Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach. (English) Zbl 1222.47132

Using the translation along trajectories method in infinite dimensions [R. Bader and 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.


47J35 Nonlinear evolution equations
47J15 Abstract bifurcation theory involving nonlinear operators
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47H11 Degree theory for nonlinear operators


Zbl 1034.34072
Full Text: DOI


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