Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach
Using the translation along trajectories method in infinite dimensions [\textit{R. Bader} and \textit{W. Kryszewski}, Nonlinear Anal., Theory Methods Appl. 54, No.~4, A, 707--754 (2003; Zbl 1034.34072)] with subsequent application of averaging and degree theories, the author derives existence criteria of periodic solutions to the abstract periodic evolution problem of the form \[ \dot{u}(t)=-Au(t)+F(t,u(t)),\quad t>0;\;u(t)\in M,\;t\geq 0;\;u(0)=u(\tau), \] with a linear operator \(A:D(A)\to E\), so that \(-A\) generates a \(C_0\)-semigroup of compact linear contractions on a Banach space \(E\), \(M\subset E\) is a closed convex cone and \(F: [0,+\infty)\times M\to E\) is a continuous mapping satisfying the tangency condition \(F(t, \overline{u})\in T_M(\overline{u})\) for all \(\overline{u}\in M\), \(t \geq 0\), where \(T_M(\overline{u})\) is the tangent cone to \(M\) at \(\overline{u}\). Aiming at applications to partial differential equations and systems, the author considers the problem in a more general case, when \(F\) is tangent to \(M\), i.e., \(F\) may take values outside \(M\), and without the assumption that nonlinear perturbations are positive.
scientific article; zbMATH DE number 5900634
scientific article