Consider the $N$-coupled nonlinear elliptic system \begin{equation}\tag{${\rm P}$} \begin{cases} \displaystyle -\Delta U_j+ U_j=\mu U_j^3+\beta U_j\sum_{k\neq j} U_k^2 \quad \text{in } \Omega,\\ U_j> 0 \quad\text{in } \Omega,\quad U_j=0 \quad \text{on } \partial\Omega,\ j=1, \ldots, N. \end{cases} \end{equation} where $\Omega$ is a smooth and bounded (or unbounded if $\Omega$ is radially symmetric) domain in $\mathbb R^n$, $n\leq3$. By using a $Z_N$ index theory, we prove the existence of multiple solutions of (P) and show the dependence of multiplicity results on the coupling constant $\beta$.

By using Mawhin's continuation theorem of coincidence degree theory and linear inequality, we establish the existence of eight positive periodic solutions for three species non-autonomous Lotka-Volterra cooperative systems with harvesting terms. An example is given to illustrate the effectiveness of our results.

By using the theory of coincidence degree and some refined analysis techniques, we study a general kind of periodic solutions to $p$-Laplacian neutral functional differential equation with multiple deviating arguments. A general analysis method to tackle with such equations is formed. Some new and universal results on the existence of periodic solutions are obtained, meanwhile, some known results in the literatures are improved. An example is provided as an application to our theorems.

It is proved that the extensible beam equation in Ball's model admits periodic solutions near equilibrium states if subject to external periodic force of high frequency. The approach is based on translation along trajectories, averaging method and homotopy invariants such as topological degree and fixed point index.

In this paper, first we obtain some results for structure of positive solutions set of some nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of the nonlinear operator equation. The method to show our main results is the global bifurcation theory.

The existence and localization of strong (Carathéodory) solutions is obtained for a second-order Floquet problem in a Banach space. The combination of applied degree arguments and bounding (Liapunov-like) functions allows some solutions to escape from a given set. The problems concern both semilinear differential equations and inclusions. The main theorem for upper-Carathéodory inclusions is separately improved for Marchaud inclusions (i.e. for globally upper semicontinuous right-hand sides) in the form of corollary. Three illustrative examples are supplied.

Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in cylindrical pipe with inflow and outflow in the two-dimensional case is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence of solutions is proved by the Leray-Schauder fixed point theorem.

For a nonautonomous dynamics with discrete time defined by a sequence of linear operators in a Banach space, we establish the robustness of polynomial contractions and of polynomial dichotomies under sufficiently small linear perturbations. In addition, we consider the general case of nonuniform polynomial behavior.

We give a construction to remove coincidence points of continuous maps on graphs ($1$-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidence index, Nielsen coincidence number, or coincidence Reidemeister trace in this setting, and the results of our previous paper ``A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles'' are invalid.

The notion of hemi-convex multifunctions is introduced. It is shown that each convex multifunction is hemi-convex, but the converse is not true. Some fixed point results for hemi-convex multifunctions are also proved.