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Problems for ordinary differential equations with transmission conditions. (English) Zbl 1062.34094
Consider a boundary-functional problem with transmission conditions for the ordinary differential-operator equation on $[-1,1]$ $$ L(\lambda)u:=\lambda^2u(x)-a(x)u''(x)+(Bu)(x)=f(x)\quad x\in(-1,0)\cup(0,1) $$ with boundary-functional conditions $$ L_1u:=\alpha_1 u^{(m_1)}(-1)+\beta_1 u^{(m_1)}(-0)+\sum_{k=1}^{n_1}\delta_{1k}u^{(m_1)}(x_{1k})+T_1u=f_1, $$ $$ L_2u:=\alpha_2 u^{(m_2)}(+0)+\beta_2u^{(m_2)}(1)+\sum_{k=1}^{n_2}\delta_{2k}u^{(m_2)}(x_{2k})+T_2u=f_2, $$ and transmission conditions $$ L_\nu u:=\alpha_\nu u^{(m_\nu)}(-0)+\beta_\nu u^{(m_\nu)}(+0)+\sum_{k=1}^{n_\nu}\delta_{\nu k}u^{(m_\nu)}(x_{\nu k})+T_\nu u=f_\nu, $$ where $\nu=3,4$, $a(x)=a_1$ at $[-1,0)$, $a(x)=a_2$ at $(0,1]$, $a_1,a_2,\alpha_\nu,\beta_\nu,\delta_{\nu k},f_\nu$ are complex numbers $x_{\nu k}\in(-1,0)\cup(0,1)$, $B$ is a linear operator, and $T_\nu$ is a linear functional in the space $L_q(-1,1)$. The authors prove an isomorphism, coerciveness with respect to the spectral parameter, completeness and an Abel basis for a system of root functions. The obtained results in the article are new even in case of Sobolev spaces without weight.

34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34-02Research monographs (ordinary differential equations)
34B10Nonlocal and multipoint boundary value problems for ODE
47E05Ordinary differential operators
47N20Applications of operator theory to differential and integral equations
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