On the Fredholm solvability of inverse problems (Q1278157)

In a Banach space \(X\), the linear inverse problem of recovering \(p\in X \) from the given \(u_1:=u(T)\) is studied, where \(u(t)\) is the solution to the Cauchy problem \[ u'(t)=Au(t)(t)p(t), 0<t<T,\qquad u(0)=u_0, \] and \(A\) is a linear closed operator in \(X\) with a dense domain of definition. Solvability conditions for this inverse problem are obtained and the uniqueness is discussed.