Study of a complete abstract differential equation of elliptic type with variable operator coefficients. I. (Q928354)

The authors consider the abstract differential equation of second-order \[ u''(t)+ p(t)u'(t)+ q(t)u(t)-\lambda u(t)= f(t),\quad t\in(0,\delta),\quad u(0)= \varphi,\;u'(\delta)=\psi, \] where \(\lambda\) is a positive real number, \(q(t)\) are closed linear operators and \(p(t)\) are bounded linear operators. The existence, uniqueness and maximal regularity results under appropriate differentiability assumptions combining those of Yagi and Da Prato-Grisvard are obtained. An example is given.