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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(λ)u:=λ 2 u(x)-a(x)u '' (x)+(Bu)(x)=f(x)x(-1,0)(0,1)

with boundary-functional conditions

L 1 u:=α 1 u (m 1 ) (-1)+β 1 u (m 1 ) (-0)+ k=1 n 1 δ 1k u (m 1 ) (x 1k )+T 1 u=f 1 ,
L 2 u:=α 2 u (m 2 ) (+0)+β 2 u (m 2 ) (1)+ k=1 n 2 δ 2k u (m 2 ) (x 2k )+T 2 u=f 2 ,

and transmission conditions

L ν u:=α ν u (m ν ) (-0)+β ν u (m ν ) (+0)+ k=1 n ν δ νk u (m ν ) (x νk )+T ν u=f ν ,

where ν=3,4, a(x)=a 1 at [-1,0), a(x)=a 2 at (0,1], a 1 ,a 2 ,α ν ,β ν ,δ νk ,f ν are complex numbers x νk (-1,0)(0,1), B is a linear operator, and T ν 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.

MSC:
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