We study asymptotic properties of solutions for a system of second differential equations with $p$-Laplacian. The main purpose is to investigate lower estimates of singular solutions of second order differential equations with $p$-Laplacian ${\left(A\left(t\right){\Phi }_p\left(y^{\text{'}}\right)\right)}^{\text{'}}+B\left(t\right)g\left(y^{\text{'}}\right)+R\left(t\right)f\left(y\right)=e\left(t\right)$. Furthermore, we obtain results for a scalar equation.

By using Mawhin’s continuation theorem, the existence of even solutions with minimum positive period for a class of higher order nonlinear Duffing differential equations is studied.

The work characterizes when is the equation $y^{\left(n\right)}+\mu p\left(x\right)y=0$ eventually disconjugate for every value of $\mu$ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $q$, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $y$ such that $y^{\left(i\right)}>0$, $i=0,...,q-1$, ${(-1)}^{i-q}{y}^{\left(i\right)}>0$, $i=q,...,n$. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems$$\left\{\begin{array}{cc}-{\left[\phi \left(u^{\text{'}}\right)\right]}^{\text{'}}=\lambda u^p\left(1-{\displaystyle \frac{u}{N}}\right)\hfill & \text{in}(-L,L),\hfill \\ u(-L)=u\left(L\right)=0,\hfill \end{array}\right.$$ where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi \left(u\right)$ is either $\phi \left(u\right)=u$ or $\phi \left(u\right)=u/\sqrt{1-u^2}$. We prove that the corresponding bifurcation curve is $\subset$-shape. Thus, the exact multiplicity of positive solutions can be obtained.

Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind $$y^{\text{'}\text{'}}\left(x\right)+\frac{k}{x}{y}^{\text{'}}\left(x\right)+g\left(x\right){\mathrm{e}}^{ny}=0.$$ Then we consider two types of functions $g\left(x\right)$ and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered...

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

We consider the nonlinear Dirichlet problem $$-u^{\text{'}\text{'}}-r\left(x\right){\left|u\right|}^{\sigma }u=\lambda u\text{in}(0,\infty ),u\left(0\right)=0\text{and}\underset{x\to \infty }{lim}u\left(x\right)=0,$$ and develop conditions for the function $r$ such that the considered problem has a positive classical solution. Moreover, we present some results showing that $\lambda =0$ is a bifurcation point in $W^{1,2}(0,\infty )$ and in $L^p(0,\infty )(2\le p\le \infty )$.

Fuzzy cellular neural networks with time-varying delays are considered. Some sufficient conditions for the existence and exponential stability of periodic solutions are obtained by using the continuation theorem based on the coincidence degree and the differential inequality technique. The sufficient conditions are easy to use in pattern recognition and automatic control. Finally, an example is given to show the feasibility and effectiveness of our methods.

We prove the existence of a positive solution to the BVP $${\left(\Phi \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=f(t,u\left(t\right)),u^{\text{'}}\left(0\right)=u\left(1\right)=0,$$ imposing some conditions on Φ and f. In particular, we assume $\Phi \left(t\right)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_{\infty }$ bound for the solution is provided by the $L_{\infty }$ norm of any test function with negative energy.