Bifurcation of fixed points from a manifold of trivial fixed points in the infinite-dimensional case

We obtain a necessary as well as a sufficient condition for the existence of bifurcation points of a coincidence equation, and, in particular, of a parametrized fixed point problem. In both cases the trivial solutions are assumed to form a finite-dimensional submanifold of a Banach manifold. An application is given to a delay differential equation on a manifold: we detect periodic solutions that rotate close to an equilibrium point. To Albrecht Dold and Edward Fadell, superb mathematicians and first rate friends     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

On the periodic solutions of certain class of seventh-order differential equations

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

Sur la suite de polynômes orthogonaux associée à la formeu=δ c +λ(x−c) −1 L

We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

     

Existence of periodic solutions for the Lotka-Volterra type systems

In this paper we prove the existence of nonstationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the S¹-degree due to Dylawerski et al. [G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz, An S¹-equivariant degree and the Fuller index, Ann. Polon. Math. 63 (1991) 243-280].     

Almost periodic weak solutions to forced pendulum type equations without friction

Summary We obtain sufficient conditions for the existence of almost periodic weak solutions to equations of the frictionless pendulum type with forcing.     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

Extrapolation with spline-collocation methods for two-point boundary-value problems I: Proposals and justifications

Asymptotic expansions for the error in some spline interpolation schemes are used to derive asymptotic expansions for the truncation errors in some spline-collocation methods for two-point boundary-value problems. This raises the possibility of using Richardson extrapolation or iterated deferred corrections to develop efficient high-order algorithms based on low-order collocation in analogy with similar codes based on low-order finite difference methods; some specific such procedures are proposed.This research was supported in part by the United States Office of Naval Research under Contract N00014-67-A-0126-0015.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Hankel Operators on Weighted Fock Spaces

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations.     

Convergence results for the MATLAB ode23 routine

We prove convergence results on finite time intervals, as the user-defined tolerance τ→0, for a class of adaptive timestepping ODE solvers that includes the ode23 routine supplied in MATLAB Version 4.2. In contrast to existing theories, these convergence results hold with error constants that are uniform in the neighbourhood of equilibria; such uniformity is crucial for the derivation of results concerning the numerical approximation of dynamical systems. For linear problems the error estimates are uniform on compact sets of initial data. The analysis relies upon the identification of explicit embedded Runge-Kutta pairs for which all but the leading order terms of the expansion of the local error estimate areO(∥f(u∥)²). This work was partially supported by NSF Grant DMS-95-04879.     

Order conditions for canonical Runge-Kutta-Nyström methods

We are concerned with Runge-Kutta-Nyström methods for the integration of second order systems of the special formd²y/dt²=f(y). If the functionf is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nyström formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nyström formulae.This research has been supported by Junta de Castilla y León under project 1031-89 and by Dirección General de Investigación Científica y Técnica under project PB89-0351.     

Analysis of the dynamics of local error control via a piecewise continuous residual

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.     

Perturbation of Dirichlet forms by measures

Perturbations of a Dirichlet form by measures are studied. The perturbed form –_–+₊ is defined for _– in a suitable Kato class and ₊ absolutely continuous with respect to capacity. L _p-properties of the corresponding semigroups are derived by approximating _– by functions. For treating ₊, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L _p -L _q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ₁ the same is shown to be true for the perturbed semigroup, for a large class of measures.