Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

This paper is concerned with the oscillatory properties of even order advanced type dynamic equation with mixed nonlinearities of the form $$\bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t) \bigr) \bigr]^\Delta+ p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr) \bigr) +\sum_{i=1}^kp_i(t) \varPhi_{\alpha_i} \bigl(x\bigl(\delta(t)\bigr) \bigr)=0 $$ on an arbitrary time scale $\mathbb{T}$ , where Φ _?(u)=|u|^??1 u. We present some new oscillation criteria for the equation by introducing parameter functions, establishing a new lemma, using a Hardy-Littlewood-Pólya inequality and an arithmetic-geometric mean inequality and developing a generalized Riccati technique. Our results extend and supplement some known results in the literature. Several examples are given to illustrate our main results.     

Comparison and oscillatory behavior for certain second order nonlinear dynamic equations

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale $\mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $\mathbb{T}$ . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

On Dynamical Adjoint Functor

We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules. As an illustration, we give explicit $U\bigl(\mathfrak{sl}(n)\bigr)$ - and $U_\hbar\bigl(\mathfrak{sl}(n)\bigr)$ -invariant star product on projective spaces.     

An Integral Test on Time-Dependent Local Extinction for Super-coalescing Brownian Motion with Lebesgue Initial Measure

This paper concerns the almost sure time-dependent local extinction behavior for super-coalescing Brownian motion X with (1+β)-stable branching and Lebesgue initial measure on ?. We first give a representation of X using excursions of a continuous-state branching process and Arratia’s coalescing Brownian flow. For any nonnegative, nondecreasing, and right-continuous function g, let $$\tau:=\sup\bigl\{t\geq0: X_t\bigl(\bigl[-g(t),g(t)\bigr]\bigr )>0 \bigr \}.$$ We prove that ?{τ=∞}=0 or 1 according as the integral $\int_{1}^{\infty}\! g(t)t^{-1-1/\beta} dt$ is finite or infinite.     

Global Attractor for a Generalized Discrete Nonlinear Schrödinger Equation

A generalized discrete nonlinear Schrödinger equation $$i\dot{u}_n(t)+\sum_{m=-\infty}^{+\infty} J(n-m)u_m(t)+g\bigl(u_n(t)\bigr)+i\gamma u_n(t)=f_n,\quad n\in\mathbb{Z}, $$ with long-range interactions in weighted spaces \(\ell_{\mathbf{{q}}}^{2}\) is considered. Under suitable assumptions on the coupling constants J(m), the damping γ and the weight \(\mathbf{{q}}=(q_{n})_{n\in \mathbb{Z}}\) , the existence of a global attractor is proved.     

Fractional eigenvalues

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$ for large values of $p$ and derive the limit equation as $p\rightarrow \infty $ . Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.     

Interior-point algorithms for P_{*}(\kappa )-LCP based on a new class of kernel functions

In this paper, we propose interior-point algorithms for $P_* (\kappa )$ -linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functions. We show that if a strictly feasible starting point is available, then the new algorithm has $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n}\log n \log \frac{n\mu ^0}{\epsilon }\bigr )$ and $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n} \log \frac{n\mu ^0}{\epsilon }\bigr )$ iteration complexity for large- and small-update methods, respectively. These are the best known complexity results for such methods.     

Existence of triple symmetric positive solutions for four-point boundary-value problem with one-dimensional p-Laplacian

In this paper, we consider the four-point boundary value problem for one-dimensional p-Laplacian $$\bigl(\phi_{p}(u'(t))\bigr)'+q(t)f\bigl(t,u(t),u'(t)\bigr)=0,\quad t\in(0,1),$$ subject to the boundary conditions $$u(0)-\beta u'(\xi)=0,\qquad u(1)+\beta u'(\eta)=0,$$ where φ _p (s)=|s| ^p?2 s. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.     

On the Uniqueness of the Kendall Generalized Convolution

Kendall (Foundations of a theory of random sets, in Harding, E.F., Kendall, D.G. (eds.), pp. 322?C376, Willey, New York, 1974) showed that the operation $\diamond_{1}\colon \mathcal{P}_{+}^{2}\rightarrow \mathcal{P}_{+}$ given by $$\delta_x\diamond_1\delta_1=x\pi_2+(1-x)\delta_1,$$ where x??[0,1] and ?? _?? is the Pareto distribution with the density function ?? s ^????1 on the set [1,??), defines a generalized convolution on ?₊. Kucharczak and Urbanik (Quasi-stable functions, Bull. Pol. Acad. Sci., Math. 22(3):263?C268, 1974) noticed that also the following operation $$\delta_x\diamond_{\alpha}\delta_1=x^{\alpha}\pi_{2\alpha}+\bigl(1-x^{\alpha}\bigr)\delta_1$$ defines generalized convolutions on ?₊. In this paper, we show that ? _?? convolutions are the only possible convolutions defined by the convex linear combination of two fixed measures. To be precise, we show that if ? :?²??? is a generalized convolution defined by $$\delta_x\diamond \delta_1=p(x)\lambda_1+\bigl(1-p(x)\bigr)\lambda_2,$$ for some fixed probability measures ?? ₁,?? ₂ and some continuous function p :[0,1]??[0,1], p(0)=0=1?p(1), then there exists an ??>0 such that p(x)=x ^?? , ?=? _?? , ?? ₁=?? _2?? and ?? ₂=?? ₁. We present a similar result also for the corresponding weak generalized convolution.     

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Diophantine properties of IETs and general systems: quantitative proximality and connectivity

Three properties of dynamical systems (recurrence, connectivity and proximality) are quantified by introducing and studying the gauges (measurable functions) corresponding to each of these properties. The properties of the proximality gauge are related to the results in the active field of shrinking targets. The emphasis in the present paper is on the IETs (interval exchange transformations) $( \mathcal {I},T)$ , $\mathcal {I}=[0,1)$ . In particular, we prove that if an IET T is ergodic (relative to the Lebesgue measure λ), then the equality A1 $$ \liminf_{n\to\infty} \, n\, \bigl|T^n(x)-y \bigr|=0 $$ holds for λ×λ-a.a. $(x,y)\in \mathcal {I}^{2}$ . The ergodicity assumption is essential: the result does not extend to all minimal IETs. Also, the factor? n? in (A1) is optimal (e.g., it cannot be replaced by n?ln(ln(lnn))). On the other hand, for Lebesgue almost all 3-IETs $( \mathcal {I},T)$ we prove that for all ?>0 A2 $$ \liminf_{n\to\infty} \, n^ \epsilon \bigl |T^n(x)-T^n(y)\bigr| = \infty,\quad\text{for Lebesgue a.a.} \ (x,y)\in \mathcal {I}^2. $$ This should be contrasted with the equality lim?inf _n→∞?|T ⁿ (x)?T ⁿ (y)|=0, for a.a. $(x,y)\in \mathcal {I}^{2}$ , which holds since $( \mathcal {I}^{2}, T\times T)$ is ergodic (because generic 3-IETs $( \mathcal {I},T)$ are weakly mixing). We introduce the notion of τ-entropy of an IET which is related to obtaining estimates of type (A2). We also prove that no 3-IET is strongly topologically mixing.     

Extended Euler congruence

The matricial Euler congruence $\mathop{\mathrm{Tr}}\bigl(A^{p^{n}}\bigr)\equiv\mathop{\mathrm{Tr}}\bigl(A^{p^{n-1}}\bigr)$ modulo p ⁿ , previously announced in  Arnold, Japanese J. Math. 1(1), 1–24, 2006 for A∈M _N (?), is given a proof based on extending it to the ring of Witt vectors of length n.     

Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C ^1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D _?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C ¹ we can improve this to full regularity provided additional structure conditions are satisfied.     

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X _t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ(λ)=?αλ+βλ ²+∫_(0,+∞)(e ^?λx ?1+λx)n(dx), where α=?ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫_(0,+∞) x ² n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W _t =e ^?αt ∥X _t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W _∞ such that W _t →W _∞, $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$ . Moreover, W ^? and G ₁(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$ . Moreover W ^? and G ₂(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Mi?o? in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Mi?o?. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya’s 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960’s and early 1970’s.     

Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms

This paper deals with the damped superlinear oscillator $$x'' + a(t)\phi_p\bigl(x' \bigr) + b(t)\phi_q\bigl(x'\bigr)+ \omega^2x = 0, $$ where a(t) and b(t) are continuous and nonnegative for t≥0; p and q are real numbers greater than or equal to 2; ? _r (x′)=|x′| ^r?2 x′. This equation is a generalization of nonlinear ship rolling motion with Froude’s expression, which is very familiar in marine engineering, ocean engineering and so on. Our concern is to establish a necessary and sufficient condition for the equilibrium to be globally asymptotically stable. In particular, the effect of the damping coefficients a(t), b(t) and the nonlinear functions ? _p (x′), ? _q (x′) on the global asymptotic stability is discussed. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation $$u' + \omega^{p-2}a(t)\phi_p(u) + \omega^{q-2}b(t)\phi_q(u) + 1 = 0 $$ is divergent or convergent. In addition, explicit sufficient conditions and explicit necessary conditions are given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, some examples are included to illustrate our results. Finally, our results are extended to be applied to a more complicated model.     

Stability in nonlinear neutral integro-differential equations with variable delay using fixed point theory

The nonlinear neutral integro-differential equation $$\frac{d}{dt}x ( t ) =-\int_{t-\tau ( t ) }^{t}a ( t,s ) g \bigl( x ( s ) \bigr) ds+\frac{d}{dt}G \bigl( t,x \bigl( t-\tau ( t ) \bigr) \bigr) , $$ with variable delay τ(t)≥0 is investigated. We find suitable conditions for τ, a, g and G so that for a given continuous initial function ψ a mapping P for the above equation can be defined on a carefully chosen complete metric space $S_{\psi }^{0}$ in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Burton (Proc. Am. Math. Soc. 132:3679–3687, 2004), Becker and Burton (Proc. R. Soc. Edinb., A 136:245–275, 2006) and Jin and Luo (Comput. Math. Appl. 57:1080–1088, 2009).     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.