Direct and inverse problems for a differential equation in a space with a cone (Q1594238)

This article deals with the Cauchy problem \[ {dv\over dt}= Av+ \phi(t)p\quad (0< t\leq t_1),\quad v(0)= v_0,\tag{1} \] and the inverse problem \[ {dv\over dt}= Av+ \phi(t)p\quad (0< t\leq t_1),\quad v(0)= v_0,\quad g(v(t))= \psi(t),\tag{2} \] where \(A\) is a linear unbounded operator which generates a differentiable for \(t>0\) and locally integrable semigroup \(T(t)\), \(p\in E\), \(g\in E^*\), \(E\) a Banach space ordered by a cone \(K\); furthermore, in problem (1), \(v(t)\) is an unknown function and \(\phi(t)\) a given continuous scalar function, and, in problem (2), \(v(t)\) and \(\phi(t)\) are unknown function, \(\psi(t)\) is a given scalar function. Basic results are a theorem about existence and uniqueness of a solution \(v(t)\) of (1) and the formula \[ v(t)= T(t) v_0+ \int^t_0 T(t- s) \phi(s) p \,ds \] for this solution and the theorem about existence and uniqueness of a solution \(v(t,\phi(t))\) of (2).