We prove existence results for solutions of periodic boundary value problems concerning the $n$th-order differential equation with $p$-Laplacian ${\left[\phi \left(x^{\left(n-1\right)}\left(t\right)\right)\right]}^{\text{'}}=f\left(t,x\left(t\right),x\text{'}\left(t\right),\mathrm{...},x^{\left(n-1\right)}\left(t\right)\right)$ and the boundary value conditions $x^{\left(i\right)}\left(0\right)=x^{\left(i\right)}\left(T\right)$, $i=0,\mathrm{...},n-1$. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that $f$ may be a polynomial and the degree of some variables among $x_0,x_1,\mathrm{...},x_{n-1}$ in the function $f\left(t,x_0,x_1,\mathrm{...},x_{n-1}\right)$ is allowed to be greater than $1$.

We deal with a mathematical model for a four-component chemical reaction-diffusion process. The model is described by a system of strongly coupled reaction-diffusion equations with different diffusion rates. The existence of the global solution of this reaction-diffusion system in unbounded domain is proved by using semigroup theory and estimates on the growth of solutions.

Exploiting minmax characterizations for nonlinear and nonoverdamped eigenvalue problems, we prove the existence of a countable set of eigenvalues converging to $\infty$ and inclusion theorems for a rational spectral problem governing mechanical vibrations of a tube bundle immersed in an incompressible viscous fluid. The paper demonstrates that the variational characterization of eigenvalues is a powerful tool for studying nonoverdamped eigenproblems, and that the appropriate enumeration of the eigenvalues is of predominant importance, whereas the natural ordering of the eigenvalues may yield false conclusions.

We extend, sharpen, or give independent proofs of classical maximum principles. We also concentrate on maximum principles for equations of higher order. All proofs (except for one) are derived via comparison principles. The two parts maybe read independently, but the whole paper is not self-contained.

The nonlinear parabolic equations describing motion of incompressible media are investigated. The rheological equations of most general type are considered. The deviator of the stress tensor is expressed as a nonlinear continuous positive definite operator applied to the rate of strain tensor. The global-in-time estimate of solution of initial boundary value problem is obtained. This estimate is valid for systems of equations of any non-Newtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possible to prove the existence of global-in-time solutions of two-dimensional initial boundary value problems for generalized linear viscoelastic liquids, that is, for liquids with linear integral rheological equation, and for third-grade liquids.

The generalized resolvents for a certain class of perturbed symmetric operators with equal and finite deficiency indices are investigated. Using the Weinstein-Aronszajn formula, we give a classification of the spectrum.