Higher order nonlinear degenerate differential operator equations
The authors present optimal regularity results in \(L^p(0,a;E)\) for nonlocal boundary value problems for higher order nonlinear degenerate differential operator equations of the type \[ (-1)^m u^{[2m]}(x) +\sum_{i=0}^{2m-1} A_i(x) u^{[i]}(x)+Au(x)=F(x,u, u^{[1]},\dots,u^{[2m-1]}) \] with boundary conditions \[ \sum_{i=0}^{m_k}\alpha_{ki}u^{[i]}(0) +\beta_{ki}u^{[i]}(a)=0,\quad k=1,2,\dots,2m. \] Here, \(u^{[i]}=\left(x^\gamma {d\over {dx}}\right)^i u,\) \(0<\gamma<1-1/p,\) \(0\leq m_k\leq 2m-1,\) \(\alpha_{ki}\) and \(\beta_{ki}\) are complex numbers, while \(A_i\) and \(A\) are possibly linear operators in a Banach space \(E\).
scientific article; zbMATH DE number 5615123