### A Construction of Stable Subharmonic Orbits in Monotone Time-periodic Dynamical Systems.

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A simple proof is given of a basic surjectivity result for monotone operators. The proof is based on the dynamical systems method (DSM).

We generalize a Theorem of Koldunov [2] and prove that a disjointness proserving quasi-linear operator between Resz spaces has the Hammerstein property.

In order to save CPU-time in solving large systems of equations in function spaces we decompose the large system in subsystems and solve the subsystems by an appropriate method. We give a sufficient condition for the convergence of the corresponding procedure and apply the approach to differential algebraic systems.

In this paper we present a new theorem for monotone including iteration methods. The conditions for the operators considered are affine-invariant and no topological properties neither of the linear spaces nor of the operators are used. Furthermore, no inverse-isotony is demanded. As examples we treat some systems of nonlinear ordinary differential equations with two-point boundary conditions.

We prove an intermediate value theorem for certain quasimonotone increasing functions in ordered Banach spaces, under the assumption that each nonempty order bounded chain has a supremum.

In the paper [13] we proved a fixed point theorem for an operator $\mathcal{A}$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: $$m(\mathcal{A}x-\mathcal{A}y)\prec Am(x-y).$$ The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$.

Maps $f$ defined on the interior of the standard non-negative cone $K$ in ${\mathbb{R}}^{N}$ which are both homogeneous of degree $1$ and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson’s part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that the extension must have at least...

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.

We consider the existence of at least one positive solution to the dynamic boundary value problem $$\begin{array}{cccc}\hfill -{y}^{\Delta \Delta}\left(t\right)& =\lambda f(t,y\left(t\right))\text{,}\phantom{\rule{4.0pt}{0ex}}t\in {[0,T]}_{\mathbb{T}}y\left(0\right)\hfill & \hfill ={\int}_{{\tau}_{1}}^{{\tau}_{2}}{F}_{1}(s,y\left(s\right))\Delta sy\left({\sigma}^{2}\left(T\right)\right)& ={\int}_{{\tau}_{3}}^{{\tau}_{4}}{F}_{2}(s,y\left(s\right))\Delta s,\hfill \end{array}$$ where $\mathbb{T}$ is an arbitrary time scale with $0<{\tau}_{1}<{\tau}_{2}<{\sigma}^{2}\left(T\right)$ and $0<{\tau}_{3}<{\tau}_{4}<{\sigma}^{2}\left(T\right)$ satisfying ${\tau}_{1}$, ${\tau}_{2}$, ${\tau}_{3}$, ${\tau}_{4}\in \mathbb{T}$, and where the boundary conditions at $t=0$ and $t={\sigma}^{2}\left(T\right)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.