It is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.

We study the quasistatic behaviour of an elastoviscoplastic material submitted to a cyclic load. We prove the weak convergence of the solution of the stress evolution problem towards a periodical one when t → + ∞.

A necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

Structure conditions on a strongly nonlinear operator A(u) are given, under whichthe initial-Dirichlet-boundary value problem for has weak solutions.

In the context of reliability theory, two definitions are given for coherent functions of n variables, where both function and variables can take any of l possible levels. The enumeration problem for such functions is discussed and several recursive bounds are derived. In the case of l = 2 (the Dedekind problem) a recursive upper bound is derived which is better than the previous best explicit upper bound forn < 15, and also provides a systematic improvement on this bound for larger values of n.

We obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one-dimensional motion of a non-newtonian fluid. Precise estimates for the support of these solutions are obtained, and the optimality of our hypotheses is discussed.

When solving practically the neutral type equations the derivatives are replaced by finite differences while the members of integral type are replaced by quadrature formulae. The paper deals with the convergence of a natural class of methods applied to the Cauchy problem for functional-differential neutral type equations. It is not obligatory for the approximated operators to be compact.

Absolutely square integrable solutions are determined for the equation = λ y where the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions as x approaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

The paper presents sufficient conditions on the coefficients of second and fourth order differential equations to ensure that there exists at least one pair of conjugate points on an interval (a, b), −∞≦ a <b ≦ ∞. Oscillation criteria related to the equation (p(x)y″)″ + q(x)y = 0, 0 < x < ∞, are proved with no sign restrictions on q(x).

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.

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

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.

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.

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.