Stability of certain nonautonomous difference equations

In this paper, the stability of a class of nonautonomous nonlinear difference equations that are determined by functions which satisfy certain contractive type condition is studied in metric spaces, and its variant problem in ordered Banach spaces is also investigated. Several examples are given to illustrate their applications.     

Continuation method for -sublinear mappings

Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies

for all , , where . Thompson's metric is among the main tools we are using.     

Cationic liposomes modified with non-ionic surfactants as effective non-viral carrier for gene transfer

A defined change in formulation components affects the physical and chemical characteristics of cationic liposomes (CLs) carriers in many ways. Therefore, a great degree of control can be exercised over the structure by modifying the CLs with various materials, leading to new innovations for carrier improvement. In the present study, surface modifications of cationic liposomes with non-ionic surfactants—sorbitan monoesters serials (Span 85, 80, 40 and 20) were carried out for developing a new gene transfer carrier. Span modified cationic liposomes (Sp-CLs) were prepared by reverse phase evaporation method (RPV) and self-assemble complexes of antisense oligonucleotides/surfactant modifying cationic liposomes were prepared by auto-coacervation through electrostatic effect. Characterization of Sp-CLs and the self-assembled complex was performed by electron microscope, particle size, zeta potential, turbidity and agarose electrophoresis. Furthermore, in vitro cellular uptake experiment showed that Span plays a role in enhancing the cellular uptake of encapsulated oligonucleotides mediated by Sp-CLs by the endocytosis-dependent route. CLs modified with Span 40 significantly facilitated the cellular uptake by COS-7 cells and HeLa cells; also showed some positive effect on gene expression. That suggests it is a potential non-viral carrier for efficient gene transfer.     

Path stability and nonlinear weak ergodic theorems

Let be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iterates in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.

     

Some contractive type mappings and their applications to difference equations

We discuss a class of discrete dynamic systems in a complete metric space (M, d) defined by mappings which satisfy various types of contractive type conditions. Their variants in ordered Banach space are investigated and applied to solve the global asymptotic stability of the equilibriums of some discrete dynamic systems.     

Thompson’s metric and global stability of difference equations

We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation

Omega Limit Sets in Positive Cones

The omega limit sets of a nonlinear operator T which is definedon a positive cone and satisfies certain ray-contractive typeconditions are discussed. Under the assumption that the restrictionof T to a compact subset is surjective, the following alternativesare proved: the omega limit set of a point in the cone eitherconsists of a fixed point or forms a 2-cycle. In addition, newproofs and extensions to relevant results are given. 2000 MathematicsSubject Classification 47H07, 47H09.     

Horizontally slotted photonic crystal nanobeam cavity with embedded active nanopillars for ultrafast direct modulation

A horizontally slotted photonic crystal nanobeam cavity with an embedded active nanopillar structure is proposed for ultrafast direct modulation. By designing the thicknesses of both the nanobeam and the horizontal slot layer, the quality factor (Q factor) and the mode volume (Vn ) of the proposed cavity can be engineered independently. As a result, the spontaneous emission (SpE) rate is enhanced with a small Vn of 2.4 while the SpE rate and the cavity photon lifetime have an optimal Q factor of ～ 1000. In our simulation, the modulation bandwidth could be enhanced up to 170 GHz with different emission linewidths of the active nanopillar.     

Stability of Positive Fixed Points of Nonlinear Operators

This paper considers the stability of positive fixed points of nonlinear operators under purturbations. Through the use of the Thompson's metric, we obtain the results in ordered Banach spaces by proving the corresponding results in metric spaces first. An application to the existence of periodic solutions for a parametrized system of ordinary differential equations is given.     

The existence of a fixed point for the sum of two monotone operators

Let A and H be two operators defined in an ordered Banach space such that