On completeness of systems of the eigen and adjoint functions for certain systems of linear integro-differential equations (Q2768854)

The author studies completeness properties of eigenfunctions and associate functions of two spectral problems for Dirac-type integro-differential system NEWLINE\[NEWLINE-iBy'+Q(x)y+\int_{0}^{x}M(x,t)y(t) dt=\lambda y.\tag{1}NEWLINE\]NEWLINE Here \(y\) is a 2-component vector, \(B\) is a diagonal \((2\times 2)\)-matrix with nonzero elements of different signs; the elements of the main diagonal of \((2\times 2)\)-matrix \(Q(x)\) equal zero and the other ones are integrable functions on \([0,1]\); the elements of matrix \(M(x,t)\) are essentially bounded functions on the set \(\Omega =\{0\leq t\leq x\leq 1\}\); \(\lambda\) is the spectral parameter. Two kinds of boundary conditions involving spectral parameter are considered: NEWLINE\[NEWLINE[y_1-hy_2]_{x=0}=0,\quad[a(\lambda)y_1+b(\lambda)y_2]_{x=1}=0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE[y_1-hy_2]_{x=0}=0,\quad[a(\lambda)y_1^2+2b(\lambda)y_1y_2+c(\lambda)y_2^2] _{x=1/2}=0.NEWLINE\]NEWLINE Here \(a(\lambda),b(\lambda),c(\lambda)\) are polynomials, and \(h\neq 0\). The author generalizes his earlier results [Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, No. 8, 24-28 (1999; Zbl 0947.47036)] relevant to the case where the boundary conditions do not depend on \(\lambda \). He also makes some critical remarks concerning the main statement of \textit{E. I. Tarapova's} paper [Teor. Funkts. Funkts. Anal. Prilozh 31, 157-160 (1979; Zbl 0418.34028)], where the analogous problem for the Sturm-Liouville operator has been studied.