Weakly degenerating differential equations with unbounded operator coefficients
The author considers the boundary value problem \[ t^{2\alpha}u''(t)- Au(t)= f(t),\quad 0< t< 1,\;0<\alpha< 1,\;u(0)= u(1)=0, \] formulated in a Banach space \(E\) where \(u(t)\) and \(f(t)\) stand for continuous and continuously differentiable functions on \((0,1)\) with values in \(E\), while \(A\) is a linear operator. The main aim is to show that this problem has a unique solution if the domain of \(A\) is dense in \(E\) and for all \(\lambda\leq 0\) the operator \((\lambda I-A)\) has a bounded inverse satisfying \(\|(\lambda I-A)^{-1}\|\leq M(1+| \lambda|)^{-1}\). Here, \(M\) signifies a certain constant. To this end, the author replaces first the operator \(A\) by a scalar factor \(\lambda\) and makes the change of variable \(\tau= (1-\alpha)^{-1}t^{1-\alpha}\) to reduce the given differential equation into a Bessel equation. Then, by Green function method, the author writes an explicit expression of the solution to the Bessel equation through which one obtains a solution to the original problem.
weakly degenerating differential equations
scientific article; zbMATH DE number 1175124