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.

We consider the boundary value problem involving the one dimensional $p$-Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.

We study the existence and positivity of solutions of a highly nonlinear periodic differential equation. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ a modification of Krasnoselskii’s fixed point theorem introduced by T. A. Burton ([4], Theorem 3) to show the existence and positivity of solutions of the equation.

The system of nonlinear differential equations $$x^{\text{'}}+p_1\left(t\right)x^{{\alpha }_1}+q_1\left(t\right)y^{{\beta }_1}=0,y^{\text{'}}+p_2\left(t\right)x^{{\alpha }_2}+q_2\left(t\right)y^{{\beta }_2}=0,A$$ is under consideration, where ${\alpha }_i$ and ${\beta }_i$ are positive constants and $p_i\left(t\right)$ and $q_i\left(t\right)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $\left(x\right(t),y(t\left)\right)$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t\to \infty$, which can be...

The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2n+1)}=f(t,u,u^{\text{'}},...,u^{\left(2n\right)})$, where $f:ℝ\times {ℝ}^{2n+1}\to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f(t,x₀,x₁,...,x_{2n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].