We derive an integral over the $m$-dimensional unit hypercube that generalizes Bessel’s integral for $J_n(x)$. The integrand is $G(x\psi(t)) \exp(−2\pi i n \cdot t)$, where $G$ is analytic, and $\psi(t) = e^{2\pi it_1} + \ldots + e^{2\pi it_m} + e^{−2\pi i(t_1+...+t_m)}$, while n is a set of non-negative integers. In particular, we consider the case when $G$ is a hypergeometric function $_{p}F_q$.

In this paper we study the abstract Cauchy problem for quasi-linear evolution equation $u'(t) = A(u(t)) u(t)$, where $\{ A(w); w \in W \}$ is a family of closed linear operators in a real Banach space $X$ such that $D(A(w)) = Y$ for $w \in W$, and $W$ is an open subset of another Banach space $Y$ which is continuously embedded in $X$. The purpose of this paper is not only to establish a ‘global’ well-posedness theorem without assuming that $Y$ is dense in $X$ but also to propose a new type of dissipativity condition which is closely related with the continuous dependence of solutions on initial data.

In this paper the authors give the weighted weak $(1,1)$ boundedness and the weighted $L^p$ boundedness for the Littlewood-Paley operators with complex parameters.

In this paper we prove that generalized Jordan derivations and generalized Jordan triple derivation of 2−torsion free semiprime rings are generalized derivations.

We establish some properties concerning the linear form $\mathcal{B}[\nu]$ which is symmetric $D$-semiclassical of class 1. An integral representation is obtained. A connection with the $D$-classical Bessel one is discussed.

Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, sharp estimates of the radii of convergence balls of Newton-like methods for operator equations are given in Banach space. New results can be used to analyze the convergence of other developed Newton iterative methods.

In this paper we obtain a convolution identity for the coefficients $B_n(\alpha,\theta,q)$ defined by \[ \sum_{n=-\infty}^{\infty} B_{n}(\alpha,\theta,q) x^{n} = \frac{\prod\limits_{n=1}^{\infty} (1 + 2x q^n \cos \theta + x^2 q^{2n})}{\prod\limits_{n=1}^{\infty} (1 + \alpha q^n xe^{i\theta})}, \] using the well-known Ramanujan’s $_{1}\psi_1$-summation formula. The work presented here complements the works of K.-W. Yang, S. Bhargava, C. Adiga and D. D. Somashekara and of H. M. Srivastava.

We illustrate a general method, which is useful for the solution of integro-differential equations, and apply the technique to solve the betatron equations of motion with the inclusion of wake field effects.

A product of doubling measures on the real line can be defined in such a way that another doubling measure on the line is obtained. It follows that doubling measures on the line form a semiring.

We consider the following differential equation on a time scale $T$ \[ y^{\Delta \Delta}(t) + P(t, y(\sigma(t))) = 0, \; t \in [a,b] \cap T \] subject to conjugate boundary conditions \[ y(a) = 0, \qquad y(\sigma^{2}(b)) = 0 \] where $a,b \in T$ and $a \lt \sigma(b)$. By using different fixed point theorems, criteria are established for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained.

Metric noncommutative geometry, initiated by Alain Connes, has known some great recent developments under the impulsion of Rieffel and the introduction of the category of compact quantum metric spaces topologized thanks to the to quantum Rieffel-Gromov-Hausdorff distance. In this paper, we undertake the first step to generalize such results and constructions to locally compact quantum metric spaces. Our present work shows how to generalize the construction of the bounded-Lipschitz metric on the state space of a C*-algebra which need not be unital, such that the topology induced by this distance on the state space is the weak* topology. In doing so we obtain some results on a state space picture of the strict topology of a C*-algebra.

Various characterizations are given of real and complex Hilbert space operators with the numerical range in a given closed half plane.

The boundedness of multilinear commutators of singular integral operators with non-smooth kernels on spaces of homogeneous type is obtained.

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of $T$-periodic solutions for the first order neutral functional differential equation with a deviating argument of the form \[ (x(t) + Bx(t-\delta))' = g_1(t,x(t)) + g_2(t, x(t-\tau(t))) + p(t). \]

In this paper the strong convergence for weighted sums of linearly negative quadrant dependent(LNQD) arrays is discussed. The central limit theorem for weighted sums of LNQD variables and linear process based on LNQD variables is also considered. Finally the results on i.i.d. of Li et al. ([7]) in LNQD setting are obtained.

By introducing the concept of weakly completely continuous subspaces of operator ideals, it will be given some characterizations of this concept, specially in terms of relative weak compactness of all point evaluations related to that subspace. Also it is shown that the only Banach spaces such that all closed subspace of an operator ideal between them has this property, are reflexive Banach spaces.

In this paper, we give connections between distribution cosine functions (defined in [10]) and almost-distribution cosine functions (introduced in [13]). We prove several equalities involving trigonometric convolution products and distribution cosine functions as well as some relations between distribution cosine functions and ultradistribution semigroups.

Let $R$ be any ring and let $M$ be any right $R$-module. $M$ is called hollow-lifting if every submodule $N$ of $M$ such that $M/N$ is hollow has a coessential submodule that is a direct summand of $M$. We prove that every amply supplemented hollow-lifting module with finite hollow dimension is lifting. It is also shown that a direct sum of two relatively projective hollowlifting modules is hollow-lifting.

The concept of $\mathcal{I}$-convergence is a generalization of statistical convergence and it is dependent on the notion of the ideal $\mathcal{I}$ of subsets of the set $\mathbb{N}$ of positive integers. In this paper we prove a decomposition theorem for $\mathcal{I}$-convergent sequences and we introduce the notions of $\mathcal{I}$ Cauchy sequence and $\mathcal{I}^{*}$-Cauchy sequence, and then study their certain properties.