The existence of the Hopf bifurcation for parabolic functional equations with delay of maximum order in spatial derivatives is proved. An application to an integrodifferential equation with a singular kernel is given.

Necessary and sufficient conditions have been found to force all solutions of the equation $${\left(r\left(t\right)y^{\text{'}}\left(t\right)\right)}^{(n-1)}+a\left(t\right)h\left(y\left(g\left(t\right)\right)\right)=f\left(t\right),$$ to behave in peculiar ways. These results are then extended to the elliptic equation $${\left|x\right|}^{p-1}\Delta y\left(\right|x\left|\right)+a\left(\right|x\left|\right)h\left(y\right(g\left(\right|x\left|\right)\left)\right)=f\left(\right|x\left|\right)$$ where $\Delta$ is the Laplace operator and $p\ge 3$ is an integer.

Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\overline{x}=\varphi \left(x\right)$, $\overline{y}=L\left(x\right)y$, $\overline{z}=M\left(x\right)z,...$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\text{'}}=f(x,y,z,z^{\text{'}})$) and certain differential equations with deviation (if $z=y\left(\xi \right(x\left)\right)$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the...

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.

A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for...

We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation $$u^{\text{'}\text{'}}\left(t\right)=p_0\left(t\right)u\left(t\right)+{\int }_0^{\omega }p(t,s)u\left(\tau (t,s)\right)\mathrm{d}s+q\left(t\right),$$ and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.

We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Our main intention in this regard is to expose precise mathematical conditions for critical paths to be minimum solutions in a variety of situations of interest in Physics. Our claim is that, with a few elementary techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models...