Study of solvability and estimation of the number of solutions for one class of singular integral equations (Q5932617)

The author considers some class of singular integral equations equivalent to a homogeneous linear conjugator problem with a matrix-function of a special form. For simplicity the author studies integral equations and the corresponding matrix-functions on a simple smooth closed contour \(\Gamma\) dividing the complex plane into two domains \(D^+\) and \(D^-\) \((0\in D^+, \infty\in D^-)\) and studies them in the class \(H_{\mu}(\Gamma)\). The singular integral equation \[ K\varphi(t) \equiv a_0(t)\varphi(t) + a_1(t)S[k_1\varphi](t) + \cdots + a_n(t)S[k_n\varphi](t) = f(t) \] is considered on \(\Gamma\) where the right-hand side is \[ f(t) = 2M_{1,m_1}(t)a_1(t) + \cdots + 2M_{n,m_n}(t)a_n(t). \] Here \(M_{k,m_k}(t)\), \(k = 1,\dots, n\), is a polynomial of degree \(m_k\), \(a_0(t)\), \(a_i(t)\), and \(k_i(t)\), \(i = 1,\dots,n\), are \(H_{\mu}\)-continuous functions of points of the contour, \(S[\omega](t)\) is a singular operator. Suppose that \(\varphi(t)\) is a solution to the above-mentioned equation (in this case the equation is solvable). The following homogeneous linear conjugation problem can arrive: \[ \begin{gathered} \mathbf\Phi^+(t) = G(t)\mathbf\Phi^-(t),\quad t\in\Gamma,\quad \Big(\mathbf\Phi(z) = \big(\varphi_1(z),\varphi_2(z),\dots, \varphi_{n}(z)\big)\Big), \\ G = G_0 - E,\quad G_0 = \|g_{ij}\|, \quad g_{ij}= 2k_ia_j/\Delta_{1n},\quad i,j = 1,\dots, n, \end{gathered} \] where \(\det G_0 = 0\) and \(\det G = (-1)^n\Delta_n\). Thus if \(\varphi(t)\) is a solution to the equation then the functions \[ \varphi^+_i = P[k_i\varphi] - M_{i,m_i},\quad \varphi^-_i = Q[k_i\varphi] + M_{i,m_i} \] determine a solution to the homogeneous linear conjugation problem whose order at infinity equals \(\max(m_1,\dots, m_n)\). The main result of the article is as follows: Theorem 1. Let \(\kappa\) be the index of the singular integral equation, let \(\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_n\) be the partial indices of the matrix-function, and let \(\kappa_1\geq\kappa_2\geq \cdots \geq \kappa_n\) be the partial indices of the second equation in the homogeneous linear conjugation problem whose possible values lie in the interval \(\lambda_n\leq\kappa_n\leq\cdots\leq \kappa_1\leq\lambda_1\). If \(\lambda_n\geq 0\) then the equation \(K\varphi = 0\) has \(\kappa\) linearly independent solutions constituting \(n\) series of \(\kappa_1, \kappa_2, \dots, \kappa_n\) solutions for \(\lambda_n>0\). If \(\lambda_1\geq\cdots\geq \lambda_s > 0 \geq \lambda_{s + 1}\geq\cdots\geq \lambda_n\) then the equation has at most \(\sum_{i=1}^s\lambda_i\) linearly independent solutions. If \(\lambda_1 \leq 0\) then the equation has no nontrivial solutions.