Let $L$ be a nonnegative self-adjoint operator on $L^2(\mathbb{R}^n)$ satisfying the reinforced $(p_L, p_L')$ off-diagonal estimates, where $p_L\in[1,2)$ and $p_L'$ denotes its conjugate exponent. Assume that $p\in(0,1]$ and the weight $w$ satisfies the reverse Hölder inequality of order $(p'_L/p)'$. In particular, if the heat kernels of the semigroups $\{e^{-tL}\}_{t\gt 0}$ satisfy the Gaussian upper bounds, then $p_L=1$ and hence $w\in A_\infty({\mathbb R}^n)$. In this paper, the authors introduce the weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ associated with the operator $L$, via the Lusin area function associated with the heat semigroup generated by $L$. Characterizations of $H^p_{L,\,w}(\mathbb{R}^n)$, in terms of the atom and the molecule, are obtained. As applications, the boundedness of singular integrals such as spectral multipliers, square functions and Riesz transforms on weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ are investigated. Even for the Schrödinger operator $-\Delta+V$ with $0\le V\in L_{\rm{loc}}^1 (\mathbb{R}^n)$, the obtained results in this paper essentially improve the known results by extending the narrow range of the weights into the whole $A_\infty(\mathbb{R}^n)$ weights.

Poisson equation is frequently encountered in mathematical modeling for scientific and engineering applications. Fast Poisson numerical solvers for 2D and 3D problems are, thus, highly requested. In this paper, we consider solving the Poisson equation $\nabla^{2}u = f(x,y)$ in the Cartesian domain $\Omega = [-1,1] \times [-1,1]$, subject to all types of boundary conditions, discretized with the Chebyshev pseudospectral method. The main purpose of this paper is to propose a reflexive decomposition scheme for orthogonally decoupling the linear system obtained from the discretization into independent subsystems via the exploration of a special reflexive property inherent in the second-order Chebyshev collocation derivative matrix. The decomposition will introduce coarse-grain parallelism suitable for parallel computations. This approach can be applied to more general linear elliptic problems discretized with the Chebyshev pseudospectral method, so long as the discretized problems possess reflexive property. Numerical examples with error analysis are presented to demonstrate the validity and advantage of the proposed approach.

In this paper, Stokes' theorem is proved by the Kurzweil-Henstock approach. Sufficient conditions for the existence of the exterior derivative of a $k$-form in $\mathbb{R}^n$ are given. Concepts of strong differentiability are used in sufficient conditions.

Let $\varphi: \mathbb{C}^n \to\mathbb{R}$ be a $C^2$ plurisubharmonic function on $\mathbb{C}^n$. Suppose that there exist $C_1, C_2 \gt 0$ such that $\sup_{\mathbb{C}^n} |\bar\partial \partial\varphi| \lt C_1$ and $H_{\varphi}(\xi,\xi)(z) \geq C_2 |\xi|^2$ for $\xi \in \mathbb{R}^{2n}$ and $z \in \mathbb{C}^n$, where $H_{\varphi}(\xi,\xi)(z)$ is the real Hessian of $\varphi$ at $z$. We prove $L^{p, \varphi}$ estimates for $\bar\partial$ on $\mathbb{C}^n$ for all $p \in [1,\infty]$. Moreover, by using the estimates for $\bar\partial$, we characterize boundedness and compactness of Hankel operators with anti-holomorphic symbols on generalized Fock spaces on $\mathbb{C}^n$.

Let $H(\mathbb{D})$ denote the space of all holomorphic functions on the unit disk $\mathbb{D}$ of $\mathbb{C}$. Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$, $n$ be a positive integer and $g\in H(\mathbb{D})$. In this paper, we investigate the boundedness and compactness of a generalized integration operator $$ I^{(n)}_{g,\varphi}f(z) = \int^z_0 f^{(n)}(\varphi(\zeta)) g(\zeta) d\zeta,\ \ z \in \mathbb{D},$$ between Bloch-type spaces and $F(p,q,s)$ spaces.

In this paper, a DC infinite programming problem with inequality constraints is considered. By using the method of Fenchel conjugate functions, a dual scheme for the DC infinite programming problem is introduced. Then, under suitable conditions, weak and strong duality assertions are obtained. Moreover, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of the DC infinite programming problem are given. As applications, the proposed approach is applied to conic programming problems.

The goal of the paper is to investigate the Hölder continuity of the solution map to a parametric optimal control problem which is governed by elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a programming problem and parametric variational inequality, we get sufficient conditions under which the solution map is Hölder continuous in parameters.

In this paper, we introduce some gap functions for set-valued mixed variational inequalities under suitable conditions. We further use these gap functions to study global error bounds for the solutions of set-valued mixed variational inequalities in Hilbert spaces. The results presented in this paper generalize and improve some corresponding known results in literatures.

In a real locally convex Hausdorff topological vector space, we first introduce the concept of nearly $E$-subconvexlikeness of set-valued maps via improvement set and obtain an alternative theorem. Furthermore, under the assumption of nearly subconvexlikeness, we establish scalarization theorem, Lagrange multiplier theorem, weak $E$-saddle point criteria and weak $E$-duality for weak $E$-optimal solution in vector optimization with set-valued maps. We also propose some examples to illustrate the main results.

We prove a Lewent type determinantal inequality: Let $A_i$, $i=1,\ldots, n$, be (strictly) contractive trace class operators over a separable Hilbert space. Then \[ \left|\det\left(\frac{I+\displaystyle\sum_{i=1}^n\lambda_iA_i}{I-\displaystyle\sum_{i=1}^n\lambda_iA_i}\right)\right|\le\prod_{i=1}^n\det\left(\frac{I+|A_i|}{I-|A_i|}\right)^{\lambda_i}, \] where $\sum_{i=1}^n \lambda_i = 1$, $\lambda_i \ge 0$, $i=1,\ldots, n$, are (scalar) weights and $|A| = (A^*A)^{1/2}$.

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For $A\subseteq \mathbb{Z}_m$ and $n \in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(\mathbb{Z}_m \setminus A,n) = \hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$. As a corollary, we prove that there exists $A\subseteq \mathbb{Z}_m$ such that $\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)=\hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$ if and only if $2d \mid m$, where $d=(k_1,m)(k_2,m)/(k_1,k_2,m)^2$. We also pose several problems for further research.

The aim of this paper is to introduce and study a new class of the extended general nonlinear quasi-variational inequalities and a new class of the extended general Wiener-Hopf equations. The equivalence between the extended general nonlinear quasi-variational inequalities and the fixed point problems, and as well as the extended general Wiener-Hopf equations is established. Then by using these equivalences, we discuss the existence and uniqueness of a solution of the extended general nonlinear quasi-variational inequalities. Applying the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping $S$, we define some new $p$-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping $S$ which is also a unique solution of the extended general nonlinear quasi-variational inequalities. The convergence analysis of the suggested iterative schemes under some suitable conditions is studied. We also suggest and analyze a class of extended general projection dynamical systems associated with the extended general nonlinear quasi-variational inequalities. We show that the trajectory of the solution of the extended general projection dynamical system converges globally exponential to a unique solution of the extended general nonlinear quasi-variational inequalities. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.

In this paper, we study a semilinear parabolic equation $$u_t = \Delta u + \int_0^t u^p ds - ku^q, \quad x \in \Omega ,\quad t\gt 0$$ with boundary condition $u(x,t) = \int_\Omega f(x,y) u^l(y,t) dy$ for $x \in \partial\Omega$, $t \gt 0$, where $p$, $q$, $l$, $k\gt 0$. The blow-up criteria and the blow-up rate are obtained under some appropriate assumptions.

In this paper, we give several new sufficient conditions for the existence of infinitely many homoclinic orbits of the second-order ordinary $p$-Laplacian system $$ \frac{d}{dt} \left(|\dot{u}(t)|^{p-2} \dot{u}(t)\right) - a(t) |u(t)|^{p-2} u(t) + \nabla W(t,u(t)) = 0, $$ where $p \gt 1, \ t\in {\mathbb{R}},\ u\in {\mathbb{R}}^{N}$, $a\in C({\mathbb{R}}, {\mathbb{R}})$ and $W\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$ are no periodic in $t$, which greatly improve the known results due to Rabinowitz and Willem.

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of $R$, or $R$ has zero characteristic and there exists a natural number $n\gt 1$ such that $\frac{1}{n}\in R$, then $R$ has a maximal subring. It is shown that if $R$ is a reduced ring with $|R|\gt 2^{2^{\aleph_0}}$ or $J(R)\neq 0$, then any $R$-algebra has a maximal subring. Residually finite rings without maximal subrings are fully characterized. It is observed that every uncountable $UFD$ has a maximal subring. The existence of maximal subrings in a noetherian integral domain $R$, in relation to either the cardinality of the set of divisors of some of its elements or the height of its maximal ideals, is also investigated.

In this paper, we investigate fixed points of meromorphic functions $f(z)$ for difference Riccati equations, and obtain some estimates of exponents of convergence of fixed points of $f(z)$ and shifts $f(z+n)$, differences $\triangle f(z) = f(z+1)-f(z)$ and divided differences $\frac{\triangle f(z)}{f(z)}$.

Using critical point theory, we study the multiplicity of solutions for some periodic and Neumann boundary value problems involving the discrete $p(\cdot)$-Laplacian.

In this paper, we consider a triple hierarchical variational inequality defined over the common solution set of minimization and mixed equilibrium problems. Combining the hybrid steepest-descent method, viscosity approximation method and averaged mapping approach to the gradient-projection algorithm, we propose two iterative methods: implicit one and explicit one, to compute the approximate solutions of our problem. The convergence analysis of the sequences generated by the proposed methods is also established.

Recently, many authors studied the relations between the squared norm of the second fundamental form (extrinsic invariant) and the warping function (intrinsic invariant) for warped product submanifolds (see [1, 7, 14]). Inspired by those relations we establish a general sharp inequality, namely $\|h\|^2 \geq 2s[\|\nabla lnf\|^2+\alpha ^2 -\beta^2]$, for contact CR-warped products of nearly trans-Sasakian manifolds. Our inequality generalizes all derived inequalities for contact CR-warped products either in any contact metric manifold. The equality case is also handled.