Regular boundary value problems with a discontinuous coefficient, functional-multipoint conditions, and a linear spectral parameter (Q616524)

The paper is concerned with the equation \[ L(\lambda)u:=\lambda u(x)+a(x)u^{(m)}(x)+Bu|_x=f(x),\quad x\in [-1,0)\cup (0,1], \] with some multipoint boundary-transmission conditions, which contain the spectral parameter \(\lambda\) and linear functionals in the direct sum space \(X=L_q(-1,0)+L_q(0,1)\), \(1<q<\infty\). Here, \(m\) is a positive integer, \(a\) is a piecewise constant scalar function and \(B\) is an abstract linear operator in \(X\). The authors find sectors in the complex plane which depend on \(m\), where the operator solution associated to the above problem is an isomorphism between appropriate Sobolev spaces. The coerciveness of the problem with respect to the spectral parameter and the space variable is also proved.