Multipoint boundary value problem for an evolutionary equation with parameters in Banach space (Q1605564)

The authors deal with the differential equation \[ \frac{dy(t)}{dt}=Ay(t)+f(t),\quad t\in[0,T], \] where \(A\) is a closed operator with a dense domain of definition \({\mathcal D}(A)\) in a Banach space \(\mathcal B\), \(f(t)\) is a step function \(f(t)=a_k\), \(t_k\leq t<t_{k+1}\), \(k=0,1,\dots,n\), with \(t_0=0\), \(t_{n+1}=T\). The problem is to find a function \(f(t)\) and the unknown parameters \(a_k\in{\mathcal B}\) such that the solution \(y(t)\) to the differential equation takes the given values \(y(t_k)=y_k,k=0,1,\dots,n\). They derive necessary and sufficient conditions for existence and uniqueness of a solution to this multipoint boundary value problem for an evolutionary equation with parameters in a Banach space.