On the solvability of a complete second-order differential equation in Banach space - Gershtein, L. M. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
For the complete second-order differential equation with unbounded operator coefficients $u'' + A(t)u' + B(i)u = f, \quad u(0) = u_o, \quad u'(0)=u_1$ the Cauchy problem is studied. By using the "coinmutant method", we construct the coercive solution of this problem is in the Holder space in the case where the operator $В$ has the same "strength" as the operator $А^2$.