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UDC 517.948.34
ON THE SOLVABILITY OF AN INTEGRO-DIFFERENTIAL EQUATION
Yu. K. LANDO
The integro-differential (i.d.) equation
\[ Au=\lambda u+f, \tag{1} \]
where
\[ Au=\sum_{i=0}^{n}\left[p_{n-i}(x)u^{(i)}(x)+\int_{0}^{1}K_{n-i}(x,y)u^{(i)}(y)\,dy\right],\qquad p_0(x)\ne0, \]
is solvable for arbitrary (continuous) right-hand sides \(f(x)\) if the corresponding homogeneous equation \(Au=\lambda u\) has \(n\) linearly independent solutions. If the number of such solutions is greater than the order \(n\) of the equation, then it may turn out to be unsolvable [1, 2]. It is therefore natural to call the value of the parameter \(\lambda\) a regular value of the i.d. equation (1) if the corresponding homogeneous equation has \(n\) linearly independent solutions, and an eigenvalue if the number of such solutions is greater than \(n\).
In order to find solvability conditions for the equation in the case of an eigenvalue, consider the Cauchy problem:
\[ Au=\lambda u+f,\qquad u^{(i)}(0)=\gamma_i\quad (i=0,1,\ldots,n-1). \tag{2} \]
This problem has been studied by many authors, in particular by Ya. V. Bykov [3], V. V. Vasil’ev [4, 5], T. I. Vigranenko [6], L. E. Krivoshein [7], and V. N. Nikolenko [8]. The investigation was carried out by reducing the problem to a resolving integral equation, or by the so-called method of A. I. Nekrasov [9]. (See also the survey article of M. M. Vainberg [10], which contains an extensive bibliography.)
If the value \(\lambda\) is regular for the Cauchy problem at some point \(a\) of the segment \([0,1]\), i.e., if the homogeneous Cauchy problem \(Au=\lambda u,\ u^{(i)}(a)=0\) has only the trivial solution, then the value \(\lambda\) is regular for the i.d. equation (1).
Consequently, the value \(\lambda\) can be an eigenvalue for the i.d. equation (1) if it is an eigenvalue for the Cauchy problem at every point of the segment \([0,1]\). Therefore, in what follows it is assumed that the value \(\lambda\) is an eigenvalue for the Cauchy problem (2).
Unlike the works of the authors mentioned above, in the present article a Lagrange identity is derived and an adjoint problem is constructed, which makes it possible to obtain, in the case of an eigenvalue \(\lambda\), necessary and sufficient solvability conditions both for the Cauchy problem (2) and for equation (1).
Assuming the functions \(u\) and \(z\) to belong to the space \(C^n[0,1]\), and the coefficients \(p_i(x)\) and kernels \(K_i(x,y)\) to be sufficiently smooth, and integrating by parts, we obtain the Lagrange identity
\[ \int_{0}^{1}\overline{z}(Au-\lambda u)\,dx = \sum_{k=1}^{n}\overline{v}_k[z(x)]u^{(k-1)}(x)\Big|_{x=0}^{x=1} + \int_{0}^{1}u\,\overline{(A^*z-\overline{\lambda}z)}\,dx, \tag{3} \]
where
\[ \overline{A^*z} = \sum_{i=0}^{n}(-1)^i \left[ p_{n-i}\overline{z} + \int_{0}^{1}K_{n-i}(y,x)\overline{z}(y)\,dy \right]^{(i)}, \]
\[ \overline{v}_k(z) = \sum_{i=0}^{n-k}(-1)^i \left[ p_{n-i-k}\overline{z} + \int_{0}^{1}K_{n-i-k}(y,x)\overline{z}(y)\,dy \right]^{(i)} \qquad (k=1,2,\ldots,n). \]
Denote
\[ -v_{k0}(z)=v_k[z(x)]_{|x=0},\qquad v_{k1}(z)=v_k[z(x)]_{|x=1}. \]
We shall call the problem
\[ A^*z=\overline{\lambda}z,\qquad v_{k1}(z)=0\quad (k=1,2,\ldots,n) \tag{4} \]
the adjoint of the Cauchy problem (2). Put in (2) \(\gamma_i=0\), i.e., consider the problem
\[ Au=\lambda u+f,\qquad u^{(i)}(0)=0\quad (i=0,1,\ldots,n-1). \tag{5} \]
Suppose that \(\lambda=0\) is a regular value of problem (2), and let \(T(x,s)\) be the Green’s function of problem (5) for \(\lambda=0\). Then \(\overline{T}(s,x)\) is the Green’s function of the adjoint problem (4) [11]. Problem (5) is equivalent to the integral equation
\[ u(x)=\lambda\int_0^1 T(x,s)u(s)\,ds+\int_0^1 T(x,s)f(s)\,ds, \tag{6} \]
and the adjoint problem (4) is equivalent to the adjoint integral equation
\[ z(x)=\frac{1}{\overline{\lambda}}\int_0^1 \overline{T}(s,x)z(s)\,ds. \tag{7} \]
It follows from this that the ranks \(p\) of the eigenvalues of the adjoint problems (2) and (4) are equal. If \(z_i,\ i=1,2,\ldots,p\), are eigenfunctions of the adjoint problem, then from (7) it follows that
\[ \frac{1}{\lambda}\int_0^1 \overline{z_i}(x) \left[\int_0^1 T(x,s)f(s)\,ds\right]dx = \int_0^1 f(s)\overline{z_i}(s)\,ds. \]
Therefore, for the solvability of problem (5) it is necessary and sufficient that the eigenfunctions of the adjoint problem be orthogonal to the right-hand side \(f(x)\):
\[ \int_0^1 f\overline{z_i}\,dx=0. \]
Let \(\varphi\) be an arbitrary function from the space \(C^n[0,1]\) satisfying the initial conditions (2), \(\varphi^{(i)}(0)=\gamma_i\).
Seeking the solution of problem (2) in the form \(u=y+\varphi\), we obtain, for determining the function \(y\), the problem
\[ \begin{cases} Ay=\lambda y-(A\varphi-\lambda\varphi)+f,\\ y^{(i)}(0)=0\quad (i=0,1,\ldots,n-1). \end{cases} \]
For its solvability, and hence also for the solvability of problem (2), it is necessary and sufficient that the orthogonality conditions
\[ \int_0^1 (A\varphi-\lambda\varphi-f)\overline{z_i}\,dx=0 \quad (i=1,2,\ldots,p) \]
be satisfied.
From Lagrange’s identity (3) it follows that
\[ \int_0^1 (A\varphi-\lambda\varphi)\overline{z_i}\,dx = \sum_{k=1}^n \gamma_{k-1}\overline{v_{k0}(z_i)}. \]
Hence the validity of the following theorems follows.
Theorem 1. In order that the Cauchy problem (2) be solvable, it is necessary and sufficient that the initial values \(\gamma_k\) satisfy the system of equations
\[ \sum_{k=1}^n \gamma_{k-1}\overline{v_{k0}(z_i)} = \int_0^1 f\overline{z_i}\,dx \quad (i=1,2,\ldots,p). \]
Theorem 2. Equation (1) is solvable if and only if the rank of the matrix \(\|\overline{v}_{k0}(z_i)\|\) \((i=1,2,\ldots,p;\ k=1,2,\ldots,n)\) does not change when the column of the numbers
\[
\left\{\int_0^1 \overline{f}z_i\,dx\quad (i=1,2,\ldots,p)\right.
\]
is appended to it.
Theorem 3. The number \(m\) of linearly independent solutions of the homogeneous equation \(Au=\lambda u\) is equal to \(n+p-q\), where \(n\) is the order of the equation; \(p\) is the rank of the eigenvalue \(\lambda\) of the Cauchy problem; \(q\) is the rank of the matrix \(\|\overline{v}_{k0}(z_i)\|\).
Theorem 4. If the value \(\lambda\) is regular for equation (1), then it is solvable for arbitrary (continuous) right-hand sides.
Theorem 5. If \(m=n+p\), then equation (1) is solvable if and only if the eigenfunctions of the adjoint problem (4) are orthogonal to the right-hand side \(f(x)\).
The formulated theorems can be illustrated by the following examples.
For the equation
\[
u''-\frac{1}{2\pi}\int_0^{2\pi}u(y)\,dy=\lambda u+f
\]
the value \(\lambda=-1\) is an eigenvalue, \(p=1,\ q=0,\ m=3\). Here \(1,\ \cos x,\ \sin x\) are solutions of the homogeneous equation; \(1-\cos x\) is an eigenfunction of the Cauchy problem and of the adjoint problem. The equation will be solvable if its right-hand side \(f\) is orthogonal to \(1-\cos x\), and is not solvable (for example, if \(f=1\)) otherwise.
For the equation
\[
u''+u-6\int_0^1 u(y)\,dy=\lambda u+f
\]
the value \(\lambda=1\) is regular, \(p=1,\ q=1,\ m=2\). The solutions of the homogeneous equation are \(2x-1,\ x^2\). The eigenfunction of the Cauchy problem is \(x^2\), and the eigenfunction of the adjoint problem is \(-(x-1)^2\). The nonhomogeneous equation is always solvable:
\[
u=\int_0^x (x-t)f_1(t)\,dt-\int_0^1\left[\int_0^y (y-t)f_1(t)\,dt\right]dy+C_1(2x-1)+C_2x^2.
\]
Let us note that the number of linearly independent solutions of the equation \(Au=\lambda u\) and of the adjoint equation \(A^*z=\overline{\lambda}z\) may be different; \(A^{**}\) is not always equal to \(A\). The spectrum of equation (1) (the set of eigenvalues of equation (1)) is Fredholm. Thus, for example, the Volterra equation has no eigenvalues.
If the functions \(p_i(x),\ K_i(x,y)\) are not sufficiently smooth, then instead of the adjoint problem one may consider the adjoint integral equation (7) and obtain analogous results.
References
- Landau Yu. K. Scientific Notes of the Minsk Pedagogical Institute, vol. 5, phys.-math. series, 1956, pp. 49–58.
- Landau Yu. K. Reports of the Academy of Sciences of the BSSR, 6, No. 10, 616–619, 1962.
- Bykov Ya. V. On Some Problems in the Theory of Integro-Differential Equations. Frunze, 1957.
- Vasil’ev V. V. Reports of the Academy of Sciences of the USSR, 100, 849–852, 1955.
- Vasil’ev V. V. Proceedings of Higher Educational Institutions, Mathematics, No. 6, 29–38, 1963.
- Vitrenko T. I. Proceedings of Higher Educational Institutions, Mathematics, No. 5, 6–18, 1961.
- Keroshen L. E. Approximate Methods for Solving Ordinary Integro-Differential Equations. Frunze, 1962.
- Nikolenko V. N. Uspekhi Matematicheskikh Nauk, 7:5, 225–228, 1952.
- Nekrasov A. I. Collected Works, 1. Publishing House of the Academy of Sciences of the USSR, 1961.
- Vainberg M. M. Itogi Nauki. 1962. Moscow, INI AN SSSR, 1964, pp. 5–37.
- Landau Yu. K. Proceedings of the Academy of Sciences of the BSSR, phys.-tech. series, 11–21, No. 4, 1960.
Received by the editors
April 23, 1966.
Minsk Pedagogical Institute
named after A. M. Gorky