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On a Certain Class of Integro-Differential Equations
E. I. Grudo
In [1] an integro-differential equation of the form
\[ x\frac{du}{dx}=\lambda u(x)+f_1(x,u(x))+\int_0^x g_1(x,t,u(t))\,dt \tag{1} \]
\[ (\lambda=\mathrm{const}), \]
was considered, where \(f_1(x,u)\), \(g_1(x,t,u)\) are functions holomorphic in a neighborhood of \(x=t=u=0\), \(f_1(0,0)=\dfrac{\partial f_1(0,0)}{\partial u}=0\), and it was shown that if \(\lambda\) is not equal to a positive integer, then equation (1) has a unique solution holomorphic in a neighborhood of \(x=0\). If \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) is not equal to a positive integer, then equation (1) has a solution holomorphic in powers of \(x\) and \(Cx^\lambda\) (\(C\) is an arbitrary constant).
In the present note we shall consider an integro-differential equation of the form
\[ x\frac{du}{dx}=\lambda u(x)+f\left(x,u(x),\int_0^x g(x,t,u(t))\,dt\right) \]
\[ (\lambda=\mathrm{const}), \tag{2} \]
where \(f(x,u,z)\) and \(g(x,t,u)\) are functions holomorphic in a neighborhood of \(x=t=u=z=0\), \(f(0,0,0)=\dfrac{\partial f(0,0,0)}{\partial u}=0\). Without loss of generality we shall assume that \(g(x,t,0)\equiv 0\). Let
\[ f(x,u,z)=\sum_{i,j,k=0}^{\infty} f_{ijk}x^i u^j z^k,\quad f_{010}=f_{000}=0, \]
\[ g(x,t,u)=\sum_{i,j,k=0}^{\infty} g_{ijk}x^i t^j u^{k+1}. \]
We shall first consider equation (2) in the cases \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) is not equal to a positive integer, and \(\operatorname{Re}(\lambda)=0\), but \(\operatorname{Im}(\lambda)\ne 0\). The path of integration in these cases in (2) is chosen so that \(x^\lambda\to 0\) as \(x\to 0\) along this path, and so that it contains no closed curves inside which the point \(x=0\) lies.
We shall seek a formal solution of equation (2) in the cases under consideration in the form of the series
\[ u(x)=\sum_{i+j=1}^{\infty} a_{ij}x^{i+j\lambda}\quad (i\geqslant 0,\ j\geqslant 0). \tag{3} \]
To determine the coefficients \(a_{ij}\) we have the relations
\[ (i+j\lambda)a_{ij}=\lambda a_{ij}+\widetilde P_{ij}(a_{mn}), \tag{4} \]
where \(\widetilde P_{ij}(a_{mn})\) are polynomials in the \(a_{mn}\) for which \(m+n<i+j\), \(m\leqslant i\), \(n\leqslant j\); the coefficients of these polynomials are polynomials in a finite number (for finite \(i\) and \(j\)) of the coefficients \(f_{ijk}\) and \(g_{ijk}\). These relations make it possible to determine all \(a_{ij}\) \((i+j\geqslant 1)\), since in the cases under consideration \(i+(j-1)\lambda\ne 0\) for all \(i\geqslant 0\) and \(j\geqslant 0\), except for \(i=0\), \(j=1\), and the coefficient \(a_{01}\) remains arbitrary: \(a_{01}=C\).
Further, it is obvious that for \(j\ne 0\)
\[ a_{ij}=C^j b_{ij}, \]
where \(b_{ij}\) does not depend on \(C\), and \(b_{01}=1\). The coefficients \(a_{i0}\), however, do not depend on \(C\). Therefore the series (3) may be written in the form
\[ u(x)=\sum_{i+j=1}^{\infty} b_{ij}x^i(Cx^\lambda)^j, \tag{5} \]
where \(b_{i0}=a_{i0}\) \((i=1,2,\ldots)\).
We shall now prove, in the cases under consideration, the convergence of the series (5) when \(|x|\), \(|Cx^\lambda|\) are sufficiently small.
First of all, it is obvious that there exists a constant number \(B>0\) such that
\[ |i+(j-1)\lambda|\geqslant \frac{1}{B} \]
for all \(i\geqslant 0\), \(j\geqslant 0\), except for \(i=0\), \(j=1\), since by hypothesis \(\lambda\) is such that either \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) is not equal to a positive integer, or \(\operatorname{Re}(\lambda)\leqslant 0\), but \(\operatorname{Im}(\lambda)\ne 0\).
Let now \(G(x,t,u)\) be a majorant of the series \(g(x,t,u)\), and \(F(x,u,z)\) a majorant of the series \(f(x,u,z)\). We shall assume that
\[ F(x,u,z)=\sum_{i+j+k=1}^{\infty} F_{ijk}x^i u^j z^k,\qquad F_{010}=0, \]
\[ G(x,t,u)=\sum_{i+j+k=0}^{\infty} G_{ijk}x^i t^j u^{k+1}. \]
Consider the equation
\[ U=T+BF[x,U,xBG(x,x,U)], \tag{6} \]
where \(T\) is a parameter. This equation has, in a neighborhood of \(x=T=0\), a holomorphic solution
\[ U=\sum_{i+j=1}^{\infty} B_{ij}x^iT^j,\qquad B_{01}=1\quad (i\geqslant 0,\ j\geqslant 0) \tag{7} \]
with nonnegative coefficients \(B_{ij}\).
We shall show that
\[ |b_{ij}| \leqslant B_{ij} \tag{8} \]
for all \(i+j>1,\ i\geqslant 0,\ j\geqslant 0\).
Let inequality (8) hold for \(i+j<p\). We shall then verify that it also holds for \(i+j=p\). Substituting series (5) into
\[ \int_0^x g(x,t,u(t))\,dt, \]
we obtain the series
\[ \sum_{i+j=2}^{\infty} q_{ij}x^i(Cx^\lambda)^j, \]
where the \(q_{ij}\) are polynomials in the \(b_{i'j'}\), for which \(i'+j'<i+j,\ i'\leqslant i,\ j'\leqslant j\). Substituting the series
\[ U=\sum_{i+j=1}^{\infty} B_{ij}x^i(Cx^\lambda)^j \]
into the series \(xG(x,x,U)\), we obtain the series
\[ \sum_{i+j=2}^{\infty} Q_{ij}x^i(Cx^\lambda)^j, \]
where the \(Q_{ij}\) are nonnegative and are polynomials in the \(B_{i'j'}\), for which \(i'+j'<i+j,\ i'\leqslant i,\ j'\leqslant j\). Obviously,
\[ |q_{ij}| \leqslant BQ_{ij} \]
for \(i+j\leqslant p\). Further, obviously, we have
\[ b_{ij}=-\frac{1}{i+(j-1)\lambda}\,P_{ij}(b_{i'j'},q_{i''j''}), \]
where \(P_{ij}\) is a polynomial in the \(b_{i'j'}\) and \(q_{i''j''}\), for which
\(i'+j'<i+j,\ i'\leqslant i,\ j'\leqslant j,\ i''+j''\leqslant i+j,\ i''\leqslant i,\ j''\leqslant j\). It is also obvious that
\[ B_{ij}=B\overline{P}_{ij}(B_{i'j'},BQ_{i''j''}), \]
where \(\overline{P}_{ij}\) is a polynomial in the \(B_{i'j'}\) and \(BQ_{i''j''}\), for which
\(i'+j'<i+j,\ i'\leqslant i,\ j'\leqslant j,\ i''+j''\leqslant i+j,\ i''\leqslant i,\ j''\leqslant j\). The polynomial \(\overline{P}_{ij}\) is the coefficient of \(x^iT^j\) in the series obtained by substituting series (7) into the series
\(F[x,U,xBG(x,x,U)]\). It is easy to see that
\[ |P_{ij}| \leqslant \overline{P}_{ij} \]
for \(i+j=p\), and therefore
\[ |b_{ij}| \leqslant B_{ij} \]
for \(i+j=p\). Since
\[ b_{10}=\frac{f_{100}}{1-\lambda},\quad b_{01}=1,\quad B_{10}=BF_{100},\quad B_{01}=1, \]
it follows that
\[ |b_{10}| \leqslant B_{10},\quad b_{01}=B_{01}, \]
and hence (8) holds for all \(i+j\geqslant 1,\ i\geqslant 0,\ j\geqslant 0\). By virtue of (8) and the convergence of series (7), the convergence of series (5) follows for sufficiently small \(|x|,\ |Cx^\lambda|\). Thus, it holds that
Theorem 1. If \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) is not equal to a positive integer, or \(\operatorname{Re}(\lambda)\leq 0\), but \(\operatorname{Im}(\lambda)\ne 0\), then the integro-differential equation (2) has a solution (5), holomorphic in powers of \(x\) and \(Cx^\lambda\), for \(|x|\) and \(|Cx^\lambda|\) sufficiently small; moreover, the path of integration in equation (2) is chosen in the manner indicated above.
Let us now consider the case \(\operatorname{Re}(\lambda)\leq 0\), \(\operatorname{Im}(\lambda)=0\). In this case the path of integration is taken so that it contains no closed curves inside which the point \(x=0\) lies. Then equation (2) has the formal solution
\[ u(x)=\sum_{i=1}^{\infty} b_i x^i . \tag{9} \]
To determine the coefficients \(b_i\) we have the formulas
\[ b_i=\frac{1}{i-\lambda}\,P_i(b_m), \]
where \(P_i(b_m)\) is a polynomial in the \(b_m\), for which \(m<i\), and in a finite number of coefficients \(f_{ij}\) and \(g_{ijk}\).
There exists such a constant number \(B>0\) that, in our case,
\[ \frac{1}{i-\lambda}\leq B \]
for all \(i\geq 1\).
Let us then consider the equation
\[ U=BF[x,U,xBG(x,x,U)], \]
which has, in a neighborhood of \(x=0\), a holomorphic solution
\[ U=\sum_{i=1}^{\infty} B_i x^i \]
with nonnegative \(B_i\). It is easy to see that
\[ |b_i|\leq B_i\quad (i=1,2,\ldots), \]
whence follows the convergence of the series (9) in a neighborhood of \(x=0\). Therefore the following holds.
Theorem 2. The integro-differential equation (2), in the case \(\operatorname{Re}(\lambda)\leq 0\), \(\operatorname{Im}(\lambda)=0\), has a unique solution (9) holomorphic in a neighborhood of \(x=0\).
Let now \(\lambda\) be equal to a positive integer. Then, for \(i=\lambda,\ j=0\), the coefficient \(a_{\lambda 0}\) in the series (3) cannot be determined if \(\widetilde P_{\lambda 0}(a_{mn})\ne 0\). In this case equation (2) has no formal, and hence no holomorphic, solution of the form (5). If \(\widetilde P_{\lambda 0}(a_{mn})=0\), then in this case the coefficient \(a_{\lambda 0}\) may be taken arbitrarily, while all the other coefficients are determined uniquely. Hence, in the case \(\widetilde P_{\lambda 0}(a_{mn})=0\), equation (2) has a formal solution (5), and we may assume, without loss of generality, that \(b_{\lambda 0}=0\).
Let us now prove the convergence of this series
\[ u(x)=\sum_{i+j=1}^{\infty} b_{ij}x^i(Cx^\lambda)^j,\quad b_{\lambda 0}=0,\quad b_{01}=1\ (i\geq 0,\ j\geq 0). \tag{10} \]
In the case of positive integer \(\lambda\) there exists a constant number \(B>0\) such that
\[ |i+(j-1)\lambda|\geqslant \frac{1}{B} \]
for all \(i\geqslant 0,\ j\geqslant 0\), except for the cases \(i=0,\ j=1\) and \(i=\lambda,\ j=0\). Considering now equation (6) and its solution (7), it is easy to see that
\[ |b_{ij}| \leqslant B_{ij} \]
for all \(i+j\geqslant 1\). Therefore the convergence of series (10) for sufficiently small \(|x|\), \(|Cx^\lambda|\) is established, and with it is established
Theorem 3. If the integro-differential equation (2) in the case of positive integer \(\lambda\) has a formal solution of the form (10), then this series (10) converges for sufficiently small \(|x|\), \(|Cx^\lambda|\), and, consequently, equation (2) has an infinite set of holomorphic solutions (10) in a neighborhood of \(x=0\), where the path of integration is taken as in the case \(\operatorname{Re}(\lambda)\leqslant 0,\ \operatorname{Im}(\lambda)=0\).
We now give a certain generalization of the preceding theorems. Consider the integro-differential equation
\[ x\frac{du}{dx}=\lambda u(x)+f\left[x,u(x),\int_0^x g_1(x,t,u(t))\,dt,\ldots,\int_0^x g_n(x,t,u(t))\,dt\right], \tag{11} \]
where \(f(x,u,z_1,\ldots,z_n)\), \(g_1(x,t,u),\ldots,g_n(x,t,u)\) are functions holomorphic in a neighborhood of \(x=u=t=z_1=\cdots=z_n=0\), and
\[ f(0,0,0,\ldots,0)=0,\quad \frac{\partial f(0,0,0,\ldots,0)}{\partial u}=0,\quad g_1(x,t,0)=\cdots=g_n(x,t,0)=0. \]
The paths of integration are taken in accordance with \(\lambda\), as above.
It is easy to see that equation (11) has, in the case \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) not equal to a positive integer, or in the case \(\operatorname{Re}(\lambda)\leqslant 0\), but \(\operatorname{Im}(\lambda)\ne 0\), a formal solution of the form (5)
\[ u(x)=\sum_{i+j=1}^{\infty} b_{ij}x^i(Cx^\lambda)^j,\quad b_{01}=1\quad (i\geqslant 0,\ j\geqslant 0); \tag{12} \]
in the case of positive integer \(\lambda\) it either has no formal solution of the form (10), or it has it
\[ u(x)=\sum_{i+j=1}^{\infty} b_{ij}x^i(Cx^\lambda)^j,\quad b_{\lambda 0}=0,\ b_{01}=1; \tag{13} \]
in the case \(\operatorname{Re}(\lambda)\leqslant 0,\ \operatorname{Im}(\lambda)=0\) it has a formal solution of the form (9)
\[ u(x)=\sum_{i=1}^{\infty} b_i x^i. \tag{14} \]
Let \(F(x,u,z_1,\ldots,z_n)\) and \(G_1(x,t,u),\ldots,G_n(x,t,u)\) be majorants of, respectively, the series \(f(x,u,z_1,\ldots,z_n)\) and \(g_1(x,t,u),\ldots,g_n(x,t,u)\), where
\[ F(0,0,0,\ldots,0)=\frac{\partial F(0,0,0,\ldots,0)}{\partial u}=0, \]
\[ G_1(x,t,0)=\cdots=G_n(x,t,0)=0. \]
To prove the convergence of the series (12), (13), (14), consider the equation
\[ U=T+BF[x,U,xBG_1(x,x,U),\ldots,xBG_n(x,x,U)]. \tag{15} \]
where \(B>0\) is the same as above. Equation (15) has, in a neighborhood of \(x=T=0\), a holomorphic solution
\[ U=\sum_{i+j=1}^{\infty} B_{ij}x^iT^j,\qquad B_{01}=1\quad (i\geq 0,\ j\geq 0) \]
with nonnegative coefficients \(B_{ij}\). It is easy to see that
\[ |b_{ij}| \leq B_{ij} \]
for all \(i+j\geq 1\) for the series (12), (13), and
\[ |b_i|\leq B_{i0} \]
for all \(i\geq 1\) for the series (14). Therefore the convergence of the series (12), (13), (14) for sufficiently small \(|x|\), \(|Cx^\lambda|\) is established. Thus, we have
Theorem 4. If \(\operatorname{Re}(\lambda)>0\), but \(\lambda\) is not equal to a positive integer, or if \(\operatorname{Re}(\lambda)\leq 0\), but \(\operatorname{Im}(\lambda)\neq 0\), then the integro-differential equation (11) has a solution (12), holomorphic in the powers of \(x\) and \(Cx^\lambda\), for sufficiently small \(|x|\), \(|Cx^\lambda|\).
If \(\lambda\) is equal to a positive integer, then equation (11) either has no formal solution of the form (13), or has one. In the latter case the series (13) converges for sufficiently small \(|x|\), \(|Cx^\lambda|\), and hence equation (11) has an infinite set of solutions (13), holomorphic in a neighborhood of \(x=0\).
If \(\operatorname{Re}(\lambda)\leq 0\), \(\operatorname{Im}(\lambda)=0\), then equation (11) has a unique solution (14), holomorphic in a neighborhood of \(x=0\).
In all these cases the paths of integration in equation (11) are taken as in the corresponding cases of equation (2).
Literature
- Horn I. Über eine nicht lineare Volterrasche Integralgleichung, Jahresber. Deutsch. Mat.—Ver., 23, 1914.
Received by the editors
October 25, 1964.
Institute of Mathematics, Academy of Sciences of the BSSR