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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.

MSC:

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

Citations:

Zbl 1034.34072
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References:

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