Some new conditions on coefficient functions $A_{i}(z)$, which will guarantee all nontrivial solutions of $f^{(n)} + A_{n-1}(z) f^{(n-1)} + \cdots + A_{0}(z)f = 0$ are of infinite order, are found in this paper. The first condition involves two classes of extremal functions for some inequalities about finite asymptotic values and deficient values. The second condition assumes that a coefficient itself is a nontrivial solution of another differential equation $w'' + P(z)w = 0$, where $P(z)$ is a polynomial.

Let $(\overline{M}^{n+1}, \overline{F})$ be a complete simply connected Randers space with $\overline{F}(x,Y) = \overline{a}(x,Y)+ \overline{b}(x,Y)$, where $\overline{a}(x,Y) = \sqrt{\overline{a}_{ij}(x) Y^i Y^j}$ and $\overline{b}(x,Y) = \overline{b}_i(x) Y^i$ are a Riemannian metric and a $1$-form on the smooth $(n+1)$-dimensional manifold $\overline{M}$ respectively. Assume the $1$-form $\overline{b}$ is parallel with respect to $\overline{a}$ and the sectional curvature $\overline{K}_{\overline{M}}$ of $\overline{M}$ with respect to $\overline{a}$ satisfies $\delta(n) \leq \overline{K}_{\overline{M}} \leq 1$. In this paper, we study the compact hypersurface $(M,F)$ of the Randers space $(\overline{M}^{n+1}, \overline{F})$ with constant mean curvature $|H|$ and prove that if the norm square $S$ of the second fundamental form of $(M,F)$ with respect to the Finsler metric $\overline{F}$ satisfies a certain inequality, then $S = n|H|^2$ and $M$ is the unit sphere or equality holds. In that case, we describe all $M$ that satisfy this equality, which generalizes the result of [8] from the Riemannian case to the Randers space.

In this paper, we mainly investigate exponential stability of traveling wavefronts for delayed $2D$ lattice differential equation with nonlocal interaction. For all non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where $c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these traveling waves are asymptotically stable, when the initial perturbation around the traveling waves decay exponentially at far fields, but can be allowed arbitrarily large in other locations. Our approach adopted in this paper is the weighted energy method and the squeezing technique with the help of Gronwall's inequality. Furthermore, from stability result, we prove the uniqueness (up to shift) of the traveling wavefront. Our results can apply to the discrete diffusive Mackey-Glass model and the dicrete diffusive Nicholson's blowflies model on $2D$ lattices.

We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation\[ (-\Delta)^\alpha_p u + V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.

In this article, we study the existence and multiplicity of nontrivial solutions to the problem \[\begin{cases} -\epsilon^{2} F(x,D^{2}u) = f(x,u) + \psi(Du) &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\partial \Omega$},\end{cases}\]where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$, $n \gt 2$. We show that the problem possesses nontrivial solutions for small value of $\epsilon$ provided $f$ and $\psi$ are continuous and $f$ has a positive zero. We employ degree theory arguments and Liouville type theorem for the multiplicity of the solutions.

This paper is concerned with the existence of families of time-periodic solutions for the nonlinear wave equations with Hamiltonian perturbations on one-dimensional tori. We obtain the result by a new method: a para-differential conjugation together with a classical iteration scheme, which have been used for the nonlinear Schrödinger equation in [22]. Avoiding the use of KAM theorem and Nash-Moser iteration method, though a para-differential conjugation, an equivalent form of the investigated nonlinear wave equations can be obtained, while the frequencies are fixed in a Cantor-like set whose complement has small measure. Applying the non-resonant conditions on each finite-dimensional subspaces, solutions can be constructed to the block diagonal equation on the finite subspace by a classical iteration scheme.

In this paper, we obtain a conditional variational principle for the topological entropy of level sets of Birkhoff averages for maps with the gluing orbit property. Our result can be easily extended to flows.

In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS (nonlinear Schrödinger equations):\[ \mathrm{i} \partial_{t} u - \Delta u + V * u + \partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad x \in \mathbb{T}^d, \; t \in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an $\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0 = u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm $\|u(t)\|_s$ remains bounded by $C_s \varepsilon$.

Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T_*$ be the maximal Calderón-Zygmund operator and $\vec{b} := (b_1,\ldots,b_m)$ be a finite family of $\widetilde{\operatorname{RBMO}}(\mu)$ functions. In this paper, the authors establish the boundedness of the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ on the Lebesgue space $L^p(\mu)$ with $p \in (1, \infty)$. For $\vec{b} = (b_1,\ldots,b_m)$ being a finite family of Orlicz type functions, the weak type endpoint estimate for the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ is also presented. The main tool to deal with these estimates is the smoothing technique.

The aim of this paper is to show the equivalence of two classes of nonsmooth functions. We also compare optimality conditions which have been stated for these classes.

In this paper, we propose a method for solving some well-known classes of fractional Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The method is proposed by utilizing Haar wavelets in conjunction with Adomian's decomposition method. The operational matrices for the Haar wavelets are derived and constructed. Procedure of implementation and convergence analysis of the method are presented. The method is tested on the fractional standard Lane-Emden equation and the fractional isothermal gas spheres equation. We compare the results produce by present method with some well-known results to show the accuracy and applicability of the method.

In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by parabolic equations. The time discretization is based on the finite difference methods. The state and co-state variables are approximated by the nonconforming $EQ_1^{\operatorname{rot}}$ elements, and the control variable is approximated by the piecewise constant element, respectively. Some superclose properties are obtained for the above three variables. Moreover, for the state and co-state, the convergence and superconvergence results are achieved in $L^2$-norm and the broken energy norm, respectively.