Uniqueness criterion for solutions of nonlocal problems on a finite interval for abstract singular equations
Let \(E\) be a complex Banach space and let \(A,B\) be linear closed operators on \(E\) whose domains are not necessary dense. Consider the equation \[ B(u''(t)+\frac{k}{t}u'(t))=Au(t),\quad 0<t<1. \] The following cases are considered \begin{itemize} \item \(k\geq0\) with Neumann boundary condition \(u'(0)=0\) and non-local condition \(\int_0^1t^k(1-t^2)^{\alpha-1}u(t)dt=0\) (\(\alpha>0\)) \item \(k<1\) with Dirichlet boundary condition \(u(0)=0\) and non-local condition \(\int_0^1t(1-t^2)^{\beta-1}u(t)dt=0\) (\(\beta>0\)) \item \(k\geq0\) with Neumann boundary condition \(u'(0)=0\) and non-local condition \(a\int_0^1t^ku(t)dt+bu'(1)=0\) (\(a,b\ne0\)) \item \(k<1\) with Dirichlet boundary condition \(u(0)=0\) and non-local condition \(a\int_0^1t^ku(t)dt+b\lim_{t\to1}(t^{k-1}u(t))'=0\) \end{itemize} For each case a uniqueness criterion is established. Applications to the equation \(t^\gamma v''+bt^{\gamma-1}v'=Av\) are given.
scientific article; zbMATH DE number 7488476
scientific article