Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

First- and Second-Order Integral Functionals of the Calculus of Variations which Exhibit the Lavrentiev Phenomenon

We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations $$\mathcal{J}(x) = \int\limits_0^1 {L(t,x(t),\dot x(t))dt \to \inf , x(0) = x_0 } , x(1) = x_1$$ ,when the infimum of the problem in the class of absolutely continuous functionsW _1,1[0, 1] is strictly less than the infimum of the same problem in the class of Lipshitzian functionsW _1,∞[0, 1]. We suggest an approach to constructing new classes of integrands which exhibit the Lavrentiev phenomenon (Theorem 2.1). A similar method is used to construct (Theorem 3.1) a class of autonomousC ¹-differentiable integrandsL(x, .x, ..x) of the calculus of variations which are regular, i.e., convex, coercive w.r.t. ..x, and exhibit theW _2,1–W _2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem $$\begin{array}{*{20}c} {\mathcal{J}(x( \cdot )) = \int\limits_0^1 {L(x(t),\dot x(t),\ddot x(t)) dt \to \inf ,} } \\ {x(0) = x_0 , \dot x(0) = \upsilon _0 , x(1) = x_1 , \dot x(1) = \upsilon _1 } \\ \end{array}$$ ,the infimum of this problem over the spaceW _{2, 1}[0, 1] is strictly less than its infimum over the spaceW _2,∞[0, 1]. This provides a negative answer to the question of whether functionals with regular autonomous second-order integrands should only have minimizers with essentially bounded second derivative.     

Global Gradient Estimates for Nonlinear Elliptic Equations with Vanishing Neumann Data in a Convex Domain

In this paper, we obtain the following global L ^q estimates

$$\left|\mathbf{f}\right|^{p } \in L^{q}({\Omega}) \Rightarrow \left|\nabla u\right|^{p } \in L^{q}({\Omega}) \quad \text{for any} ~~q\ge 1 $$

in a convex domain Ω of weak solutions for nonlinear elliptic equations of p-Laplacian type with vanishing Neumann data

$$\begin{array}{@{}rcl@{}} \text{div} \left( \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \right) & =& \text{div} \left( | \mathbf{f}|^{p-2} \mathbf{f} \right) \quad\text{in} ~~{\Omega},\\ \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \cdot \mathbf{\nu} &=& | \mathbf{f}|^{p -2} \mathbf{f}\cdot \mathbf{\nu} \quad \quad \text{on}~~ \partial{\Omega}, \end{array} $$

where ν is the outwardpointing unit normal to ?Ω. Our argument is based on the works of Banerjee and Lewis (Nonlinear Anal 100:78–85, 2014), Kinnunen and Zhou (Comm Partial Differential Equations 24(11&12):2043–2068, 1999, Differential and Integral Equations 14(4):475–492, 2001), and Byun, Wang, and Zhou (Comm Pure Appl Math 57(10):1283–1310, 2004, J Funct Anal 20(3):617–637, 2007). In the proof of the above result, we only focus on the boundary case while the interior case can be obtained as a corollary.

     

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.     

Nonlinear evolution inclusions with one-sided perron right-hand side

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form $ {x}^{\prime}(t)\in Ax(t)+F\left( {x(t)} \right) $ , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X^* is uniformly convex and F is one-sided Perron with sublinear growth, then, we prove a variant of the well known Filippov-Pli? theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $ K\subseteq \overline{D(A)} $ are established. As applications, we derive ε - δ lower semicontinuity of the solution map and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

Analytic Solutions of Systems

In this paper, functional series solutions of the nonlinear analytic system for the unknown state variable x(t), and functional series solutions of the analytic infinite-dimension

Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation

For a general differential system $ \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} $ , where X _d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u _d ’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.     

Multiple positive solutions of singular dirichlet second-order boundary-value problems with derivative dependence

The existence of multiple positive solutions for the singular Dirichlet boundary-value problem

Nonoscillation and Exponential Stability of Delay Differential Equations with Oscillating Coefficients

We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Let X be a real Banach space and I a nonempty interval. Let \(K:I\rightsquigarrow X\) be a multi-function with the graph \(\mathcal {K} \). We give here a characterization for \(\mathcal {K} \) to be approximate/near weakly invariant with respect to the differential inclusion \(x^{\prime }(t)\in F(t, x(t))\) by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârj? et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârj? et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion \(x^{\prime }(t)\in F(t,x(t))\).     

Lower and Upper Bounds for the Blow-Up Time of a Class of Damped Fourth-Order Nonlinear Evolution Equations

This paper is concerned with the study of the nonlinear damped wave equation

$$u_{tt}+{\Delta}^{2}u-{\Delta} u-\omega{\Delta} u_{t}+\alpha(t)u_{t}=\left\vert u\right\vert^{p-2}u, $$

in a bounded domain with smooth boundary. The blow-up of solutions are investigated under some conditions. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.

     

Bifurcations of Cycles in Systems of Differential Equations with a Finite Symmetry Group – II

One-parameter bifurcations of periodic solutions of differential equations in ?ⁿ with a finite symmetry group Γ are studied. The following three types of periodic solutions x(t) with the symmetry group H $\subseteq $ Γ are considered separately. ? F-cycles: H consists of transformations that do not change the periodic solution, h(x(t)) ≡ x(t); ? S-cycles: H consists of transformations that shift the phase of the solution, $$h\left( {x\left( t \right)} \right) \equiv x\left( {t + {\tau }\left( h \right)} \right)\quad \left( {{\tau }\left( h \right) \ne 0\,{if}\,h \ne e} \right)$$ ? FS-cycles: H consists of transformations of both F and S types. In the present paper bifurcations of F-cycles at double real multipliers and all codimension one bifurcations of S-cycles were studied. In the present paper a more complicated case of a double pair of complex multipliers for F-cycles is considered and bifurcations of FS-cycles are shortly discussed.     

Attractor for the nonlinear Schr?dinger equation with a nonlocal nonlinear term

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrödinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor \( \mathcal{A} \) for the nonlocal equation in the strong topology of H ¹(Ω). We also prove that the global attractor is regular, i.e., \( \mathcal{A} \subset {H^2}\left( \Omega \right) \), assuming that f(x) is of class C ². Furthermore, we estimate the number of the determining modes for this equation.     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

Small Time Asymptotics on the Diagonal for Hörmander’s Type Hypoelliptic Operators

We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x ₀. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x ₀, then so is also the original control problem, and in this case we show that the fundamental solution blows up as \(\phantom {\dot {i}\!}t^{-\mathcal {N}/2}\), where \(\phantom {\dot {i}\!}\mathcal {N}\) is a number determined by the Lie algebra at x ₀ of the fields, that define the hypoelliptic operator.     

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 0 is continuous in ? ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.