The paper is concerned with a nonlinear system of delayed differential equations as a generalization of an equation describing a simple model of the fluctuation of biological populations. The dependence of the behavior of monotone solutions on the coefficients and delays is studied and optimal sufficient conditions are derived for the existence of increasing and unbounded solutions and for the existence of increasing and convergent solutions. Inequalities estimating such solutions with some given increasing functions are derived as well. The results are compared with the linear case illustrated by examples, and open problems are formulated.

This paper is mainly concerned with controlled time fractional differential equations of Sobolev type in Caputo and Riemann-Liouville fractional derivatives with the order in $(1,2)$ respectively. By properties on some corresponding fractional resolvent operators family, we first establish sufficient conditions for the existence of mild solutions to these controlled time fractional differential equations of Sobolev type. Then, we present the existence of optimal controls of systems governed by corresponding time fractional differential equations of Sobolev type via setting up approximating minimizing sequences of suitable functions twice.

Let $\Omega$ be a smooth bounded domain in $\mathbb R^2$. For $\varepsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $$ -\Delta u=\frac{1}{\varepsilon^2}(1-|u|^2)u \ \text{ in $\Omega$} $$ such that on $\partial \Omega$ u satisfies $|u|=1$ and $u\wedge \partial_\nu u=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions, we use a variational reduction method very similar to the one used in [12]. We obtain the exact same result as the authors of the aforementioned article obtained for the Neumann problem. This is because the renormalized energy for the Neumann problem and for the semi-stiff problem are the same. In particular, if $\Omega$ is simply connected a solution with degree one on the boundary always exists and if $\Omega$ is not simply connected, then for any $k\geq 1$ a solution with $k$ vortices of degree one exists.

The one-dimensional $p$-Laplacian eigenvalue problem \begin{equation*} \begin{cases} -(|y'|^{p-2}y')'=(p-1)(\lambda -q(x))|y|^{p-2}y,\\ y(0)=y(1)=0, \end{cases} \end{equation*} is considered in this paper. We derive its normalized eigenfunction expansion by using a Prüfer-type substitution. Employing some theories in Banach spaces, we discuss the basis property related to these eigenfunctions as an application.