By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: $x_i^{\prime }(t)=x_i(t)[r_i(t)-{\sum }_{j=\mathrm{1}}^n{a}_{ij}(t)x_j(t)-{\sum }_{j=\mathrm{1}}^n\mathrm{‍}{b}_{ij}(t){\int }_{-{\tau }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(\xi )x_j(t+\xi )d\xi -{\sum }_{j=\mathrm{1}}^n\mathrm{‍}{c}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{g}_{ij}(\xi )x_j^{\prime }(t+\xi )d\xi ],\hspace{0.17em}\hspace{0.17em}\mathrm{\Delta }{x}_i(t_k)=-I_{ik}(x_i(t_k)),\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}k=\mathrm{1,2},\dots$$.\mathrm{}$ Some verifiable criteria are established easily.

We study the dynamical behavior of solutions of an n-dimensional nonlinear Schrödinger equation with potential and linear derivative terms under the presence of phenomenological damping. This equation is a general version of the dissipative Gross-Pitaevskii equation including terms with first-order derivatives in the spatial coordinates which allow for rotational contributions. We obtain conditions for the existence of a global attractor and find bounds for its dimension.

We study a new kind of asymptotic behaviour near $t=\mathrm{0}$ for the nonautonomous system of two linear differential equations: $\mathbf{x}\mathrm{\text{'}}(t)=A(t)\mathbf{x}(t)$, $t\in (\mathrm{0},t_{\mathrm{0}}]$, where the matrix-valued function $A=A(t)$ has a kind of singularity at $t=\mathrm{0}$. It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that $\parallel \mathbf{x}(t){\parallel }_{\mathrm{2}}\to \mathrm{0}$ as $t\to \mathrm{0}$ and the length of the solution curve of $\mathbf{x}$ is finite (resp., infinite) for every $\mathbf{x}\ne \mathbf{0}$. It is characterized in terms of certain asymptotic behaviour of the eigenvalues of $A(t)$ near $t=\mathrm{0}$. Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at $t=\mathrm{0}$.

This paper studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established.

We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results.

We study the asymptotic behavior at small diffusivity of the solutions,$u^{\epsilon }$, to a convection-diffusion equation in a rectangular domain $\Omega$. The diffusiveequation is supplemented with a Dirichlet boundary condition, which is smoothalong the edges and continuous at the corners. To resolve the discrepancy, on $\partial \Omega$, between $u^{\epsilon }$ and the corresponding limit solution, $u^0$, we propose asymptotic expansionsof $u^{\epsilon }$ at any arbitrary, but fixed, order. In order to manage some singulareffects near the four corners of $\Omega$, the so-called elliptic and ordinary corner correctorsare added in the asymptotic expansions as well as the parabolic and classicalboundary layer functions. Then, performing the energy estimates on the differenceof $u^{\epsilon }$ and the proposed expansions, the validity of our asymptotic expansions isestablished in suitable Sobolev spaces.

Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the beam grows up to ∞ as $t\to \infty$ under some assumptions on the forcing term. Our approach is to reduce the multidimensional problems to one-dimensional problems for fourth-order ordinary differential inequalities.

We derive some simple sufficient conditions on the amplitude $a(x)$, the phase $\phi (x),$ and the instantaneous frequency $\omega (x)$ such that the so-called chirp function $y(x)=a(x)S(\phi (x))$ is fractal oscillatory near a point $x=x_0$, where ${\phi }^{\prime }(x)=\omega (x)$ and $S=S(t)$ is a periodic function on $ℝ$. It means that $y(x)$ oscillates near $x=x_0$, and its graph $\mathrm{\Gamma }(y)$ is a fractal curve in ${ℝ}^2$ such that its box-counting dimension equals a prescribed real number $s\in [1,2)$ and the $s$-dimensional upper and lower Minkowski contents of $\mathrm{\Gamma }(y)$ are strictly positive and finite. It numerically determines the order of concentration of oscillations of $y(x)$ near $x=x_0$. Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions.