An inverse problem for a second-order differential equation in a Banach space
The author considers in a Banach space \(E\) the differential equation \[ d^2v / dt^2 = Av + f(t) + p, \] with \(0 \leq t \leq t_1\). Here, \(A\) is a linear unbounded operator generating a cosine operator function, \(f\) is continuous on \([0, t_1]\) with values in \(E\), and \(p\) is an unknown element of \(E\). The boundary conditions \[ v(0) = v_0, \quad v'(0) = v'_0, \quad\text{and} \quad v(t_1) = v_1, \] are given. The problem considered here is to find a pair \((v(t), p)\) satisfying the equation (in the classical sense) and the boundary conditions. The author obtains necessary and sufficient conditions for the unique solvability of this problem.
linear unbounded operator generating a cosine operator function
scientific article; zbMATH DE number 2163693