\(C_{0}\)-operator Laplace integral and boundary value problems for operator degenerate equations (Q416853)

Consider the differential equation \[ a(t)u''(t) + b(t)u'(t) = Au(t) + f(t) \] in a Banach space, with \(a(t) \geq 0,\) \(a, b\) continuously differentiable in \((0, 1),\) \(a, a'\), \(b, b'\) bounded, \(-A\) the generator of a strongly continuous semigroup and \(f\) continuous and bounded in \((0, 1)\). The problem is to find bounded solutions in \((0, 1)\). There is a boundary condition at \(t = 1\) and, since \(a\) may vanish at \(t = 0\), a boundary condition there may or may not be imposed (depending on the singularity). The authors give conditions on the coefficients which ensure that the scalar problem \((A = \lambda)\) is well posed. Then, they use an approach due to Maslov to express the solution of the operator problem as an operator Laplace transform of the solution of the corresponding scalar problem.