In [1], the author provided an example which shows that there is a linear functional bounded uniformly on all atoms in $H^1(\mathbb{R}^n)$, and it can not be extended to a bounded functional on $H^1(\mathbb{R}^n)$. In this note, we first give a new atomic decomposition, where the decomposition converges in $L^2(\mathbb{R}^n)$ rather than only in the distribution sense. Then using this decomposition, we prove that for $0 \lt p \leq 1$, $T$ is a linear operator which is bounded on $L^{2}(\mathbb{R}^n)$, then $T$ can be extended to a bounded operator from $H^{p}(\mathbb{R}^n)$ to $L^{p}(\mathbb{R})$ if and only if $T$ is bounded uniformly on all $(p,2)$-atoms in $L^{p}(\mathbb{R}^n)$. A similar result from $H^{p}(\mathbb{R}^n)$ to $H^{p}(\mathbb{R}^n)$ is also obtained. These results still hold for the product Hardy space and Hardy space on spaces of homogeneous type.

In this paper, we establish the existence of multiple positive solutions for singular integral equations. The proof is based on a general existence principle established using a nonlinear alternative principle of Leray--Schauder type and a well-known fixed point theorem in cones. As an application, we consider higher order nonlinear singular conjugate boundary value problems. Some recent results in the literature are generalized and improved.

In this paper, we first introduce the weak convexlike condition which generalizes the convexlike concept due to Fan. Next, using the separation theorem for convex sets, we will prove a non-compact generalization of Fan's minimax theorem by relaxing the concavelike assumption to the weak concavelike condition. Also we give some examples which show that the convex and concave assumptions on Kneser's minimax theorem can not be relaxed with the quasi-convex and quasi-concave conditions simultaneously, and the previous minimax theorems can not be available.

We define the generalized Roper-Suffridge extension operator $\Phi_{n, \beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)$ on Reinhardt domain $D_p$ as \[ \Phi_{n, \beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)(z) = \bigg(f(z_1), \bigg(\frac{f(z_1)}{z_1}\bigg)^{\beta_2} \Big(f'(z_1)\Big)^{\gamma_2} z_2, \cdots, \bigg(\frac{f(z_1)}{z_1}\bigg)^{\beta_n} \Big(f'(z_1)\Big)^{\gamma_n} z_n \bigg) \] for $z = (z_1, z_2, \cdots, z_n) \in D_p$, where $D_p = \Big\{(z_1, z_2, \cdots, z_n) \in C^n: \sum\limits_{j=1}^{n} |z_j|^{p_j} \lt 1 \Big\}$, $p = (p_1, p_2, \cdots, p_n)$, $p_j \gt 0$, $0 \leq \gamma_j \leq 1 - \beta_j$, $0 \leq \beta_j \leq 1$, $j = 1,2,\cdots,n$, and we choose the branch of the power functions such that $\Big(\frac{f(z_1)}{z_1}\Big)^{\beta_j}|_{z_1=0} = 1$ and $(f'(z_1))^{\gamma_j}|_{z_1=0}=1$, $j = 2,\cdots,n$. In the present paper, we show that the operator $\Phi_{n,\beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)$ preserves almost spirallike mapping of type $\beta$ and order $\alpha$ and spirallike mapping of type $\beta$ and order $\alpha$ on $D_p$ for some suitable constants $\beta_j, \gamma_j, p_j$. The results improve the corresponding results of earlier authors.

In this paper we determine all rank 2 geometries for the Mathieu group $M_{23}$ for which object stabilizers are maximal subgroups.

In this paper, we develop stability criteria in terms of two measures for perturbed setvalued delay integro-differential equations with fixed moments of impulsive effects. Variational Lyapunov method is employed to establish the stability properties of the perturbed system via a comparison result which connects the solutions of perturbed system and the unperturbed one through the solutions of a comparison system.

The purpose of the paper is to derive various properties and characteristics of certain subclass of analytic functions using multiplier transformations and the method of differential subordination.

In this paper we present a new algorithm of steepest descent type. A new technique for steplength computation and a monotone strategy are provided in the framework of the Barzilai and Borwein method. In contrast with Barzilai and Borwein approach's in which the steplength is computed by means of a simple approximation of the Hessian in the form of scalar multiple of identity and an interpretation of the secant equation, the new proposed algorithm considers another approximation of the Hessian based on the weak secant equation. By incorporating a simple monotone strategy, the resulting algorithm belongs to the class of monotone gradient methods with linearly convergence. Numerical results suggest that for non-quadratic minimization problem, the new method clearly outperforms the Barzilai-Borwein method.

This paper is concerned with the asymptotic behavior of the solution $u_\varepsilon$ of a p-Ginzburg-Landau system with the radial initial-boundary data. The author proves that the zeros of $u_\varepsilon$ in the parabolic domain $B_1(0) \times (0,T]$ locate near the axial line $\{0\} \times (0,T]$. In addition, the author also consider the Holder convergence of the solution when the parameter $\varepsilon$ tends to zero. The convergence is derived by establishing a uniform gradient estimate for the regularized solution of the system.

Let $R$ be a Noetherian ring, let $M$ and $N$ be finitely generated modules and let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of $R$. Let $s$ be an integer such that $\mathfrak{b}_{\mathfrak{p}} \subseteq \sqrt{\mbox{Ann} \mbox{H}^i_{\mathfrak{a}_{\mathfrak{p}}}(M_{\mathfrak{p}} ,N_{\mathfrak{p}})}$ for all $i \le s$ and all prime ideal $\mathfrak{p}$ of $R$. Then we show the following statements hold:

(1) If $\mbox{H}^i_{\mathfrak{b}}(N) = 0$ for all $i \lt s$, then $\mbox{H}_{\mathfrak{a}}^{i}(M,N)$ is finitely generated for all $i \leq s$.

(2) $\mathfrak{b} \subseteq \sqrt{\mbox{Ann} \mbox{H}_{\mathfrak{a}}^{2}(M,N)}$.

These statements generalize the corresponding results which are shown in [6] and [1] for standard local cohomology module.

We investigate the Hyers-Ulam stability of the Jensen functional equation for mappings from linear spaces into multi-normed spaces. We then establish an asymptotic behavior of the Jensen equation in the framework of multi-normed spaces which are somewhat similar to the operator sequence spaces and have some connections with operator spaces and Banach lattices.

A viscosity method for a hierarchical fixed point approach to variational inequality problems is presented. This method is used to solve variational inequalities where the involving operators are complements of nonexpansive mappigs and the solutions are sought in the set of the fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings.

Considered in this article is the Cauchy problem of a generalized Korteweg-de Vries equation $$\left\{\begin{array}{l} u_{t} + u_{xxx} + uu_{x} + |D_{x}|^{2\alpha}u = 0, \quad t \in \mathbb{R}^{+},\, x \in \mathbb{R}, \\ u(x,0) = \varphi(x) \end{array}\right. $$ with $0 \leq \alpha \le 1$. The local well-posedness of the Cauchy problem in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R})$ for $s \in (\frac{\alpha-3}{2(2-\alpha)}, 0]$ is proved. In addition, the mapping that associated to appropriate initial-data the corresponding solution is analytic as a function between appropriate Banach spaces.

In this article, motivated by the work of Brown, Shields, and Zeller[2], we first consider the representation problem of zero by exponential sums in several complex variables. Then we give a complete characterization of dominating sets for bounded real harmonic functions, denoted by $h^\infty(\Re{B_n})$, on the open unit ball $\Re{B_n}$ in $\mathbb{R}^n$, $n \geq 3$, and for bounded real $\mathcal{M}$-harmonic functions, denoted by $\mathcal{M} h^\infty(B_n)$, on the open unit ball $B_n$ in $\mathbb{C}^n$, $n \ge 2$.

The purpose of this paper is to introduce and consider a shrinking projection method of proximal-type for finding a common element of the set $EP$ of solutions of a generalized equilibrium problem, the set $F(S) \cap F(\widetilde S)$ of common fixed points of a pair of relatively nonexpansive mappings $S,\widetilde S$ and the set $T^{-1}0$ of zeros of a maximal monotone operator $T$ in a uniformly smooth and uniformly convex Banach space. It is proven that under appropriate conditions, the sequence generated by the shrinking projection method of proximal-type, converges strongly to some point in $EP \cap F(S) \cap F(\widetilde S) \cap T^{-1}0$. This new result represents the improvement, generalization and development of the previously known ones in the literature.

Two iterative algorithms for nonexpansive mappings on Hadamard manifolds, which are extensions of the well-known Halpern's and Mann's algorithms in Euclidean spaces, are proposed and proved to be convergent to a fixed point of the mapping. Some numerical examples are provided.

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant surfaces in Lorentzian Kaehler surfaces to slant submanifolds in neutral Kaehler manifolds; moreover, we characterize slant submanifolds with parallel canonical structures. By applying the results obtained in the first part we completely classify slant surfaces with parallel mean curvature vector and minimal slant surfaces in the Lorentzian complex plane in the second part of this article.

The initial boundary value problem for a system of nonlinear wave equations of Kirchhoff type with strong damping in a bounded domain is considered. The existence, asymptotic behavior and blow-up of solutions are discussed under some conditions. The decay estimates of the energy function and the estimates for the lifespan of solutions are given.

In this paper, we obtain a precise estimation of the hyper order of solutions for a class of second order linear differential equations, and investigate the condition of the existence of nontrivial subnormal solution for a class of second order periodic equations. These results generalize the results of Gundersen, Steinbart [5] and Wittich [10]. Our methods of proofs are different from the methods applied in [5, 10].

In this paper we investigate $(n+1)(n \geq 5)$-dimensional contact $CR$-submanifolds $M$ of $(n-1)$ contact $CR$-dimension in a $(2m+1)$-dimensional unit sphere $S^{2m+1}$ which satisfy the condition $h(FX,Y) - h(X,FY) = g(FX,Y) \zeta$ for a normal vector field $\zeta$ to $M$, where $h$ and $F$ denote the second fundamental form and a skew-symmetric endomorphism (defined by (2.3)) acting on tangent space of $M$, respectively.

In this paper, we consider the multipoint boundary value problem for the one-dimensional $p$-Laplacian $$\left(\phi_p(u'(t))\right)' + q(t) f\left(t,u(t),u'(t)\right) = 0,~~~~\ \ \ \ \ \ t \in (0,1), $$ subject to the boundary value conditions: $$ u(0) = \sum_{i=1}^{m-2} a_{i} u(\xi_{i}),\ \ \ \ \ \ u(1) = \sum_{i=1}^{m-2} b_{i} u(\xi_{i}), $$ where $\phi_{p}(s) = |s|^{p-2}s$, $p \gt 1$, $\xi_{i} \in (0,1)$ with $0 \lt \xi_{1} \lt \xi_{2} \lt \cdots \lt \xi_{m-2} \lt 1$ and $a_{i}, b_{i} \in [0,1)$, $0 \leq \sum\limits_{i=1}^{m-2} a_{i} \lt 1$, $0 \leq \sum\limits_{i=1}^{m-2} b_{i} \lt 1$. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term $f$ is involved with the first-order derivative explicitly.

Let $X$ be a compact Hausdorff space which satisfies the first axiom of countability, and $C_+(X)$ the set of all continuous functions from $X$ to $[0,\infty)$. If $\varphi : C_{+}(X) \rightarrow C_{+}(X)$ is a bijective map which preserves the order in both directions, then there exists a homeomorphism $\omega : X \mathcal{\rightarrow } X$ and for each $x \in X$ a bijective, increasing map $m_{x} : [0,\infty) \mathcal{\rightarrow } [0,\infty)$ such that $\varphi(f)(x) = m_{x}(f(\omega(x)))$, for all $x \in X$ and $f \in C_{+}(X)$.

Normalized systems are constructed with respect to wave and Dunkl operators. Non-trivial solutions can thus be built to the equation $Dv(x) + \lambda v(x) = 0$, where $D$ is either the wave operator or the Dunkl operators and $\lambda \in \mathbb{C}$.

We give explicit parameterizations of rotation hypersurfaces in Lorentz-Minkowski space $L^{n+1}$. Then we obtain rotation hypersurfaces in Lorentz-Minkowski space $L^{n+1}$ with constant mean curvature. In particular, we determine nonplanar rotation hypersurfaces with zero mean curvature, namely, generalized catenoids of $L^{n+1}$. In the case the rotation axis is light-like, the generalized catenoids generalize Enneper's surfaces of the 2nd and 3rd kind.

In this paper, we deal with weak and strong solutions to $F$-implicit generalized vector variational inequalities and $F$-implicit generalized (weak) vector variational inequalities. Several results of the existence for the weak solutions and strong solutions to both problems are derived.

In this paper, we introduce the concepts of $C$-quasiconcave mappings and properly $C$-quasiconcave mappings in abstract convex spaces. By using the Fan-Browder type fixed point theorem and the maximal element theorem, we establish some minimax theorems for vector-valued mappings in abstract convex spaces. We also give some examples to illustrate our results.

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $J$ be the duality mapping of $E$ and let $C$ be a nonempty closed convex subset of $E$. Then, a mapping $S: C \to C$ is said to be nonspreading [23] if \[ \phi(Sx,Sy) + \phi(Sy,Sx) \le \phi(Sx,y) + \phi(Sy,x) \] for all $x,y \in C$, where $\phi(x,y) = \|x\|^2 - 2\langle x,Jy \rangle + \|y\|^2$ for all $x,y \in E$. In this paper, we prove that every nonspreading mapping of $C$ into itself has a fixed point in $C$ if and only if $C$ is bounded. This theorem extends Ray's theorem [27] in a Hilbert space to that in a Banach space.