Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekeland's variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

The Weierstrass necessary condition for a multidimensional basic variational problem due to Hadamard is strengthened.

We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, then

for p > 1 (0 < p < 1), respectively.

In this paper, we provide sufficient conditions which guarantee the uniform stability as well as asymptotic stability of the positive equilibrium for a food limited population model with time delay.

In this paper, we prove the existence of the global smooth solution for the equation of a viscoelastic model with relaxation in time under the only assumption that the C0-norm of the initial data is small, without smallness hypothesis for the C1-norm.

We classify completely integrable holonomic systems of first-order differential equations for one real-valued function by equivalence under the group of point transformations in the sense of Sophus Lie. In order to pursue the classification, we use the notion of one parameter Legendrian unfoldings which induces a special class of divergent diagrams of map germs which are called integral diagrams. Our normal forms are represented by integral diagrams.

We study the large-time behaviour of solutions to the Korteweg-de Vries-Burgers equation with bore-like initial data. This work relies on the methods of Amick, Bona and Schonbeck to obtain sharp rates of temporal decay of certain norms of the solution, thus obtaining an improvement over results of Naumkin and Shishmarev.

The paper discusses the asymptotic behaviour of weak solutions u(t, x), as t → ∞, to the boundary value problem for one-dimensional viscoelastic equations with singular memory. The changes of phase are admitted for the problem. One of our results is that ut(t, ·)⇀0 weakly in L2(0,1) as t → ∞.

Forms of the colour algebra introduced by Domokos and Kövesi-Domokos are studied by relating them to the well-known Cayley–Dickson algebras. Automorphisms groups and derivation algebras of these algebras are also determined.

We show the existence of a timelike periodic trajectory for a time-dependent Lorentzian metric on ℝn × ℝ.

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

It is shown that under the action of a geometric subgroup of and , for a germ f satisfying a certain finiteness condition, given a germ p, if the tangent spaces of f and f + p are equal for all t ∈ [0, 1], then f and f + p are -equivalent.

We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem −Δu + λu = up in antipodal invariant domains including ball domains with Neumann boundary conditions. Here p is the critical Sobolev exponent (N + 2)/(N − 2). The shape of the solutions and the location of the peaks are also studied.

A sequence of lemmas leads to a two-fold characterisation of the syntactic monoid in the title. Some alternatives as well as special cases, in particular when the code consists of a singleton, are considered.

Weighted Poincaré inequalities are established in any bounded domain D in ℝn (n ≧ 2), and their connection with the Minkowski content of ∂D is explored.

Most of the development of shape theory was in the so-called outer shape theory, where the shape of spaces is described with the help of some outside objects.

This paper belongs to the so-called inner shape theory, in which the shape of spaces is described intrinsically without the use of any outside gadgets. We give a description of shape theory that does not need absolute neighbourhood retracts. We prove that the category ℋN whose objects are topological spaces and whose morphisms are proximate homotopy classes of proximate nets is naturally equivalent to the shape category h. The description of the category ℋN for compact metric spaces was given earlier by José M. R. Sanjurjo. We also give three applications of this new approach to shape theory.

A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.

Results are obtained on the existence of positive solutions to the following elliptic system:

in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

In this work we study reaction–diffusion systems in fractional power spaces Xα which are embedded in L∞. We prove that the solution operators T(t) to these problems are globally defined, point dissipative, locally bounded and compact. That ensures the existence of global attractors. We also find a set containing the range of every function in the attractor, providing good estimates on asymptotic concentrations. This is done under very few hypotheses on the reaction term. These hypotheses are natural and easy to verify in many applications. The tools employed are the theory of invariant regions for systems of parabolic partial differential equations, the notion of contracting sets and the variation of constants formula. Several examples are considered to emphasise the applicability of these techniques.