In this note we propose an exact simulation algorithm for the solution of (1)$$\mathrm{d}{X}_t=\mathrm{d}{W}_t+\overline{b}\left(X_t\right)\mathrm{d}t,X_0=x,$$d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where $\overline{b}$b̅is a smooth real function except at point 0 where $\overline{b}(0+)\ne \overline{b}(0-)$b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)$$\mathrm{d}{X}_t^{\beta }=\mathrm{d}{W}_t+\overline{b}\left(X_t^{\beta }\right)\mathrm{d}t+\beta \mathrm{d}{L}_t^0\left(X^{\beta }\right),X_0=x$$d X t β = d W t + b̅ ( X t β ) d t + β d L t 0 ( X β ) ,   X 0 = x towardsX solution of (1) as β ≠ 0 tends to 0. In this note, we show that this convergence...

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...

The example is constructed of the C1-smooth skew product of interval maps possessing the one-dimensional ramified continuum (containing no arcs homeomorphic to the circle) with an infinite set of ramification points as the global attractor.

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.

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$$$\left\{\begin{array}{c}{u}^{}\hfill \end{array}\right.$$

We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $$\begin{array}{cc}& u^{\text{'}\text{'}}+Au-\Delta u^{\text{'}}+{\left|v\right|}^{\rho +2}{\left|u\right|}^{\rho }u=f_1\hfill \\ & v^{\text{'}\text{'}}+Av-\Delta v^{\text{'}}+{\left|u\right|}^{\rho +2}{\left|v\right|}^{\rho }v=f_2*\hfill \end{array}$$ where $A$ is the pseudo-Laplacian operator.

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 $L^p(0,\infty )(2\le p\le \infty )$.