In a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

A generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

The inequality considered in this paper is

where N is the real-valued symmetric differential expression defined by

General properties of this inequality are considered which result in giving an alternative account of a previously considered inequality

to which (*) reduces in the case p = q = 0, r = 1.

Inequality (*) is also an extension of the inequality

as given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

A new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

This paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined by

where r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

This paper obtains, under certain general conditions on the coefficient q, a best-possible upper bound on the real parameter λ for the differential equation

to have a non-trivial solution in the integrable-square space L2 (a, ∞).

Let Vrs, s = 1 …, K be Hermitian operators on Hilbert spaces Hr, r = 1, …, k. For x = x1⊗…⊗xk ∊ H1⊗…⊗Hk we define Δx by the formal determinantal expansion Δx = ⊗det{Vrsxr}. Δ is extended to all of H by linearity and continuity. The paper presents results concerning positivity properties of Δ on decomposable tensors.

It is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.

Uniformly valid asymptotic approximations are presented for solutions of the angular equations associated with the problem of diffraction by a plane angular sector. Error estimates are provided for all approximations. The asymptotic variable is related to the number of zeros of the solutions of the angular equations and expressions for the eigenvalues of the equations are presented in decoupled form.

Formulae are derived for the numbers of completely 0-simple and completely simple semigroups with m ℒ-classes, n ℜ-classes and underlying finite group G.

In this paper we study the stabilization problem for non autonomous control processes in Hilbert spaces. We prove that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite.

An application to control processes with delays in control is presented.

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.

Let S be a Clifford semigroup with identity. The weak containment question is posed for S, and answered affirmatively when each of the maximal groups Se in S is amenable. The amenability of S itself is characterised in terms of PL(S), the set of normalised positive definite functions on S arising from the left regular representation of S. A type of mean associated with PL(S) and satisfying a condition weaker than left invariance is introduced.