In this article, we investigate the existence of solutions and Levitin-Polyak well-posedness for a class of system of parametric strong quasi-equilibrium problems (SPSQEP) involving set-valued mappings in Hausdorff topological vector spaces. The existence of solutions to the problem (SPSQEP) are presented, and then the notions of Levitin-Polyak well-posedness and generalized Levitin-Polyak well-posedness for (SPSQEP) are introduced. Moreover, some metric characterizations of these well-posedness are derived under quite mild conditions. The relationships between these well-posedness of (SPSQEP) and the existence and uniqueness of its solutions are established. Finally, some examples are given to illustrate the presented results.

Let $G$ be a homogeneous nilpotent Lie group. In this paper, we introduce a new class of multiparameter weights $A^{\mathcal{F}}_p$ associated with a flag $\mathcal{F}$ on $G$ and show that such class of weights can be characterized via two type of flag maximal operators. We then prove that singular integrals with flag kernels are bounded on $L^p_w(G)$, $1 \lt p \lt \infty$, when $w \in A_{p}^{\mathcal{F}}(G)$, which extends a recent result of Nagel-Stein-Wainger in [13]. As an application, we get weighted norm inequalities for the multiparameter Marcinkiewicz multipliers on Heisenberg groups introduced in [11].

Very recently, Moudafi (Nonlinear Analysis 79 (2013) 117-121) introduced a relaxed alternating CQ-algorithm (RACQA) with weak convergence for the following convex feasibility problem: $$\mbox{Find}~x\in C,~y\in Q\hspace{0.4cm}\mbox{such that}~Ax=By, \tag{1.1}$$ where $H_1,~H_2,~H_3$ are real Hilbert spaces, $C\subset H_1$, $Q\subset H_2$ are two nonempty, closed and convex level sets, and $A:H_1\to H_3$, $B:H_2\to H_3$ are two bounded linear operators. In this paper, we will continue to consider the problem (1.1) and obtain a strongly convergent iterative sequence of Halpern-type to a solution of the problem and provide an affirmative answer to an open question posed by Moudafi in his recent work for convex feasibility problems in real Hilbert spaces. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a convex feasibility problem and common fixed points of an infinite family of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and generalize many known results in the current literature.

In this paper, we first study the lack of global null controllability and the local null controllability respectively. Based on the null controllability, we establish the existence of the time optimal control for nonlinear parabolic integrodifferential equations with internal distributive control.

In this paper we consider the proximal point algorithm to approximate a singularity of a multivalued monotone vector field on a Hadamard manifold. We study the convergence of the sequence generated by an inexact form of the algorithm. Our results extend the results of [3, 25] to Hadamard manifolds as well as the main result of [11] with more general assumptions on the control sequence. We also give some application to optimization.

In this paper, we consider the asymptotic behavior of solutions to the stochastic fractional complex Ginzburg-Landau equation with multiplicative noise in one spatial dimensions. We first transfer stochastic fractional Ginzburg-Landau equation into random equation which solutions generate a random dynamical system. Then, we consider the existence of a random attractor for the random dynamical system. At last we estimate the Hausdorff dimension of the random attractor by using linearization and Lyapunov exponents.

We study the minimax theorems for set-valued mappings with several hierarchical process, and propose three versions for minimax theorems in topological vector spaces setting. These problems arise from some minimax theorems in the vector settings. As applications, we discuss the existences of two kinds of saddle points. Our results are new or include as special cases recent existing results.

The center $C(G)$ and the periphery $P(G)$ of a connected graph $G$ consist of the vertices of minimum and maximum eccentricity, respectively. Almost-peripheral (AP) graphs are introduced as graphs $G$ with $|P(G)| = |V(G)| - 1$ (and $|C(G)| = 1$). AP graph of radius $r$ is called an $r$-AP graph. Several constructions of AP graph are given, in particular implying that for any $r\ge 1$, any graph can be embedded as an induced subgraph into some $r$-AP graph. A decomposition of AP-graphs that contain cut-vertices is presented. The $r$-embedding index $\Phi_{r}(G)$ of a graph $G$ is introduced as the minimum number of vertices which have to be added to $G$ such that the obtained graph is an $r$-AP graph. It is proved that $\Phi_{2}(G)\le 5$ holds for any non-trivial graphs and that equality holds if and only if $G$ is a complete graph.

Recently, Kim et al. [8] constructed a new method to obtain interesting identities related to Euler polynomials of higher order arising from Euler basis. In the present paper, we study to Genocchi polynomials of higher order arising from Genocchi basis by using the method of Kim et al. We also derive many interesting properties related to Genocchi polynomials of higher order.

In this paper, based on a kind of unified proper efficiency named as $E$-Benson proper efficiency, we present $E$-proper saddle points theorems and $E$-proper duality results including as weak duality and strong duality theorems of vector optimization problems with set-valued maps. Our main results unify and extend the cases of proper saddle points and proper duality as well as $\varepsilon$-proper saddle points and $\varepsilon$-proper duality.

Using the Davenport-Heilbronn circle method, we show that if $\lambda_1,\cdots,\lambda_5$ are positive real numbers, at least one of the ratios $\lambda_i/\lambda_j(1\leq i\lt j\leq 5)$ is irrational, then, for arbitrary positive integer $k\geq 4$, the integer parts of $\lambda_1 x_1^3 + \lambda_2 x_2^3 + \lambda_3 x_3^3 + \lambda_4 x_4^3 +\lambda_5 x_5^k$ are prime infinitely often for natural numbers $x_1,\cdots,x_5$.

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $({\mathcal X},d,\mu)$. The authors also prove that multilinear commutators of fractional integrals with RBMO(μ) functions are bounded on Orlicz spaces over $({\mathcal X},d,\mu)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in the Orlicz-type space ${\mathrm{Osc}_{\exp L^r}(\mu)}$, where $r\in [1,\infty)$, are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces.

In this paper, we develop a theory of weighted Hardy spaces $H^p_\omega$ on spaces of homogeneous type and prove that certain class of singular integral operators are bounded from $H^p_\omega$ to itself and from $H^p_\omega$ to $L^p_\omega$. As an application, we give weighted endpoint estimates for Nagel-Stein's NIS operators studided in [26].

In this paper we propose a system of ordinary differential equations to model the interaction among non-infected macrophages, infected macrophages, T cells and Mtb bacilli. Model analysis reveals the existence of infection-free equilibrium and the endemically infected equilibrium. And we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss a quadratic control and a linear control. The quadratic control allows for a weaker treatment that more effectively than the linear control.

In this paper we investigate boundary blow-up solutions of the problem $$ \left\{ \begin{array}{l} -\triangle _{p(x)}u+f(x,u)=\rho (x,u)+K(\left\vert x\right\vert )\left\vert \nabla u\right\vert ^{\delta (\left\vert x\right\vert )}\text{ in }\Omega \text{,} \\[12pt] \text{ }u(x)\rightarrow +\infty \text{ as }d(x,\text{ }\partial \Omega )\rightarrow 0\text{,} \end{array} \right. $$ where $-\triangle _{p(x)}u=-\mbox{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of $\rho (x,u)$ being a large perturbation (namely, $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $). In particular, we do not have the comparison principle.

In this paper, we use Lebesgue's dominated convergence theorem to give a probabilistic version of Mehler's formula for the well-known Rogers-Szegö polynomials. Applications of the probabilistic version are also given.

In this paper, we study the following fourth-order elliptic equation $$ \left\{ \begin{array}{ll} \Delta^{2}u-\Delta u+V(x)u=f(x, u), \ \ \ x\in\mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{array} \right. $$ where the potential $V\in C(\mathbb{R}^N, \mathbb{R})$ is allowed to be sign-changing. Under the weakest superquadratic conditions, we establish the existence of infinitely many solutions via variational methods for the above equation. Recent results from the literature are extended.

Let $b_1, b_2 \in {\rm CMO}(\mathbb{R}^n)$ and $T_{\sigma}$ be the bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfies the Sobolev regularity that $\sup_{\kappa\in \mathbb{Z}}\|\sigma_{\kappa}\|_{W^{s_1,\,s_2}(\mathbb{R}^{2n})}\lt \infty$ for some $s_1,\,s_2\in (n/2,\,n]$. In this paper, it is proved that the commutator defined by $$T_{\sigma,\,\vec{b}}(f_1,\,f_2)(x) = b_1(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(b_1 f_1,\,f_2)(x) + b_2(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(f_1,\,b_2 f_2)(x)$$ is a compact operator from $L^{p_1}(\mathbb{R}^n) \times L^{p_2}(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ when $p_k \in (n/s_k,\,\infty)$ $(k=1,\,2)$, $p \in (1,\,\infty)$ with $1/p = 1/p_1 + 1/p_2$.