On nonautonomous second-order differential equations on Banach space (Q5939816)

Here, the author considers the nonautonomous second-order differential equation NEWLINE\[NEWLINEu''(t)=A(t)u'(t)+B(t)u(t)+f(t),\qquad u^{(i)}(0)=x_i, \quad i=0,1, \tag{1}NEWLINE\]NEWLINE in a Banach space \(E\), with \(0\leq t\leq T\), \(A(t)\) and \(B(t)\) are linear operators on \(E\), and \(f\colon [0,T]\to E\) is a continuous mapping. He proves that, under suitable assumptions imposed on \(A(t)\) and \(B(t)\), problem (1) has a unique twice continuously differentiable solution on \([0,T]\). The idea of the proofs is to reduce (1) to a differential equation of first order and to apply \textit{N. Tanaka}'s result [Differ. Integral Equ. 9, No. 5, 1067-1106 (1996; Zbl 0942.34053)]. An application of the obtained results to the second-order Volterra integrodifferential equation NEWLINE\[NEWLINEu''(t)=B(t)u(t)+\int_0^t C(t,s)u(s) ds+f(t),\quad 0\leq t\leq T, \qquad u^{(i)}(0)=x_i, \quad i=0,1,NEWLINE\]NEWLINE is presented.