The classical inequality ‖y′‖2≦K‖y‖ ‖y″‖ and its higher order analogues are extended from the Lp spaces to the weighted spaces, for appropriate w.

In this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

Absence of non-trivial square integrable solutions of the Dirac equation

for values of λ in half-rays is proved under local conditions on q, curl b, and the radial derivative of q. The Coulomb potential is admitted. Another result does not contain any growth condition on the radial derivative of q. It states that there are no solutions of integrable square other than the trivial solution if λ ε ℝ\ [−1, 1] and

A Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

Several definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.

The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.

It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.

We study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Here

is bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.

This note presents some integrodifferential inequalities in n-independent variables which are generalizations of the integrodifferential inequalities recently established by Pachpatte in two independent variables.

We obtain an asymptotic estimate for the number of eigenvalues less than λ of an operator of Schrödinger type

defined on an unbounded domain.