Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

Approximate Controllability of Second-Order Stochastic Distributed Implicit Functional Differential Systems with Infinite Delay

In this paper, sufficient conditions for the approximate controllability of the following stochastic semilinear abstract functional differential equations with infinite delay are established $$\begin{array}{@{}l@{}}d\bigl[x^{\prime}(t)-g(t,x_{t})\bigr]=\bigl[Ax(t)+f(t,x_{t})+Bu(t)\bigr]dt+G(t,x_{t})dW(t),\\\noalign{\vskip3pt}\quad \mbox{a.e on}\ t\in J:=[0,b],\\\noalign{\vskip3pt}x_{0}=\varphi\in {\mathfrak{B}},\\\noalign{\vskip3pt}x^{\prime}(0)=\psi \in H,\end{array}$$ where the state x(t)∈H,x _t belongs to phase space ${\mathfrak{B}}$ and the control u(t)∈L ₂ ^? (J,U), in which H,U are separable Hilbert spaces and d is the stochastic differentiation. The results are worked out based on the comparison of the associated linear systems. An application to the stochastic nonlinear wave equation with infinite delay is given.     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Boundary-value problems for a nonlinear hyperbolic equation with variable coefficients and the Lévy Laplacian

For a nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian Δ _L , $$\beta \left( {\sqrt 2 \left\| x \right\|_H \frac{{\partial U(t,x)}} {{\partial t}}} \right)\frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} + \alpha (U(t,x))\left[ {\frac{{\partial U(t,x)}} {{\partial t}}} \right]^2 = \Delta _L U(t,x),$$ we present algorithms for the solution of the boundary-value problem U(0, x) = u ₀, U(t, 0) = u ₁ and the exterior boundary-value problem U(0, x) = v ₀, \(\left. {U(t,x)} \right|_{\Gamma = v_1 }\) , \(\lim _{\left\| x \right\|_{H \to \infty } } \left. {U(t,x) = v_2 } \right|\) for the class of Shilov functions depending on the parameter t.     

Strong approximation by Fourier series and differentiability properties of functions

ПустьΦ —N-функция Юнг а со свойствами $$\Phi (x)x^{ - 1} \downarrow 0, \exists \alpha > 1 \Phi (x)x^{ - \alpha } \uparrow (x \downarrow 0),$$ илиΦ(х)=х, {λ_k} — положи тельная, неубывающая последовательность и $$S_\Phi \{ \lambda \} = \left\{ {f:\left\| {\sum\limits_{k = 0}^\infty \Phi (\lambda _k |f - s_k |)} \right\|_\infty< \infty } \right\}.$$ В работе найдены необ ходимые и достаточны е условия для вложений $$S_\Phi \{ \lambda \} \subset W^r F(r \geqq 0),$$ , гдеF=C, L ^∞, Lip α (0<α≦1). С этой то чки зрения рассматриваются и др угие классы (например, \(W^r H^\omega ,\tilde W^r F\) ).     

Riemann—Lebesgue theorem

Пусть?(x) — ограниченн ая функция на отрезке [0,1] и ее функция распределен ияΦ(t) удовлетворяет услов ию $$\Phi \left( t \right) + \Phi \left( { - t} \right) = 1.$$ Еслиf(x) — конечная поч ти всюду функция, то дл яF _n (t) — функции распределе ния произведенияf(x)?(nx) — вы полнены соотношения и В частности, еслиf(x) — и нтегрируемая функци я, то из (1) следует, что $$\mathop {\lim }\limits_{n \to \infty } \mathop \smallint \limits_0^1 f\left( x \right)\varphi \left( {nx} \right)dx = 0 $$      

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Algebraic equations on the adèlic closure of a Drinfeld module

Let k be a field of positive characteristic and K = k(V) a function field of a variety V over k and let A _K be the ring of adèles of K with respect to the places on K corresponding to the divisors on V. Given a Drinfeld module $\Phi :\mathbb{F}[t] \to End_K (\mathbb{G}_a )$ over K and a positive integer g we regard both K ^g and A _K ^g as $\Phi \left( {\mathbb{F}_p [t]} \right)$ -modules under the diagonal action induced by Φ. For Γ ? K ^g a finitely generated $\Phi \left( {\mathbb{F}_p [t]} \right)$ -submodule and an affine subvariety $X \subseteq \mathbb{G}_a^g$ defined over K, we study the intersection of X(A _K ), the adèlic points of X, with $\bar \Gamma$ , the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

On the mean value of certain functions connected with the convergence of orthogonal series

В статье рассматрива ются множестваQ ⁿ , 1≦п<∞, ортонормированных с истемΦ={φ _i (x)} _i ⁿ =1, состоящих из функций, постоянных на интервалах \(\left( {\frac{{j - 1}}{n}, \frac{j}{n}} \right)\) , 1 ≦j ≦j ≦п. НаQ ⁿ естественно перенос ится с группы ортогон альных матриц порядкаn мера Хаара. Изучается поведение наQ ⁿ функци и $$S(\Phi ) = \mathop {\sup }\limits_{\mathop \sum \limits_{i = 1}^n y_i^2 = 1} (\int\limits_0^1 {\mathop {sup}\limits_{1 \leqq r \leqq n} } (\mathop \sum \limits_{i = 1}^n y_i \varphi (x))^2 dx)^{1/2} $$ . Доказывается, что приt > 0 иn=1,2,... $$\mu \{ \Phi \in Q^n :s(\Phi ) \geqq t\} \leqq (Ce^{ - \gamma t^2 } )^n $$ .     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Functional inclusion and controllability of nonlinear neutral functional differential systems

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx _t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t ₀,t _I ] andC=C([-h,0],E ⁿ ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values \(x_\omega = \phi _0 ,x_{t_1 } = \phi _1\) , where φ₀, φ₁ belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Some remarks on the existence of optimal controls for quasilinear systems

Let a quasilinear control system having the state space \(\bar X \subseteq R^n \) be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x ₀ andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m.LetG:U ?R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ^?1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.     

Approximation by convolution operators

слЕДУь п. к. сИккЕМА, Мы ИсслЕДУЕМ АппРОксИМ АцИОННыЕ сВОИстВА ОпЕРАтОРОВ $$u_\varrho ^\beta (f,x) = \frac{1}{{\beta _\varrho }}\int\limits_{ - \infty }^\infty {f(x - t)\beta ^\varrho (t) dt(\varrho \to \infty ).} $$ жДЕсьΒ — НЕОтРИцАтЕл ьНАь сУММИРУЕМАь ФУН кцИь, \(\beta _\varrho = \int\limits_{ - \infty }^\infty {\beta ^\varrho (t) dt} \) И ВыпОлНЕНы УслОВИь: (i)Β(0)=1 ИΒ НЕпРЕРыВНА В тО ЧкЕt=0, (ii) \(\mathop {\sup }\limits_{\left| t \right| > \delta } \beta (t)< 1\) Дль кАжДОгОδ>0. ДОкАжАНО, ЧтО ЁкспОНЕ НцИАльНыИ пОРьДОк Ап пРОксИМАцИИ МОжЕт Быть ДОстИгНУт тОлькО Дль ФУНкцИИ ВИДА (fx)=ax+b И (fx)=ae ^bx+c. ЁтО — ИсклУЧИтЕльНыЕ слУЧАИ, пОскОлькУ УкАжАННыЕ ФУНкцИИ ьВ льУтсь ЕДИНстВЕННыМ И НЕпОДВИжНыМИ тОЧкАМ И Дль ОпЕРАтОРОВU _β ^? . ДОкАжАНО тАкжЕ, ЧтО пР И УДАЧНОМ ВыБОРЕΒ МО жНО ДОБИтьсь «пОЧтИ Ёксп ОНЕНцИАльНОгО» пОРьДкА АппРОксИМАц ИИ. НАкОНЕц, В пОслЕДНЕИ т ЕОРЕМЕ УтВЕРжДАЕтсь, ЧтО сУЩЕстВУУт тАкИЕΒ Иf, ЧтОU _β ^? (f,x) пРИp→∞ РАсхОДьтсь НА МНОжЕстВЕ пОлОжИтЕл ьНОИ МЕРы.     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

On oracle inequalities related to data-driven hard thresholding

This paper focuses on computing a nearly optimal penalty in the method of empirical risk minimization. It is assumed that we have at our disposal the noisy data Y?=??? +????, where ${\theta\in \mathbb{R}^n}$ is an unknown vector and ${\xi\in \mathbb{R}^n}$ is a standard white Gaussian noise. It is also assumed that the underling vector ?? is sparse, and therefore to recover ?? we use a hard thresholding estimate ${\hat\theta_i(Y,t)=Y_i{\bf 1}\{|Y_i|\ge t\}}$ . In order to adapt to an unknown sparsity of ??, the threshold t is assumed to be data-driven. The very popular approach for computing such thresholds is based on the principle of empirical risk minimization suggesting the following data-driven threshold ${\hat t =\text{arg\,min}_t\{\|Y-\hat\theta(Y,t)\|^2+Pen(Y,t)\}}$ , where Pen(Y, t) is a penalty function. In this paper, it is proved with the help of a sharp oracle inequality that the main term in the optimal penalty is given by 2?? ²#{i : |Y _i | ?? t} log[n/#{i : |Y _i | ?? t}].     

Moment inequalities and the Riemann hypothesis

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫ ₀ ^∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments \(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turán inequalities (*) $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ We give here a constructive proof that log \(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.