Convergence for strongly order-preserving semiflows. (English) Zbl 0739.34040

Smith, Hal L.; Thieme, Horst R.

SIAM J. Math. Anal. 22, No. 4, 1081-1101 (1991).

The authors consider a semiflow \(\Phi\) on an ordered metric space \(X\) and a subset \(Z\) of \(X\) and of an ordered Banach space \(Y\) with strong order generated by the interior of a cone such that (1) the orders in \(X\) and \(Y\) coincide on \(Z\), (2) \(Z\) with the topology inherited from \(Y\) embeds continuously into \(X\), (3) \(Z\) is positively invariant with respect to \(\Phi\), (4) for some \(\tau\geq 0\), \(\Phi_ \tau(X)\subset Z\) and \(\Phi_ \tau:X\to Z\) is continuous and continuously differentiable on \(Z\), (5) for any equilibrium \(e\) of \(\Phi\) with \(\rho(e)\), the spectral radius of \(\Phi'_ \tau(e)\), not less than one, \(\rho(e)\) is a pole of the resolvent of \(\Phi'_ \tau(e)\) with finite rank and the null space of \(\rho(e)I-\Phi'_ \tau(e)\) is spanned by an element of the interior of the cone generating the strong order in \(Y\). Under the above assumptions the authors prove the following theorem.
Theorem. Suppose that for each \(x\in X\) the positive orbit of \(x\) has compact closure in \(X\) and that \(x\) can be approximated either from above or from below in \(X\) by a sequence \(\{x_ n\}\) such that the sum of the positive limit sets of \(x_ n\) has compact closure in \(X\). Then the interior of the set of points with singleton positive limit sets is dense in \(X\).
A similar theorem is proved for local semiflows, and applications to FDE’s and parabolic PDE’s are given.

Reviewer: M.Mrozek

MSC:

37-XX	Dynamical systems and ergodic theory
34G20	Nonlinear differential equations in abstract spaces
47H20	Semigroups of nonlinear operators

Keywords:

monotone dynamical systems; strongly order-preserving flow; convergent points; semiflow; Banach space