Existence and uniqueness of positive periodic solutions for abstract semilinear evolution equations. (Chinese. English summary) Zbl 1110.34328

Li, Xongxiang

J. Syst. Sci. Math. Sci. 25, No. 6, 720-728 (2005).

Summary: This paper discusses the existence of \(\omega\)-periodic solutions for the semilinear evolution equation \[ u'(t)+Au(t)=f\bigl(t,u(t)\bigr),\;t \in\mathbb{R}, \] in an ordered Banach space \(E\), where \(A\) is the infinitesimal generator of a positive \(C_0\)-semigroup, and \(f:\mathbb{R}\times E\to E\) is a continuous mapping which is \(\omega\)-periodic in \(t\). The existence and uniqueness of periodic solutions for the associated linear evolution equation are established, and the spectral radius of the periodic resolvent operator is accurately estimated. With the aid of this estimation, the existence and uniqueness of positive periodic solutions are obtained by using monotone iterative technique.

MSC:

34G20	Nonlinear differential equations in abstract spaces
47J35	Nonlinear evolution equations
34C25	Periodic solutions to ordinary differential equations