In this paper, we study the existence of nontrivial solutions and ground state solutions for the second order Hamiltonian systems: $$ \ddot u(t)+A(t)u(t)+\nabla F(t,u(t))=0\ \ \ \ \ \ \mbox{a.e. } t\in [0,T], $$ where $A(t)$ is a $N\times N$ symmetric matrix, continuous and $T$-periodic in $t$. Replacing the classical Ambrosetti-Rabinowitz superquadratic condition by a general superquadratic condition, we prove some existence theorems, which unify and improve some recent results in the literature.

Let $B$ be the class of meromorphic functions $f$ such that the set sing ($f^{-1}$) is bounded, where sing ($f^{-1}$) is the set of critical and asymptotic values of $f.$ Suppose that $f$ has at most finitely many poles in the complex plane, and that $L(f)-P$ and $f-P$ share $0$ CM, where $L[f]=f^{(k)}+a_{k-1}f^{(k-1)}+\cdots+a_1f'+a_0f,$ where $k$ is a positive integer and $a_0,$ $a_1,$ $\cdots,$ $a_{k-1}$ are complex numbers, $k$ is a positive integer, $P$ is a nonconstant polynomial. Then, the hyper-order of $f$ is nonnegative integer or $\infty.$ Applying this result, we obtain some uniqueness results for transcendental meromorphic functions having the same fixed points with their linear differential polynomials, where the meromorphic functions belong to $B$ and have at most finitely many poles in the complex plane. The results in this paper are concerning a conjecture of Brück [5]. An example is provided to show that the results in this paper are best possible.

In this paper, we investigate the spacelike hypersurfaces in anti-de Sitter space $\mathbb{H}^{n+1}_1(c)$ with nonzero constant $k$-th mean curvature $H_k$ and two distinct principal curvatures one of which is simple, and characterize such hypersurfaces as hyperbolic cylinders.

In this paper, some existence theorems are obtained for periodic solutions of a damped vibration problem with $(q,p)-$Laplacian by using variational methods. Our results extend some results in some known literatures.

In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $\alpha$, and convex of order $\beta$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions.

Let $m,n$ be positive integers. In this short note we prove that the set of all continuous and surjective functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ contains (excluding the $0$ function) a $\mathfrak{c}$-dimensional vector space. This result is optimal in terms of dimension.

We study entwining structures on a monoidal category $\Cc$ and their corresponding categories of entwined modules. Examples can be constructed from lax Doi-Koppinen and lax Yetter-Drinfeld structures in $\cal C$. If $\cal C$ is symmetric then lax Yetter-Drinfeld structures appear as special cases of lax Doi-Koppinen structures, at least if we work over a so-called lax Hopf algebra. In this case the corresponding categories of entwined modules are isomorphic, and this generalizes a well-known result of Caenepeel, Militaru and Zhu. In particular, our theory applies to Doi-Koppinen and Yetter-Drinfeld structures in symmetric monoidal categories. We present some examples of entwining structures in monoidal categories coming from actions and coactions of a weak Hopf algebra.

We study the mapping properties of the maps $f(z)=\frac{\overline z-1}{z-1}$, $g(z)=|z| \; f(z)$ and $h(z)=-zf(z)$ with $|z|\leq 1$, $z\not=1$.

Initial problems for Hamilton--Jacobi functional differential equations and initial boundary value problems of the Dirichlet type for parabolic equations are considered. It is proved that classical solutions of functional differential equations can be estimated by solutions of initial problems for ordinary functional differential equations. Theorems on the uniqueness of solutions are obtained as consequences of comparison results. A method of differential inequalities is used. Here, the involved operators do not satisfy the Volterra condition.

For $v\in\mathbb{R}^n$ let $K$ be a compact set in $\mathbb{R}^n$ containing a suitable smooth surface and such that the intersection $\{tv+x:t\in\mathbb{R}\}\cap K$ is a closed interval or a single point for all $x\in K$. We prove that every linear first order differential operator with constant coefficients in direction $v$ on space of Whitney functions $\mathcal E(K)$ admits a continuous linear right inverse.

In the present paper, we find new bounds on the modulii of the third and fourth Taylor-Maclaurin's coefficients of {\it bi-starlike functions of order} $\rho$ and {\it strongly bi-starlike functions of order} $\beta$. Our estimates on the third coefficient improve upon earlier estimates found in [D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis and its Applications, Kuwait; February 18-21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60].

In this paper we obtain the Fekete-Szegö inequalities for the classes $\mathcal{H}_\sigma(\varphi)$, $\mathcal{ST}_\sigma(\alpha, \varphi)$ and $\mathcal{M}_\sigma(\alpha, \varphi)$ of bi-univalent functions defined in terms of subordination. These inequalities result in the bounds of the third coefficient which improve many known results concerning different classes of bi-univalent functions.

We study positive solutions to singular boundary value problems of the form \begin{equation*} \begin{cases} -u''(t) = \lambda h(t) \frac{f(u(t))}{u(t)^\beta};~~(0,1)\\ u(0)=0=u(1), \end{cases} \end{equation*} where $\lambda >0$ is a parameter, $\beta \in (0,1), f:[0, \infty) \rightarrow (0,\infty)$ is a $C^1$ function, $\frac{f(s)}{s^\beta}$ is decreasing for $s \gg 1, h:(0,1) \rightarrow (0,\infty)$ is a continuous function and there exist $C_i>0,\alpha_i \in (0,1), i=1,2 $ such that $ h(t) \leq \frac{C_1}{t^{\alpha_1}};~ t \approx 0$ and $h(t) \leq \frac{C_2}{(1-t)^{\alpha_2}};~ t \approx 1$. We establish the uniqueness of positive solutions for $\lambda \gg 1,$ when $\alpha_i +\beta <1, i=1,2.$

We investigate when the group $\mbox{SL}_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $\mbox{SL}_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of $\mbox{SL}_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.