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UDC 517.948.32
ON THE THEORY OF SINGULAR INTEGRAL EQUATIONS
N. E. TOVMASYAN
§ 1. NORMAL SOLVABILITY OF ONE CLASS OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS
Let \(S^+\) be a finite simply connected domain bounded by the contour \(\Gamma\). By \(S^-\) we shall denote the domain which complements \(S^+ + \Gamma\) to the full plane.
Consider the following system of singular integral equations
\[ \begin{aligned} K(\omega) \equiv{}& \alpha(t_0)\omega(t_0) + \frac{\beta(t_0)}{\pi i}\int_{\Gamma}\frac{\omega(t)\,dt}{t-t_0} \\ &+ \frac{1}{2\pi i}\int_{\Gamma}K_1(t,t_0)\omega(t)\ln\left(1-\frac{t_0}{t}\right)\,dt + \frac{1}{2\pi i}\int_{\Gamma}K_2(t,t_0)\omega(t)\times \\ &\times \ln\left(1-\frac{t}{t_0}\right)\,dt + \int_{\Gamma}K_3(t,t_0)\omega(t)\,dt = f(t_0), \end{aligned} \tag{1} \]
where \(t_0=\xi_0+i\eta_0\in\Gamma\), \(t=\xi+i\eta\in\Gamma\); \(\omega(t)=(\omega_1(t),\ldots,\omega_n(t))\) is the unknown vector-function on \(\Gamma\); \(f(t)=(f_1(t),\ldots,f_n(t))\) is a given vector-function on \(\Gamma\); \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\) are square matrices of order \(n\), given on \(\Gamma\). In (1), by \(\ln(1-t/t_0)\) (respectively by \(\ln(1-t_0/t)\)), for a given \(t_0\in\Gamma\) we mean the branch continuous at the points \(t\in S^+ + \Gamma\), \(t\ne t_0\) \((t\in S^-+\Gamma,\ t\ne t_0)\) and tending to zero as \(t=0\) \((t=\infty)\). Without loss of generality, we shall always assume that the point \(t=0\in S^+\). The singular integrals in (1) and below are understood in the sense of the Cauchy principal value.
We note that boundary value problems for elliptic systems of differential equations in the plane (see [2], [3], etc.) lead precisely to singular integral equations of the form (1). Boundary value problems considered by the author in an article that will appear after the present work also lead to the same type of integral equations.
Introduce the notation:
\[ \alpha(t)+\beta(t)=S(t),\qquad \alpha(t)-\beta(t)=D(t), \tag{2} \]
\[ K_1(t,t)=\gamma(t),\qquad K_2(t,t)=\delta(t). \]
Equation (1) is called an equation of normal type if
\[ \det S(t)\ne 0,\qquad \det D(t)\ne 0 \tag{3} \]
everywhere on \(\Gamma\).
When condition (3) is fulfilled, equation (1) has been completely studied and three fundamental Noether theorems have been proved (see [1], p. 510). In the present paragraph
these same theorems are proved under conditions more general than condition (3). The new condition and the formula for the index involve not only the matrices \(\alpha(t)\) and \(\beta(t)\), but also the matrices \(K_1(t,t_0)\) and \(K_2(t,t_0)\).
We note that, if condition (3) is violated at a finite number of points on \(\Gamma\), then such equations have been studied in [4] and others. In the present paper a violation of condition (3) at all points of the boundary \(\Gamma\) is allowed.
Along with equation (1), we consider the adjoint homogeneous equation
\[ \begin{aligned} K'(\psi) \equiv{}& \psi(t_0)\alpha(t_0)-\frac{1}{\pi i}\int_\Gamma \frac{\psi(t)\beta(t)\,dt}{t-t_0}+{}\\ &+\frac{1}{2\pi i}\int_\Gamma \psi(t)K_1(t_0,t)\ln\left(1-\frac{t}{t_0}\right)\,dt+{}\\ &+\frac{1}{2\pi i}\int_\Gamma \psi(t)K_2(t_0,t)\ln\left(1-\frac{t_0}{t}\right)\,dt+{}\\ &+\int_\Gamma \psi(t)K_3(t_0,t)\,dt=0, \end{aligned} \tag{4} \]
where \(\psi=(\psi_1,\ldots,\psi_n)\) is the unknown vector.
For simplicity of exposition, we shall assume for the time being that the contour \(\Gamma\), the matrices \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\) \((j=1,2,3)\), and the vector \(f(t)\) are infinitely differentiable, and that the solution of equations (1) and (4) is sought in the class of infinitely differentiable vector-functions. At the end of this section we shall indicate what changes are needed if they are not infinitely differentiable.
Fix a point \(t\) on \(\Gamma\). Let \(\theta_1(t)\) and \(\theta_2(t)\) be matrices whose columns are linearly independent solutions of the algebraic systems
\[ S(t)X=0 \quad \text{and} \quad D(t)Y=0 \quad (X=(x_1,\ldots,x_n),\; Y=(y_1,\ldots,y_n)) \]
respectively.
Denote by \(G_1(t)\) the matrix whose columns are the linearly independent columns of the matrix \(S(t)\) at the point \(t\) and the columns of the matrix \(\gamma(t)\theta_1(t)\). Denote by \(G_2(t)\) the matrix whose columns are the linearly independent columns of the matrix \(D(t)\) at the point \(t\) and the columns of the matrix \(\delta(t)\theta_2(t)\).
Let the coefficients of system (1) satisfy the conditions:
\[ \text{the ranks of the matrices } S(t) \text{ and } D(t) \text{ do not change on } \Gamma, \tag{5} \]
\[ \det G_1(t)\ne 0,\qquad \det G_2(t)\ne 0,\qquad t\in\Gamma. \tag{6} \]
Denote by \(r_1\) the rank of the matrix \(S(t)\) and by \(r_2\) the rank of the matrix \(D(t)\). Let \(\sigma_1(t)\) and \(\sigma_2(t)\) be infinitely differentiable square matrices of order \(n\) satisfying the conditions:
\[ \det\sigma_1(t)\ne 0,\qquad \det\sigma_2(t)\ne 0,\qquad t\in\Gamma; \tag{7} \]
\[ S(t)\sigma_1^{(j)}(t)=0,\qquad D(t)\sigma_2^{(p)}(t)=0,\qquad t\in\Gamma \tag{8} \]
\[ (j=r_1+1,\ldots,n;\; p=r_2+1,\ldots,n), \]
where \(\sigma_1^{(1)},\ldots,\sigma_1^{(n)}\) and \(\sigma_2^{(1)},\ldots,\sigma_2^{(n)}\) are the columns of the matrices \(\sigma_1(t)\) and \(\sigma_2(t)\). When condition (5) is fulfilled, the matrices \(\sigma_1(t)\) and \(\sigma_2(t)\) can always be constructed.
Denote by \(\Omega_1(t)\) the matrix whose columns are the vectors
\[ S(t)\sigma_1^{(k)}(t),\; -t\gamma(t)\sigma_1^{(j)}(t) \quad (k=1,\ldots,r_1;\; j=r_1+1,\ldots,n), \]
and by \(\Omega_2(t)\) the mat-
matrix whose columns are the vectors \(D(t)\sigma_2^{(l)}(t)\), \(t\delta(t)\sigma_2^{(p)}(t)\) \((l=1,\ldots,r_2,\ p=r_2+1,\ldots,n)\). Under condition (5), condition (6) is equivalent to the condition
\[ \det \Omega_1(t)\ne 0,\qquad \det \Omega_2(t)\ne 0,\qquad t\in \Gamma . \tag{9} \]
If conditions (5) and (6) are satisfied, then the following theorems hold:
Theorem 1. The homogeneous equations \(K(\omega)=0\) and \(K'(\psi)=0\) have a finite number of linearly independent solutions. The necessary and sufficient conditions for the solvability of equation (1) are that
\[ \int_{\Gamma} f(t)\psi^{(j)}(t)\,dt=0\qquad (j=1,\ldots,k'_1), \tag{10} \]
where \(\psi^{(1)},\ldots,\psi^{(k'_1)}\) is a complete system of linearly independent solutions of the associated homogeneous equation \(K'(\psi)=0\).
Let \(k_1\) and \(k'_1\) denote the number of linearly independent solutions of the equations \(K(\omega)=0\) and \(K'(\psi)=0\).
Theorem 2. The following equality holds:
\[ k_1-k'_1=\frac{1}{2\pi}\left[\ln \frac{\det\bigl(\Omega_2(t)\sigma_1(t)\bigr)} {\det\bigl(\Omega_1(t)\sigma_2(t)\bigr)} \right]_{\Gamma}, \tag{11} \]
where the symbol \([\ ]_{\Gamma}\) denotes the increment of the expression enclosed in brackets when the contour \(\Gamma\) is traversed once in the positive direction.
We shall call the difference \(k-k'\) the index of equation (1).
It is clear that if the coefficients of equation (1) satisfy condition (3), then they satisfy conditions (5) and (6). To see that conditions (5) and (6) are broader than condition (3), take, for example, in equation (1) the matrices \(\alpha\) and \(\beta\) to be constant, so that \(\det(\alpha+\beta)(\alpha-\beta)=0\), and take as \(\gamma(t)\) and \(\delta(t)\) arbitrary matrices satisfying condition (6). It is clear that such equations form a sufficiently broad class; the coefficients of these equations satisfy conditions (5), (6), but do not satisfy condition (3).
To prove Theorems 1 and 2 we shall prove the following lemma:
Lemma 1. Let \(\varphi_1\) and \(\psi_1\) be infinitely differentiable functions on \(\Gamma\). Then they are representable in the form:
\[ \varphi_1(t_0)=\frac{1}{2}\bigl(\mu_1(t_0)+t_0\mu'_1(t_0)\bigr) +\frac{1}{2\pi i}\int_{\Gamma} \frac{\mu_1(t)-t_0\mu'_1(t)}{t-t_0}\,dt, \tag{12} \]
\[ \psi_1(t_0)=\nu_1(t_0)+\frac{1}{2}t_0\nu'_1(t_0) +\frac{1}{2\pi i}\int_{\Gamma} \frac{t\nu'_1(t)}{t-t_0}\,dt, \tag{13} \]
where \(\mu_1(t)\) and \(\nu_1(t)\) are infinitely differentiable functions on \(\Gamma\), \(\mu'_1=\dfrac{d\mu_1}{dt}\), \(\nu'_1=\dfrac{d\nu_1}{dt}\); the functions \(\mu_1(t)\) and \(\nu_1(t)\) are determined by \(\varphi_1\) and \(\psi_1\) uniquely.
Proof. Denote by \(F(\mu,z)\) the expression
\[ F(\mu,z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\mu(t)\,dt}{t-z}, \tag{14} \]
and \(F^+(\mu,t_0)\) and \(F^-(\mu,t_0)\) are the boundary values of \(F(\mu,z)\) at the point \(t_0\) from inside and outside the domain \(S^+\). The following Sokhotski–Plemelj formulas hold [1]:
\[ F^+(\mu,t_0)=\frac{1}{2}\mu(t_0)+\frac{1}{2\pi i}\int_\Gamma \frac{\mu(t)\,dt}{t-t_0}, \tag{15} \]
\[ F^-(\mu,t_0)=-\frac{1}{2}\mu(t_0)+\frac{1}{2\pi i}\int_\Gamma \frac{\mu(t)\,dt}{t-t_0}. \tag{16} \]
Let \(\Phi(z)\) be an analytic function in \(S^+\), continuous in \(S^+ + \Gamma\), and let \(\Psi(z)\) be an analytic function in \(S^-\), continuous in \(S^- + \Gamma\), and tending to zero as \(z\to\infty\). Then from \(\Phi=\Psi\) on \(\Gamma\) it follows that \(\Phi\equiv0\), \(\Psi\equiv0\). Using this fact and formulas (15) and (16), it is easy to establish that the representations (12) and (13) are equivalent to the equalities:
\[ F(\varphi_1,z)=F(\mu_1,z)\quad \text{for } z\in S^+;\qquad F(\varphi_1,z)=z\frac{dF(\mu_1,z)}{dz}\quad \text{for } z\in S^-; \tag{17} \]
\[ F(\psi_1,z)=\frac{d}{dz}F(t\nu_1,z)\quad \text{for } z\in S^+;\qquad F(\psi_1,z)=F(\nu_1,z)\quad \text{for } z\in S^-. \tag{18} \]
The equalities (17) and (18), in turn, are equivalent to the equalities:
\[ F(\mu_1,z)=F(\varphi_1,z)\quad \text{for } z\in S^+;\qquad F(\mu_1,z)= \]
\[ =-\frac{1}{2\pi i}\int_\Gamma \varphi_1(t)t^{-1}\ln\left(1-\frac{t}{z}\right)\,dt \quad \text{for } z\in S^-; \tag{19} \]
\[ -\frac{1}{2\pi i}\int_\Gamma \psi_1(t)\ln\left(1-\frac{z}{t}\right)\,dt+c =F(t\nu_1,z)\quad \text{for } z\in S^+,\qquad F(\psi_1,z)= \]
\[ =F(\nu_1,z)\quad \text{for } z\in S^-, \tag{20} \]
where \(c\) is an arbitrary constant.
From formulas (15) and (16) it follows that
\[ F^+(\mu,t_0)-F^-(\mu,t_0)=\mu(t_0). \tag{21} \]
Using (21), it is easy to see that the equalities (19) and (20) hold if and only if
\[ \mu_1(t_0)=F^+(\varphi_1,t_0)+\frac{1}{2\pi i}\int_\Gamma \varphi_1(t)t^{-1}\ln\left(1-\frac{t}{t_0}\right)\,dt, \tag{22} \]
\[ \nu_1(t_0)=-\frac{t_0^{-1}}{2\pi i}\int_\Gamma \psi_1(t)\ln\left(1-\frac{t_0}{t}\right)\,dt-F^-(\psi_1,t_0). \tag{23} \]
Lemma 1 follows from (22) and (23).
Proof of Theorems 1 and 2. The following formulas hold:
\[ \int \frac{dt}{t-t_0}\int_\Gamma \frac{\varphi(t,t_1)}{t_1-t}\,dt_1 = -\pi^2\varphi(t_0,t_0)+ \]
\[ +\int_\Gamma dt_1\int_\Gamma \frac{\varphi(t,t_1)\,dt}{(t-t_0)(t_1-t)}, \tag{24} \]
\[ \int_\Gamma k(t,t_0)\psi'_1(t)\ln\left(1-\frac{t}{t_0}\right)\,dt = \pi i\,k(t_0,t_0)\psi_1(t_0)- \]
\[ -\int_\Gamma \frac{k(t,t_0)\psi_1(t)\,dt}{t-t_0} -\int_\Gamma \frac{dk(t,t_0)}{dt}\,\psi_1(t)\ln\left(1-\frac{t}{t_0}\right)\,dt, \tag{25} \]
\[ \int_\Gamma k(t,t_0)\psi'_1(t)\ln\left(1-\frac{t_0}{t}\right)\,dt = -\pi i\,k(t_0,t_0)\psi_1(t_0)- \]
\[ -\int_\Gamma \frac{t_0 k(t,t_0)\psi_1(t)\,dt}{t(t-t_0)} -\int_\Gamma \frac{dk(t,t_0)}{dt}\,\psi_1(t)\ln\left(1-\frac{t_0}{t}\right)\,dt, \tag{26} \]
where \(\varphi(t,t_0)\), \(k(t,t_0)\), and \(\psi_1(t)\) are functions on \(\Gamma\) satisfying a Hölder condition in \(t,t_0\), while the derivatives with respect to \(t\) of the functions \(k(t,t_0)\) and \(\psi_1(t)\) are continuous;
\[ \psi'_1=\frac{d\psi_1}{dt}, \qquad t_1=\xi_1+i\eta_1\in\Gamma. \]
Formula (24) is the Poincaré—Bertrand formula (see [1], p. 124), and formulas (25) and (26) are obtained by integration by parts.
According to Lemma 1, the solution of equation (1) may be sought in the form
\[ \omega(t)=\sigma_1(t)g(t)\qquad (g(t)=(g_1(t),\ldots,g_n(t))), \tag{27} \]
where
\[ g_k(t_0)=v_k(t_0),\qquad g_p(t_0)=\frac{1}{2}t_0v'_p(t_0)+v_p(t_0)+ \]
\[ +\frac{1}{2\pi i}\int_\Gamma \frac{t\,v'_p(t)\,dt}{t-t_0} \tag{28} \]
\[ (k=1,\ldots,r_1;\quad p=r_1+1,\ldots,n), \]
\[ v(t_0)=\sigma_1^{-1}(t_0)\sigma_2(t_0)\varphi(t_0) \qquad (\varphi=(\varphi_1,\ldots,\varphi_n),\ v=(v_1,\ldots,v_n)), \tag{29} \]
\[ \varphi_j(t_0)=q_j(t_0),\qquad \varphi_r(t_0)=\frac{1}{2}\bigl(q_r(t_0)+t_0q'_r(t_0)\bigr)+ \]
\[ +\frac{1}{2\pi i}\int_\Gamma \frac{q_r(t)-t_0q'_r(t)}{t-t_0}\,dt \tag{30} \]
\[ (j=1,\ldots,r_2;\quad r=r_2+1,\ldots,n), \]
\[ q(t_0)=\sigma_2^{-1}(t_0)\mu(t_0) \qquad (q=(q_1,\ldots,q_n),\ \mu=(\mu_1,\ldots,\mu_n)), \tag{31} \]
\(\sigma_1(t)\) and \(\sigma_2(t)\) are the square matrices constructed above; \(\sigma_1^{-1}\) and \(\sigma_2^{-1}\) are the matrices inverse to \(\sigma_1\) and \(\sigma_2\).
Making in equation (1) the substitution (27)—(31) and using formulas (24)—(26), we obtain a singular equation with respect to the vector \(\mu\)
\[ \begin{aligned} &\alpha^{(1)}(t_0)\mu(t_0) +\frac{\beta^{(1)}(t_0)}{\pi i}\int_{\Gamma}\frac{\mu(t)\,dt}{t-t_0} +{}\\ &\quad+\frac{1}{2\pi i}\int_{\Gamma}K_1^{(1)}(t,t_0)\mu(t)\ln\left(1-\frac{t}{t_0}\right)dt +{}\\ &\quad+\frac{1}{2\pi i}\int_{\Gamma}K_2^{(1)}(t,t_0)\mu(t)\ln\left(1-\frac{t_0}{t}\right)dt +{}\\ &\quad+\int_{\Gamma}K_3^{(1)}(t,t_0)\mu(t)\,dt=f(t_0), \end{aligned} \tag{32} \]
where
\[ \alpha^{(1)}(t)=\frac{1}{2}\left(\Omega_1(t)\sigma_1^{-1}(t)+\Omega_2(t)\sigma_2^{-1}\right), \]
\[ \beta^{(1)}(t)=\frac{1}{2}\left(\Omega_1(t)\sigma_1^{-1}(t)-\Omega_2(t)\sigma_2^{-1}(t)\right), \]
\(K_j^{(1)}(t,t_0)\) \((j=1,2,3)\) are completely determined infinitely differentiable matrices, which are expressed in terms of the matrices \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\).
It follows from condition (6) that equation (32) is an equation of normal type. Since the matrices \(\alpha^{(1)}(t)\), \(\beta^{(1)}(t)\), \(K_j^{(1)}(t,t_0)\), the boundary \(\Gamma\), and the vector \(f\) are infinitely differentiable, all solutions of equation (32) satisfying the Hölder condition are infinitely differentiable.
Formulas (28)—(31) establish a one-to-one correspondence between the solutions of equations (32) and (1). It is known that if a condition of the form
\[ \int_{\Gamma} f(t)\psi(t)\,dt=0 \tag{33} \]
is necessary for the solvability of equation (1), then \(\psi\) is a solution of the adjoint homogeneous equation (4) (see [1], p. 223).
Since equation (32) is an equation of normal type, Theorems 1 and 2 are valid for it (see [1], p. 510). From the facts indicated above we obtain Theorems 1 and 2 for equation (1).
Let us note that in reducing equation (1) to equation (32) we used only condition (5). Suppose the coefficients of equation (1) satisfy condition (5) (we do not require condition (6)). Write equation (1) in the form \(K_1(\omega^{(1)})=0\), and equation (32) in the form \(K_2(\omega^{(2)})=f\), where \(\omega^{(1)}(t)\) and \(\omega^{(2)}(t)\) are the unknown vectors. If the coefficients of the equation \(K_2(\omega^{(2)})=f\) satisfy condition (5), then from this equation we can obtain the equation \(K_3(\omega^{(3)})=f\) in the same way as equation (32) was obtained from equation (1). If the coefficients of the equation \(K_3(\omega^{(3)})=f\) satisfy condition (5), then in an analogous way from it we can obtain the equation \(K_4(\omega^{(4)})=f\), and so on.
Definition. We shall say that the equation \(K_1(\omega^{(1)})=f\) satisfies condition \(k\), if the coefficients of the equations \(K_j(\omega^{(j)})=f\), constructed in the above manner, for \(j=1,2,\ldots,k-1\) satisfy condition (5), and for \(j=k\) satisfy condition (3).
We note that condition \(k+1\) is broader than condition \(k\), and condition 1 coincides with condition (3), while condition 2 coincides with conditions (5), (6).
Theorem 3. If the equation \(K_1(\omega^{(1)})=f\) satisfies condition \(k\), then theorem 1 is valid for it; the index of the equation \(K_1(\omega^{(1)})=f\) is equal to the index of the equation \(K_k(\omega^{(k)})=f\) \((k\) is an arbitrary fixed positive integer\().\)
Proof. If equation (1) satisfies condition \(k\), then, by the method indicated above, it can be reduced to the equation \(K_k^{*}(\omega^{(k)})=f\), which is an equation of normal type. From this equivalent reduction we obtain theorem 3. Consequently, if equation (1) satisfies condition \(k\), then its solution can always be reduced to the solution of an integral equation of normal type, and its index can be computed from the coefficients, without solving the equation itself.
Let \(\varphi(t,t_0)\) be a function of the variables \(t\) and \(t_0\) on \(\Gamma\). We shall say that \(\varphi(t,t_0)\) belongs to the class \(H\), \((\varphi(t,t_0)\in H)\), if it satisfies the Hölder condition with respect to \(t\) and \(t_0\). We shall say that \(\varphi(t,t_0)\in H_p\) if
\[ \frac{\partial^{j+k}\varphi(t,t_0)}{\partial t^j \partial t_0^k}\in H \quad \text{for } j+k\le p. \]
If each element of the matrix \(G(t,t_0)\) belongs to the class \(H_p\), then we shall write \(G(t,t_0)\in H_p\).
We now determine what changes occur in the theorems obtained if the vector \(f\), the matrices \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\), and the contour \(\Gamma\) belong to the class \(H_p\), and the solution is sought in the class \(H\). Let \(F(\mu,z)\) be defined by formula (14). Denote by \(F'^{+}(\mu,t_0)\) and \(F'^{-}(\mu,t_0)\) the boundary values at the point \(t_0\in\Gamma\) of the function \(\dfrac{d}{dz}F(\mu,z)\) from inside and from outside the domain \(S^{+}\).
Lemma 2. If \(\varphi_1\) and \(\psi_1\) are functions of class \(H\), then they are representable in the form:
\[ \varphi_1(t_0)=F^{+}(\mu_1,t_0)-F^{-}(\mu_1,t_0)t_0, \tag{34} \]
\[ \psi_1(t_0)=F'^{+}(t\nu_1,t_0)-F^{-}(\nu_1,t_0), \tag{35} \]
where \(\mu_1\) and \(\nu_1\) are functions of class \(H\), and they are determined uniquely by \(\varphi_1\) and \(\psi_1\).
Lemma 2 is proved analogously to lemma 1.
It is easy to see that if \(\mu_1\in H_1\), \(\nu_1\in H_1\), then the representations (34), (35) coincide with the representations (12), (13). Suppose equation (1) satisfies condition \(k\), the coefficients \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\) \((j=1,2,3)\) and the contour \(\Gamma\) are infinitely differentiable, \(f\in H_p\), and the solution is sought in the class \(H\). Then, using representation (34) instead of representation (12), and representation (35) instead of representation (13), in a manner analogous to the preceding case we reduce equation (1) to an equation of normal type. Hence we obtain that, for equation (1), the theorems proved are valid, and moreover all solutions of this equation belong to the class \(H_{p-k+1}\), but, generally speaking, do not belong to the class \(H_{p-k+2}\). The latter assertion says that if equation (1) satisfies condition \(k\) and \(f\in H_p\), then it is natural to take \(p\ge k-1\), and to seek the solution in the class \(H_l\), where \(0\le l\le p-k+1\). If equation (1) satisfies condition \(k\), the matrices \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\) belong to the class \(H_{2k}\), \(t(s)\in H_{2k+2}\), \(f\in H_{k-1}\), where \(t=t(s)\) is the parametric equation of the contour \(\Gamma\), and the solution of equation (1) is sought in the class \(H\), then it is proved analogously that the theorems obtained above remain valid.
§ 2. ANALOG OF THEOREMS 1 AND 3 FOR MORE GENERAL CLASSES OF SINGULAR INTEGRAL EQUATIONS
In this section we shall consider equation (1), where \(\omega(t)=(\omega_1(t),\ldots,\omega_n(t))\) is the unknown vector, \(f(t)=(f_1(t),\ldots,f_m(t))\) is a given \(m\)-dimensional vector on \(\Gamma\); \(\alpha(t)\), \(\beta(t)\), \(K_i(t,t_0)\) are given matrices containing \(m\) rows and \(n\) columns. The homogeneous equation adjoint to equation (1) will be equation (2), where \(\psi=(\psi_1,\ldots,\psi_m)\) is the unknown \(m\)-dimensional vector. Here we shall use the same notation as in § 1. For the time being we shall assume that the matrices \(\alpha(t)\), \(\beta(t)\), \(K_j(t,t_0)\) \((j=1,2,3)\), the vector \(f(t)\), and the contour \(\Gamma\) are infinitely differentiable, and the solution is sought in the class of infinitely differentiable vectors.
We impose the following restrictions on the coefficients of equation (1):
The rank of the matrices \(S(t)\) and \(D(t)\) does not change on \(\Gamma\). \(\tag{36}\)
The rank of the matrices \(G_1(t)\) and \(G_2(t)\) is equal to \(\min(m,n)\), \(t\in\Gamma\). \(\tag{37}\)
Recall that the matrices \(G_1(t)\) and \(G_2(t)\) are constructed by means of the matrices \(\alpha(t)\), \(\beta(t)\), \(\gamma(t)\), \(\delta(t)\) in § 1. In the present case these matrices contain \(m\) rows and \(n\) columns.
If the coefficients of system (1) satisfy conditions (36) and (37), then the following holds.
Theorem 4. If \(n>m\), then the homogeneous equation (1) has an infinite number of linearly independent solutions, while the adjoint homogeneous equation (2) has a finite number of linearly independent solutions. If, however, \(n<m\), then the homogeneous equation (1) has a finite number of linearly independent solutions, while the adjoint homogeneous equation has an infinite number of linearly independent solutions. Necessary and sufficient conditions for solvability of equation (1) consist in the fact that
\[ \int_\Gamma f(t)\psi(t)\,dt=0 \tag{38} \]
for all solutions \(\psi\) of the adjoint homogeneous equation (2).
Proof. The necessity of condition (38) is obvious. First let us prove the theorem when \(n>m\) and the ranks of the matrices \(S(t)\) and \(D(t)\) are equal to \(m\) everywhere on \(\Gamma\). In this case there exist infinitely differentiable square matrices of order \(n\), \(S_1(t)\) and \(D_1(t)\), satisfying the conditions:
\[ \begin{aligned} &1)\quad \det S_1(t)\ne 0,\quad \det D_1(t)\ne 0,\quad t\in\Gamma,\\ &2)\quad \text{the first } m \text{ rows of } S_1(t) \text{ and } D_1(t) \text{ are the rows of the matrices } S(t) \text{ and } D(t), \end{aligned} \tag{39} \]
respectively.
Consider the system of equations
\[ \begin{aligned} &\alpha_1(t_0)\omega(t_0)+\frac{\beta_1(t_0)}{\pi i}\int_\Gamma \frac{\omega(t)\,dt}{t-t_0}+ \\ &\quad+\frac{1}{2\pi i}\int_\Gamma \widetilde K_1(t,t_0)\omega(t)\ln\left(1-\frac{t_0}{t}\right)\,dt+ \\ &\quad+\frac{1}{2\pi i}\int_\Gamma \widetilde K_2(t,t_0)\omega(t)\ln\left(1-\frac{t}{t_0}\right)\,dt+ \\ &\quad+\int_\Gamma \widetilde K_3(t,t_0)\omega(t)\,dt=\widetilde f(t_0), \end{aligned} \tag{40} \]
where \(\omega=(\omega_1,\ldots,\omega_n)\) is the unknown vector, \(\tilde f=(f_1,\ldots,f_m,f_{m+1},\ldots,f_n)\),
\[ \alpha_1(t)=\frac{S_1(t)+D_1(t)}{2},\qquad \beta_1(t)=\frac{S_1(t)-D_1(t)}{2}, \tag{41} \]
\(f=(f_1,\ldots,f_m)\) is the right-hand side of equation (1); \(f_{m+1},\ldots,f_n\) are, for the time being, arbitrary infinitely differentiable functions; \(\widetilde K_j(t,t_0)\) is a square matrix of order \(n\), the first \(m\) rows of which coincide with the rows of the matrix \(K_j(t,t_0)\), while the remaining elements are equal to zero \((j=1,2,3)\).
We shall say that \(\omega=(\omega_1,\ldots,\omega_n)\) is a solution of equation (40) if there exist infinitely differentiable functions \(f_{m+1}(t),\ldots,f_n(t)\) such that the vector \(\omega(t)\) satisfies equation (40) for \(\tilde f=(f_1,\ldots,f_m,f_{m+1},\ldots,f_n)\).
It is clear that every solution of equation (40) is a solution of equation (1), and conversely.
Since equation (40), with respect to \(\omega\), is an equation of normal type, it follows, according to Theorem 1, that it has a solution if and only if the vector \(\tilde f\) satisfies the conditions
\[ \int_{\Gamma}\tilde f(t)\mathbf v^{(j)}(t)\,dt=0 \qquad (j=1,2,\ldots,k_2), \tag{42} \]
where \(\mathbf v^{(1)},\ldots,\mathbf v^{(k_2)}\) is a complete system of linearly independent solutions of the homogeneous equation adjoint to equation (40).
We note that the vectors \(\mathbf v^{(k)}=(v_1^{(k)},\ldots,v_n^{(k)})\) \((k=1,\ldots,k_2)\) are infinitely differentiable. Denote by \(\varphi^{(j)}\) the vector \((v_{m+1}^{(j)},\ldots,v_n^{(j)})\). Denote by \(k_3\) the number of those vectors among \(\varphi^{(1)},\ldots,\varphi^{(k_2)}\) which are linearly independent. Let these, for definiteness, be the vectors \(\varphi^{(1)},\ldots,\varphi^{(k_3)}\). Then conditions (42) are equivalent to the conditions
\[ \int_{\Gamma}\tilde f(t)\mathbf v^{(j)}(t)\,dt=0 \qquad (j=1,\ldots,k_3), \tag{43} \]
\[ \int_{\Gamma} f(t)\psi^{(j)}(t)\,dt=0 \qquad (j=1,\ldots,k_2-k_3), \tag{44} \]
where \(\psi^{(1)}(t),\ldots,\psi^{(k_2-k_3)}(t)\) are completely determined infinitely differentiable \(n\)-dimensional vectors.
Condition (43) can be satisfied by a suitable choice of the functions \(f_{m+1},\ldots,f_n\). Indeed, take the vector \((f_{m+1},\ldots,f_n)\) in the form
\[ (f_{m+1},\ldots,f_n) = c_1\overline{\varphi}^{(1)}(t)\overline{t}_s' +\cdots+ c_{k_3}\overline{\varphi}^{(k_3)}(t)\overline{t}_s', \tag{45} \]
where \(\overline{\varphi}^{(1)},\ldots,\overline{\varphi}^{(k_3)}\), \(\overline t\) are the complex conjugates of \(\varphi^{(1)},\ldots,\varphi^{(k_3)}\), \(t\), respectively, and \(c_1,\ldots,c_{k_3}\) are constants; \(\overline{t}_s'=d\overline t/ds\).
Substituting \(f_{m+1},\ldots,f_n\) from (45) into (43), to determine \(c_1,\ldots,c_{k_3}\) we obtain a system of linear algebraic equations whose determinant is nonzero. Consequently, condition (44) is necessary and sufficient for the solvability of equation (1). Since condition (44) is necessary for the solvability of equation (1), the functions \(\psi^{(1)}(t),\ldots,\psi^{(k_2-k_3)}(t)\) are solutions of the adjoint homogeneous equation (2). Consequently, condition (38) is necessary and sufficient for the solvability of equation (1), and the adjoint homogeneous equation (2) has a finite number of linearly independent solutions.
It is easy to see that for \(f=0\) there exists an infinite number of linearly independent vectors \((f_{m+1}, \ldots, f_n)\) satisfying conditions (43). Hence it follows that the homogeneous equation (1) has an infinite number of linearly independent solutions.
Let us now consider the case when \(n<m\) and the rank of the matrices \(S(t)\) and \(D(t)\) is equal to \(n\) everywhere on \(\Gamma\). Since the adjoint homogeneous equation (2) belongs to the case mentioned above, it has an infinite number of linearly independent solutions.
Denote by \(S_2(t)\) and \(D_2(t)\) infinitely differentiable square matrices of order \(n\) satisfying the conditions:
\[ \begin{gathered} 1)\ \det S_2(t)\ne 0,\quad \det D_2(t)\ne 0;\\ 2)\ \text{the first } n \text{ columns of the matrices } S_2(t) \text{ and } D_2(t) \text{ are the columns of the matrices } S(t) \text{ and } D(t), \text{ respectively.} \end{gathered} \tag{46} \]
Consider the equation
\[ \begin{aligned} &\alpha_2(t_0)\varphi(t_0)+\frac{\beta_2(t_0)}{\pi i}\int_{\Gamma}\frac{\varphi(t)\,dt}{t-t_0}+\\ &\quad+\frac{1}{2\pi i}\int_{\Gamma}M_1(t,t_0)\varphi(t)\ln\left(1-\frac{t_0}{t}\right)\,dt+\\ &\quad+\frac{1}{2\pi i}\int_{\Gamma}M_2(t,t_0)\varphi(t)\ln\left(1-\frac{t}{t_0}\right)\,dt+\\ &\quad+\int_{\Gamma}M_3(t,t_0)\varphi(t)\,dt=f(t_0), \end{aligned} \tag{47} \]
where \(\varphi=(\varphi_1,\ldots,\varphi_m)\) is the unknown vector, \(f=(f_1,\ldots,f_m)\) is the right-hand side of equation (1),
\[ \alpha_2(t)=\frac{S_2(t)+D_2(t)}{2},\qquad \beta_2(t)=\frac{S_2(t)-D_2(t)}{2}, \]
\(M_j(t,t_0)\) is a square matrix of order \(m\), whose first \(n\) columns are the columns of the matrix \(K_j(t_0,t)\), and the remaining elements are equal to zero \((j=1,2,3)\).
Equation (47) is an equation of normal type. It is easy to see that if \(\omega(t)=(\omega_1(t),\ldots,\omega_n(t))\) is a solution of equation (1), then \(\varphi=(\omega_1(t),\ldots,\omega_n(t),0,\ldots,0)\) is a solution of equation (47), and conversely, if \(\varphi=(\varphi_1,\ldots,\varphi_n,0,\ldots,0)\) is a solution of equation (47), then \(\omega=(\varphi_1(t),\ldots,\varphi_n(t))\) is a solution of equation (1). Hence it follows that the homogeneous equation (1) has a finite number of linearly independent solutions, while the nonhomogeneous equation (1) has a solution if and only if equation (47) has a solution \(\varphi=(\varphi_1,\ldots,\varphi_m)\) satisfying the conditions \(\varphi_{n+1}=0,\ldots,\varphi_m=0\).
Equation (47) has a solution if and only if the vector \(f(t)\) satisfies the conditions
\[ \int_{\Gamma} f(t)\psi^{(j)}(t)\,dt\qquad (j=1,\ldots,k_4), \tag{48} \]
where \(\psi^{(1)},\ldots,\psi^{(k_4)}\) is a complete system of linearly independent solutions of the homogeneous equation adjoint to (47).
If conditions (48) are fulfilled, then the solution of equation (47) is given by the formula
\[ \begin{aligned} \varphi(t_0)={}&\alpha_1^{(1)}(t_0)f(t_0)+ \int_{\Gamma}\frac{\beta_1^{(1)}(t,t_0)f(t)\,dt}{t-t_0}+{}\\ &+\int_{\Gamma}\beta_2^{(1)}(t,t_0)\ln\left(1-\frac{t_0}{t}\right)f(t)\,dt+{}\\ &+\int_{\Gamma}\beta_3^{(1)}(t,t_0)\ln\left(1-\frac{t}{t_0}\right)f(t)\,dt +c_1\varphi^{(1)}(t_0)+\cdots+c_{k_s}\varphi^{(k_s)}(t_0), \end{aligned} \tag{49} \]
where \(\alpha_1^{(1)}(t)\), \(\beta_j^{(1)}(t,t_0)\) \((j=1,2,3)\) are completely determined infinitely differentiable matrices independent of \(f\); \(\varphi^{(1)}(t),\ldots,\varphi^{(k_s)}(t)\) is a complete system of linearly independent solutions of the homogeneous equation (47), and \(c_1,\ldots,c_{k_s}\) are arbitrary constants.
Let \(\mu_1(t), \mu_2(t), \ldots\) be infinitely differentiable functions which form a basis in the space \(L_2(\Gamma)\).
It is known that the equation \(\varphi_k(t)\equiv 0\) on \(\Gamma\) is equivalent to the condition
\[ \int_{\Gamma}\varphi_k(t)\mu_j(t)\,dt=0 \qquad (j=1,2,\ldots). \tag{50} \]
From (48)—(50) we obtain that equation (47) has a solution of the form
\(\varphi(t)=(\varphi_1(t),\ldots,\varphi_n(t),0,\ldots,0)\) if and only if \(f(t)\) satisfies a countable number of conditions
\[ \int_{\Gamma} f(t)\psi^{(k)}(t)\,dt=0 \qquad (k=1,2,\ldots), \tag{51} \]
where \(\psi^{(k)}(t)\) are completely determined infinitely differentiable \(m\)-dimensional vectors.
Consequently, conditions (51) are necessary and sufficient for the solvability of equation (1). Hence, in particular, it follows that the functions \(\psi^{(k)}(t)\) are solutions of the adjoint homogeneous equation (2). Consequently, in the case when
\[ \operatorname{rang} S(t)=\operatorname{rang} D(t)=\min(m,n) \tag{52} \]
everywhere on \(\Gamma\), Theorem 4 is completely proved.
Let the coefficients of equation (1) satisfy conditions (36) and (37). Then, with the aid of the substitution (27)—(30), where \(\sigma_1(t_0)\) and \(\sigma_2(t_0)\) are the same matrices as in § 1, equation (1) is reduced to equation (32), which already satisfies condition (52).
Since Theorem 4 is valid for equation (32), the validity of Theorem 4 for equation (1) also follows from this equivalent reduction.
Using condition (52) instead of condition (3), analogously to the case \(m=n\) (see § 1), condition \(k\) is determined for equation (1) when \(m\ne n\), and the validity of Theorem 4 is proved for equation (1) satisfying condition \(k\). The results obtained remain in force if the same smoothness is required of the coefficients of equation (1), the boundary \(\Gamma\), the vector \(f\), and the unknown vector \(\omega\) as was required at the end of § 1. The results obtained also remain in force in the case when the contour \(\Gamma\) consists of a finite number of smooth closed contours \(\Gamma_k\), provided that the requirements stated in the paper are imposed on each contour \(\Gamma_k\) separately.
References
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N. I. Muskhelishvili. Singular Integral Equations. Fizmatgiz, Moscow, 1962.
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A. V. Bitsadze. Boundary Value Problems for Elliptic Equations of the Second Order. Nauka, Moscow, 1966.
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I. N. Vekua. New Methods for Solving Elliptic Equations. Gostekhizdat, Moscow, 1948.
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F. D. Gakhov. Boundary Value Problems. Moscow, 1958.
Received by the editors
March 1, 1966
Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR