A class of evolution equations: Existence of solutions with functional boundary conditions (Q1966208)

The evolution equation \[ \dot u(t)= Au(t)+ g(u)(t),\quad 0\leq t\leq 1, \] is considered with \(u\) taking values in a Banach space \(X\). Here, \(A\) is a linear (possibly unbounded) operator on \(X\) generating a strongly continuous semigroup, and \(g\) is a continuous map in functional spaces. The existence of a solution \(u\) is proved that satisfies for all \(t\) the phase restriction \(u(t)\in K\), with \(K\subset X\) being a fixed closed convex set, and the boundary condition \(u(0)= \varphi(u)\), with \(\varphi\) defined on a functional space and taking values in \(X\). In particular, initial and periodic conditions can be considered.