## 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

Zbl 1034.34072
Full Text:

### References:

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