In this note we demonstrate that a conjecture about a necessary and sufficient condition for the existence of a centre of cubic equations is not true.

We show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

In this work we consider the non-autonomous problem Δu = a(x)um in the unit ball B ⊂ RN, with the boundary condition u|∂B = +∞, and m > 0. Assuming that a is a continuous radial function with a(x) ˜ C0 dist(x, ∂B)−γ as dist(x, ∂B) → 0, for some C0 > 0, γ > 0, we completely determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of m and γ. The case 0 < m ≤ 1, as well as estimates for solutions to the linear problem m = 1, are a significant part of our results.

By assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

We consider a system of functional differential equations

where G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

We consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support, i.e. the permittivity ε = ε(x) and the conductivity σ = σ(x) are functions of x ∊ ℝ3. Existence, uniqueness and regularity results are established for the direct scattering problem. Then, based on existence and uniqueness results for the exterior and interior impedance boundary value problem, a method is presented for solving the inverse scattering problem.

We derive a compatible condition between coefficients and negative exponents, which is optimal for H01-solutions of strongly singular problems. This allows us to establish the existence of H01-solutions for p ≥ 3. Our proof is based on a variational argument.

In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:

$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$

$M:{\open R}_0^ + \to {\open R}^ + $

In this paper we investigate the differential geometry of 1-lightlike submanifolds in anti-de Sitter n-space as an application of the theory of Legendrian singularities. Based on some theory of lightlike submanifolds, we also introduce the notion of 1-lightlike horospherical Gauss curvature, which is important for us to study the singularities of 1-lightlike horospherical hypersurfaces. Moreover, we discuss the related geometric property of 1-lightlike horospherical hypersurfaces in anti-de Sitter n-space.

In this paper we study the quasi-static crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.

The solutions of stochastic differential equations are used to construct Markov processes on the Banach manifold C(S, M) of continuous maps from a compact metric space S into a smooth complete finite dimensional Riemannian manifold M. In the special case where S is a single point the construction gives a large class of diffusion processes on the manifold M, including (under certain curvature conditions) the Brownian motion process.

Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is driven only by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth-order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a 1½-dimensional situation.

We consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.

We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)-superlinear on ℝ+ and (p − 1)-(sub)linear on ℝ−. Using variational methods based on the critical point theory together with truncation techniques, Ekeland's variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

In abstract dynamics the instantaneous reversal of the motion of every moving particle of a system causes the system to move backwards, each particle of it along its old path, and at the same speed as before, when again in the same position. That is to say, in mathematical language, any solution remains a solution when t is changed into – t. In physical dynamics this simple and perfect reversibility fails, on account of forces depending on friction of solids; imperfect fluidity of fluids; imperfect elasticity of solids; inequalities of temperature, and consequent conduction of heat produced by stresses in solids and fluids; imperfect magnetic retentiveness; residual electric polarisation of dielectrics; generation of heat by electric currents induced by motion; diffusion of fluids, solution of solids in fluids, and other chemical changes; and absorption of radiant heat and light. Consideration of these agencies in connection with the all-pervading law of the conservation of energy proved for them by Joule, led me twenty-three years ago to the theory of the dissipation of energy, which I communicated first to the Royal Society of Edinburgh in 1852, in a paper entitled “On a Universal Tendency in Nature to the Dissipation of Mechanical Energy.’

Although the physical problem which originally led to the Mathieu equation

is now well over sixty years old, and although other problems studied in more recent years, such as the scattering of electric waves by an elliptic cylinder, depend essentially on the same equation, no thorough investigation of detail has hitherto been possible owing to the lack of tables of the functions defined by the equation. In fact, until 1924 all attempts to reduce the functions to a form suitable for calculation had failed. In that year a successful method of attack was discovered by the present writer, who then undertook the computation of the tables which are here published.

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

We study steady periodic water waves with variable vorticity and density, where we allow for stagnation points in the flow, and where we admit the capillarity effects of surface tension. Using global bifurcation theory, we extend a local curve of non-trivial solutions of the governing equations to a global continuum of solutions. Furthermore, we obtain a description of the behaviour of the solutions along this continuum.

By using the moving plane method combined with integral inequalities and Hardy's inequality, some new Liouville-type theorems for semilinear polyharmonic equations in $\mathbb{R}^N$ and in $\mathbb{R}^N_+$ are proved.

We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.