A connected system of differential equations in a Banach space (Q912271)

The author proves the existence and uniqueness of solutions of the problem \[ u'+A_ 1u+B_ 1v'=f,\quad v''+A_ 2v+B_ 2u=g,\quad u(0)=u_ 0,\quad v(0)=v_ 0,\quad v'(0)=v_ 1, \] where \(A_ i\), \(B_ i\) \((i=1,2)\) are linear (possibly unbounded) operators densely defined on a Banach space \(E\), \(f,g\) are given functions from \([0,1]\) into \(E\) and \(u,v,v_ 1\in E\). By a solution we mean a pair \((u,v)\) of functions \(u,v\) defined on \([0,1]\) with values in \(E\) such that \(u,u'\), \(A_ 1u\), \(B_ 2u,v,v'\), \(A_ 2v,B_ 1v'\) are defined and continuous on \([0,1]\).