Periodic solutions for nonlinear evolution equations in a Banach space

Abstract:

We prove an existence result for $T$-periodic mild solutions to nonlinear evolution equations of the form \[ u’(t) + Au(t) \backepsilon F(t,u(t)),\quad t \in {R_ + }.\] Here $(X,|| \cdot ||)$ is a real Banach space, $A:D(A) \subset X \to {2^X}$ is an operator with $A - aI$ $m$-accretive for some $a > 0$ and such that $- A$. generates a compact semigroup, while $F:{R_ + } \times \overline {D(A)} \to X$ is a Carathéodory mapping which is $T$-periodic with respect to its first argument and satisfies \[ \lim \limits _{r \to + \infty } \tfrac {1}{r}\sup \left \{ {||F(t,v)||;t \in {R_ + },v \in \overline {D(A)} ,||v|| \leq r} \right \} < a.\]. As a consequence, we obtain an existence theorem for $T$-periodic solutions to the porous medium equation.