Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.     

Invariant manifolds of admissible classes for semi-linear evolution equations

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to certain classes of admissible function spaces.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

On the cycle structure of certain classes of nonlinear shift registers

When m = qt, g(x_t+1, x_2t+1,…, x_(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x₁, x₂,…, x_m) = x₁ + g(x_t+1, x_2t+1,…, x_(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).     

Periodic solutions of semilinear equations of evolution of compact type

The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:

t_∂u−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)     

On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

A direct numerical procedure for the Cauchy problem for the heat equation

For the Cauchy problem, u_t = u_xx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, u_x(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υ_t = υ_xx, 0 < x, 0 < t ? T, υ_x(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

Three positive solutions to a discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ³x(t − k) = f(x(t)), x(a) = Δx(t₂) = Δ²x(b + 1) = 0. Under various assumptions on f and the integers a, t₂, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

Remarks on semilinear partial functional-differential equations with infinite delay

In this paper we will be concerned with the abstract semilinear equation x(Φ)(t) = Ax(Φ)(t) + F(t, x_t(Φ)), t ? τ, x_τ(Φ) = Φ, where A represents the infinitesimal generator of a linear C₀-semigroup in a Banach space and x_～(Φ) denotes the segment of the solution x due to the initial function Φ taken from ?∞ up to t. Conditions for existence, continuous dependence, and regularity of solutions are given which are independent of the topology of the phase space, as well as conditions which guarantee the compactness of bounded orbits, even if exp(At) is not compact for t > 0.