Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption

This paper deals with the Cauchy problem u_t − u_xx + u^p = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u₀(x); − ∞ < x < + ∞, where 0 < p < 1 and u₀(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.     

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:

u⁽⁴⁾(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,

where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

Permanence and global stability in a Lotka-Volterra predator-prey system with delays

Consider the permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system:

, with initial conditions

x_i(t) = φ_i(t) ≥ o, t ≤ t₀, and φ_i(t₀) > 0. 1 ≤ i ≤n

. We define x₀(t) = x_n+1(t)≡0 and suppose that φ_i(t), 1 ≤ i ≤ n, are bounded continuous functions on [t₀, + ∞) and γ_i, α_i, c_i > 0,γ_i,j ≥ 0, for all relevant i,j.Extending a technique of Saito, Hara and Ma[1] for n = 2 to the above system for n ≥ 2, we offer sufficient conditions for permanence and global asymptotic stability of the solutions which improve the well-known result of Gopalsamy.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

Uniform persistence for Lotka–Volterra-type delay differential systems

Consider the uniform persistence (permanence) of models governed by the following Lotka–Volterra-type delay differential system:

where each r_i(t) is a nonnegative continuous function on [0,+∞), r_i(t)0, each a_i0 and τ_ij^k(t)t, 1i,jn, 0km.In this paper, we establish sufficient conditions of the uniform persistence and contractivity for solutions (and global asymptotic stability). In particular, we extend the results in Wang and Ma (J. Math. Anal. Appl. 158 (1991) 256) for a predator–prey system and Lu and Takeuchi (Nonlinear Anal. TMA 22 (1994) 847) for a competitive system in the case n=2, to the above system with n2.     

Positive solution to a special singular second-order boundary value problem

Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem

where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.     

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

where τ0, p(t)C¹[0,∞), q(t),q_i(t),e(t)C[0,∞), p(t)>0, . Without assuming that q(t), q_i(t) and e(t) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach

We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption u_t = Δ(u^{σ + 1}) − u^β in Q = R^N × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem u_t = Δ(u^{σ + 1}), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation u_t = div(¦Du¦^σ Du) − u^β with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N.     

Discriminant analysis for locally stationary processes

In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {X_t,T} belongs to one of two categories described by two hypotheses π₁ and π₂. Here T is the length of the observed stretch. These hypotheses specify that {X_t,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π₁ and π₂, respectively. Although Gaussianity of {X_t,T} is not assumed, we use a classification criterion D( f:g), which is an approximation of the Gaussian likelihood ratio for {X_t,T} between π₁ and π₂. Then it is shown that D( f:g) is consistent, i.e., the misclassification probabilities based on D( f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D( f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D( f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D( f:g), we illuminate its infinitesimal behavior. Some numerical studies are given.     

Boundary value problems for second order nonlinear differential equations on infinite intervals

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L²(0,∞) and L²(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established.     

Parameter estimation in linear filtering

Suppose on a probability space (Ω, F, P), a partially observable random process (x_t, y_t), t ≥ 0; is given where only the second component (y_t) is observed. Furthermore assume that (x_t, y_t) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W₁(t)) and (W₂(t)): dx_t=−βx_tdt+dW₁(t), x₀=0, dy_t=αx_tdt+dW₂(t), y₀=0; α, β∞(a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter θ = (α, β). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and θ is a k-dimensional vector.     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

Rank One Perturbations with Infinitesimal Coupling

We consider a positive self-adjoint operator A and formal rank one perturbations B = A + α(φ, ·)φ, where φ ₋₂(A) but φ ₋₁ (A), with _s(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesimal values of α. In a sense we will make precise, every rank one perturbation is one of three forms: (i) φ ₋₁(A), α ; (ii) φ ₋₁, α = ∞; or (iii) the new type we consider here.     

A Chaotic Continuous Map Generates All Probability Distributions

Let ƒ be a continuous map of the compact unit interval I = [0, 1], such that ƒ², the second iterate of ƒ, is topologically transitive in I. If for some x and y in I and any t in I there exists lim(1/n) # {i ≤ n; |ƒⁱ(x) − ƒⁱ(y)| < t} for n → ∞, denote it by φ_xy(t). In the paper we consider the class (ƒ) if all φ_xy. The main results are that (ƒ) is convex and pointwise closed. Using this we show that (ƒ) is always bigger than the class (ƒ) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of ƒ), and prove that there are universal generators of probability distributions, i.e., maps ƒ such that (ƒ) is the class of all non-decreasing functions I I (contrary to this, (ƒ) for no ƒ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of (ƒ) as a measure of the degree of chaos of ƒ.     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.