In this paper pencils of partial differential operators depending polynomially on a complex parameter and corresponding boundary value problems with general boundary conditions are studied. We define a concept of ellipticity for such problems (for which the parameter-dependent symbol in general is not quasi-homogeneous) in terms of the Newton polygon and introduce related parameter-dependent norms. It is shown that this type of ellipticity leads to unique solvability of the boundary value problem and to two-sided a priori estimates for the solution.

The paper is devoted to studying a class of strongly hyperbolic systems of the first order. We show that if the characteristic roots of the full symbol are outside an open strip containing the real axis, then the homogeneous system possesses an exponential dichotomy and the inhomogeneous system is solvable in the space of time-bounded and almost periodic functions. We also discuss some results on the behavior of solutions for nonlinear equations in the neighborhood of a stationary point.

We consider boundary value problems for mixed-order systems of partial differential operators which depend on a complex parameter but which are not parameter-elliptic in the sense of Agmon and Agranovich-Vishik. Such systems are closely related to the theory of singularly perturbed problems. Under the condition of so-called weak parameter-ellipticity it is possible to construct the formal asymptotic solution which shows, in particular, the existence of boundary layers.

We consider high-order hyperbolic equations with nearly constant coefficients. The main condition imposed on the equation in question is that its symbol regarded as a polynomial in the variable dual to time does not vanish in an open strip containing the real axis. We discuss the asymptotic stability and the exponential dichotomy for homogeneous equations and the solvability in spaces of time-bounded, periodic, and almost periodic functions for inhomogeneous equations. We also state a result on linearization of the phase portrait for semilinear equations in the neighborhood of a stationary point.