Solvability of fourth‐order Sturm‐Liouville problems with abstract linear functionals in boundary‐transmission conditions

Abstract

This study deals with the solvability of one nonclassical boundary-value problem for fourth-order differential equation on two disjoint intervals I 1 = (−1, 0)and I 2 = (0, 1). The boundary conditions contain not only endpoints x = −1and x = 1but also a point of interaction x = 0, finite number internal points x jki ∈ I j and abstract linear functionals S k . So, our problem is not a pure differential one. We investigate such important properties as isomorphism, Fredholmness and coerciveness with respect to the spectral parameter. Note that the obtained results are new even in the case of the boundary conditions without internal points x jki and without abstract linear functionals S k .