorphic functions in a neighborhood of \(C = 0\).
References
- Horn J. Über eine nichtlineare Volterrasche Integralgleichung. Jahresbericht DMV, 23, 1914.
- Sato T. Journ. Math. Soc. Japan, 5, no. 2, 1963.
- Grudo É. I. On one case of the Volterra integral equation. Differential Equations, 1, no. 2, 1965.
Received by the editors
September 5, 1964
Institute of Mathematics, Academy of Sciences of the BSSR