In this paper, by introducing norm $\|x\|_{\alpha}(x\in R^{n})$, we give a multidimensional Hardy-Hilbert's integral inequality with two parameters $\alpha$, $\lambda$ and best constant factor.

We study maximal regularity on Triebel-Lizorkin spaces $\mathrm{F} _{p,q}^s(\mathbb T, X)$ for the integro-differential equation with infinite delay: ($P_2$): $u'(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), \ (0\leq t \leq2\pi$) with the periodic condition $u(0)=u(2\pi)$, where $X$ is a Banach space, $a\in {\mathrm L}^1(\mathbb R_+)$ and $f$ is an $X$-valued function. Under a suitable assumption (H3) on the Laplace transform of $a$, we give a necessary and sufficient condition for ($P_2$) to have the maximal regularity property on $\mathrm{F} _{p,q}^s(\mathbb T, X)$.

In this paper we define a general class of $n$-starlike functions with respect to a convex domain $D$ contained in the right half plane by using a generalized Sǎlǎgean operator introduced by F. M. Al-Oboudi in [2] and we give some properties of this class.

In this paper, generalizing Marrero's construction, we introduce the concept of $J_m-$Hadamard matrices, and by allowing permutations, we construct other $2^mm!-1$ $J_m$-Hadamard matrices from a given one of order $mt$; previous construction generated only other $2^m-1$ ones. We also generalize Craigen's construction of products of two Hadamard matrices to those of several Hadamard matrices and a $J_m$-Hadamard matrix, yielding generalizations of Craigen's results. Furthermore, we introduce the $J_m$-class $CJ_m$ for $m=2$ or $4k$ and study the partially ordered set $\mathfrak{M}$ of $J_m-$classes $CJ_m$. Our main result shows that $CJ_8\subsetneqq CJ_4\subsetneqq CJ_2$.

In this paper, the norm of a bounded self-adjoint integral operator $T:$ $% L^{2}(0,\infty )\rightarrow L^{2}(0,\infty )$ is obtained. As applications, a new bilinear integral inequality with a best constant factor and some particular cases such as Hilbert- type inequalities are considered.

In this paper we sharpen and generalize the Jordan’s inequality, our results unify and optimize some corresponding known results in the recent papers. As application, the obtained results are used to improve the wellknown L. Yang’s inequality.

Let $G$ be a MAP group, let $\mu$ be a bounded complex-valued Borel measure on $G$, and let $T_{\mu}$ be the corresponding convolution operator on $L^{1}(G)$. Let $X$ be a Banach space and let $S$ be a continuous linear operator on $X$. We show that every linear operator $\Phi:X \rightarrow L^{1}(G)$ such that $\Phi S = T_{\mu} \Phi$ is continuous if, and only if, the pair $(S,T_{\mu})$ has no critical eigenvalue.

In this paper, the $n-$species neutral delay competitive differential system with periodic coefficients is investigated by means of an abstract continuous theorem of $k-$set contractive operator and some analysis techniques. Sufficient conditions are obtained for periodic solution.

Some results related to Pečarić’s type generalisation of Bessel’s inequality in 2-inner product spaces are given. Applications for determinantal integral inequalities are also provided.

The main purpose of this paper is to find the conditions for the Minimal Horse-Shoe Lemma to be hold in some categories of graded modules. Using the Minimal Horse-Shoe Lemma, we prove that the category of p- Koszul modules is closed under direct sums, direct summands, extensions and cokernels.

Let $H(D)$ be an algebra of all holomorphic functions on the open unit disc $D$ and $X$ a subspace of $H(D)$. When $g$ is a function in $H(D)$, put $$ J_g(f)(z) = \int^z_0 f(\zeta)g^\prime(\zeta)d\zeta~{\rm and}~I_g(f)(z) = \int^z_0 f^\prime(\zeta)g(\zeta)d\zeta \quad (z \in D) $$ for $f$ in $X$. In this paper, we study $J[X] = \{g \in H(D)~;~J_g(f) \in X$ for all $f$ in $X \}$ and $I[X] = \{g \in H(D)~;~I_g(f) \in X$ for all $f$ in $X \}$. We apply the results to concrete spaces. For example, we study $J[X]$ and $I[X]$ when $X$ is a weighted Bloch space, a Hardy space or a Privalov space.

This paper is concerned with the anti-periodic boundary value problems for nonlinear higher order impulsive differential equations \begin{align*}\left\{\!\!\begin{array}{rcl} x^{(n)}(t)& = & f(t,x(t),x'(t),\cdots,x^{(n-1)}(t)),\;t\in [0,T],\;t\not=t_k,\;k=1,\cdots,p,\\ \Delta x^{(i)}(t_k)& \hspace{-0.3cm} = \hspace{-0.3cm}& I_{i,k}(x(t_k),x'(t_k),\cdots,x^{(n-1)}(t_k)),\;k=1,\cdots,p,i=0,\cdots,n-1,\\ x^{(i)}(0)& \hspace{-0.3cm} = \hspace{-0.3cm}& -x^{(i)}(T),\;i=0,\cdots,n-1. \end{array}\right.\end{align*} We obtain sufficient conditions for the existence of at least one solution. Examples are presented to illustrate the main results.

Let $R$ be a prime ring, with extended centroid $C$, $g$ a non-zero generalized derivation of $R$, $L$ a non-central Lie ideal of $R$, $k\geq 1$ a fixed integer. If $[g(u),u]_k=0$, for all $u$, then either $g(x)=ax$, with $a \in C$ or $R$ satisfies the standard identity $s_4$. Moreover in the latter case either $char(R)=2$ or $char(R)\neq 2$ and $g(x)=ax+xb$ , with $a,b \in Q$ and $a-b\in C$. We also prove a more generalized version by replacing $L$ with the set $[I,I]$, where $I$ is a right ideal of $R$.

In this paper, we introduce a new system of generalized cocomplementarity problems in Banach space. An iterative algorithm for finding approximate solutions of these problems is considered. Some convergence results for this iterative algorithm are derived and several existence results are also obtained.

Let $R$ be a ring, $M$ a right $R$-module and $(S,\leq)$ a strictly ordered monoid. In this paper, a necessary and sufficient condition is given for modules under which $[[M^{S,\leq}]]_{[[R^{S,\leq}]]}$, the module of generalized power series with coefficients in $M$ and exponents in $S$ is a reduced, Baer, PP. quasi-Baer module, respectively.

The Opial inequality is of great interest in differential and difference equations, and other areas of mathematics. The purpose of this paper is to generalize the Opial inequality to some time scale versions. One of these results says: \begin{align*} & \int_a^b h(x)\{|g(x)|^p|f^{\Delta^n}(x)|^q+|f(x)|^p|g^{\Delta^n}(x)|^q\}\Delta x\\ &\quad \leq{2q\over{p+q}}[({{b-a}\over {2}})^p]^n\int_a^b h(x)\{|f^{\Delta^n}(x)|^{p+q}+|g^{\Delta^n}(x)|^{p+q}\}\Delta x, \end{align*} if $p \ge 1$, $q \ge 1$ and $f,\ g \in C_{rd}([a,b], \Bbb {R})$ satisfy some suitable conditions.

Let $\alpha$ be a ring endomorphism. We introduce $\alpha$-skew Armendariz modules and $\alpha$-semicommutative modules which are generalizations of Armendariz modules and semicommutative modules, respectively. And investigate their properties. Moreover, we study the relationship between a module and its polynomial module.

Let $X$ be a reflexive and smooth Banach space which has a weakly sequentially continuous duality mapping. We consider in this paper the iteration scheme $x_{n+1}=\lambda_{n+1}y+(1-\lambda_{n+1})T_{n+1}x_n$ for infinitely many nonexpansive maps $T_1,T_2,T_3,...$ in $X$ as well as for finitely many nonexpansive retraction. We establish several strong convergence results which generalize [10, Theorem 3.3] and [10, Theorem 4.1] from Hilbert space setting to Banach space setting.

In this paper, we introduce the concept of $AFG$ rings. $R$ is said to be a left $AFG$ ring in case the left annihilator of every nonempty subset of $R$ is a finitely generated left ideal. Some characterizations of $AFG$ rings and applications are obtained.

Sierpiński graphs $S(n,3)$ are the graphs of the Tower of Hanoi puzzle with $n$ disks, while Sierpiński gasket graphs $S_n$ are the graphs naturally defined by the finite number of iterations that lead to the Sierpiński gasket. An explicit labeling of the vertices of $S_n$ is introduced. It is proved that $S_n$ is uniquely 3-colorable, that $S(n,3)$ is uniquely 3-edge-colorable, and that $\chi'(S_n)=4$, thus answering a question from~[15]. It is also shown that $S_n$ contains a 1-perfect code only for $n=1$ or $n=3$ and that every $S(n,3)$ contains a unique Hamiltonian cycle.

In this paper, using the concept of $A-$statistical convergence we are concerned with the Korovkin type approximation theory for a sequence of positive convolution operators defined on $C[a,b],$ the space of all real valued continuous functions on $[a,b].$ We also study rates of $A-$% statistical convergence of these operators.

An answer to the conjecture in [1] is given.

Recently, Erkus and Srivastava [Integral Transform. Spec. Funct.\textit{\ }% 174 (2006), 267-273] have introduced and systematically investigated a unified presentation of some families of multivariable polynomials. In this paper, we study a basic (or $q-$) analogue of these polynomials, which we construct here.

In this article, we complete the proof that the extended Shapley value has w-consistent property proposed by Hsiao, Yeh and Mo [4]. Then we suggest an axiomatization which is the parallel of Hart and Mas-Colell’s [1] axiomatization of the Shapley value by applying the w-consistency property.