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A remark on the mild solutions of non-local evolution equations. (English) Zbl 1015.37045
This paper is devoted to study the nonlocal evaluation equation \[ \begin{cases} u'(t)= Au(t)+ f(t,u(t))\\ u(0)+ g(u)= u_0,\end{cases}\qquad 0\leq t\leq T,\tag{1} \] where \(g: C([0, T],X)\to X\) is a continuous function, and \(X\) is a general Banach space. To study (1), a very common approach is to define a map \(F: C([0,T],X)\to C([0,T], X)\) by \[ F(u)(t)= T(t)[u_0- g(u)]+ \int^t_0 T(t- s) f(s,u(s)) ds, \qquad 0\leq t\leq T,\tag{2} \] and prove that \(F\) has a fixed point, which is called a mild solution of (1). Here \(T(\cdot)\) is the corresponding semigroup generated by (1).
The author addresses to the following question: Can the map \(F\) defined by (2) be a compact operator? In the case \(u_0= 0\), \(f(\cdot,\cdot)= 0\), this question becomes: Can the map defined by \[ [T(\cdot)g](u):= T(\cdot)(g(u)),\quad u\in C([0,T], X) \] be compact? The author shows that, the answer is no in general.

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
34G20 Nonlinear differential equations in abstract spaces
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