The purpose of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions associated with a family of linear operators, which are defined by means of the Hadamard product (or convolution). Some integral preserving properties are also considered.

We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in $L_1^n[a,b]$. The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation $x_n=f\left(x_{n-2}\right)/g\left(x_{n-1}\right)$, $n\in {\mathbb{N}}_0$, where $f,g\in C\left[\left(0,\infty \right),\left(0,\infty \right)\right]$. It is shown that if $f$ and $g$ are nondecreasing, then for every solution of the equation the subsequences $\left\{x_{2n}\right\}$ and $\left\{x_{2n-1}\right\}$ are eventually monotone. For the case when $f\left(x\right)=\alpha +\beta x$ and $g$ satisfies the conditions $g\left(0\right)=1$, $g$ is nondecreasing, and $x/g\left(x\right)$ is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then $f\left(x\right)=c_1/x$ and $g\left(x\right)=c_{2}x$, for some positive $c_1$ and $c_2$.

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi-variational inequalities and are shown to be continuous and directionally differentiable.

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.

The existence and uniqueness of global mild solutions are proven for a class of semilinear nonautonomous evolution equations. Moreover, it is shown that the system, under considerations, has a unique steady state. This analysis uses, essentially, the dissipativity, a subtangential condition, and the positivity of the related $C_0$-semigroup.

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function $u$, and prove that the set of bifurcation points for the solutions of the system is not $\sigma$-compact. Then, we deal with a linear system depending on a real parameter $\lambda >0$ and on a function $u$, and prove that there exists ${\lambda }^{\ast }$ such that the set of the functions $u$, such that the system admits nontrivial solutions, contains an accumulation point.

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy of $l_1$; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of ${\left\{T^{**}\left(t\right)\right\}}_{t\ge 0}$, which, in addition, is also characterized for abstract $L$- and $M$-spaces. As a corollary, it is proven that abstract $L$-spaces with no copy of $l_1$ are finite-dimensional.

The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radius ${\mu }_0$ in $L_p([t_0,\theta ];{ℝ}^m)\hspace{0.17em}(p\in (1,+\infty ))$. Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

Weighted $L^p$ for $p\in (1,\infty )$ and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. All results with no weight on spaces of homogeneous type are also new.

For any $n$, the contour integral $y={\text{cosh}}^{n+1}x\oint C^{\left(\text{cosh}\right(zs)/{(\text{sinh}z-\text{sinh}x)}^{n+1}}d\mathrm{z,}$ $s^2=-\lambda$, is associated with differential equation $d^{2}y\left(x\right)/d{x}^2+(\lambda +n(n+1)/{\text{cosh}}^{2}x)y\left(x\right)=0$. Explicit solutions for $n=1$ are obtained. For $n=1$, eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of the trigonometric functions does not seem to have been explored and it is also one of the purposes of this paper to put it on record.

The aim of this paper, firstly, is to construct generating functions of $q$-Euler numbers and polynomials of higher order by applying the fermionic $p$-adic $q$-Volkenborn integral, secondly, to define multivariate $q$-Euler zeta function (Barnes-type Hurwitz $q$-Euler zeta function) and $l$-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitz $q$-Euler zeta function and multivariate $q$-Euler $l$-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.

Let $X,Y$ be vector spaces and $k$ a fixed positive integer. It is shown that a mapping $f(kx+y)+f\left(kx-y\right)=2k^{2}f\left(x\right)+2f\left(y\right)$ for all $x,y\in X$ if and only if the mapping $f:X\to Y$ satisfies $f(x+y)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$ for all $x,y\in X$. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

Let $\left(\Omega ,𝒜,\mu \right)$ be a probability space and $ℒ\subset 𝒜$ a sub-$\sigma$-lattice of the $\sigma$-algebra $𝒜$. We study an extension of the best $\varphi$-approximation operator from an Orlicz space $L^{\varphi }$ to the space $L^{{\varphi }^{\prime }}$, where ${\varphi }^{\prime }$ denotes the derivative of the convex, but not necessarily a strictly convex function $\varphi$. We obtain convergence results when a sequence of $\sigma$-algebras ${ℬ}_{\text{n}}$ converges to ${ℬ}_{\infty }$ in a suitable way.

We estimate the essential norm of a compact weighted composition operator $u{C}_{\phi }$ acting between different Hardy spaces of the unit ball in ${ℂ}^N$. Also we will discuss a compact multiplication operator between Hardy spaces.

We give the twisted Carlitz's type $q$-Bernoulli polynomials and numbers associated with $p$-adic $q$-inetgrals and discuss their properties. Furthermore, we define the multiple twisted Carlitz's type $q$-Bernoulli polynomials and numbers and obtain the distribution relation for them.

For a certain range of the value $p$ in the nonlinear term ${\left|u\right|}^{p}u$, in this paper we mainly study the global existence and uniqueness of global self-similar solutions to the Cauchy problem for some nonlinear Schrödinger equations using the method of harmonic analysis.

A function $f$ is continuous if and only if, for each point $x_0$ in the domain, ${\lim }_{n\to \infty }f\left(x_n\right)=f\left(x_0\right)$, whenever ${\lim }_{n\to \infty }{x}_n=x_0$. This is equivalent to the statement that $\left(f\left(x_n\right)\right)$ is a convergent sequence whenever $\left(x_n\right)$ is convergent. The concept of slowly oscillating continuity is defined in the sense that a function $f$ is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, $\left(f\left(x_n\right)\right)$ is slowly oscillating whenever $\left(x_n\right)$ is slowly oscillating. A sequence $\left(x_n\right)$ of points in $\mathbf{R}$ is slowly oscillating if ${\lim }_{\lambda \to 1^{+}}{{\bar{\lim }}_n{\max }_{n+1\le k\le [\lambda n]}}_{}\left|x_k-x_n\right|=0$, where $\left[\lambda n\right]$ denotes the integer part of $\lambda n$. Using $\varepsilon >0$'s and $\delta$'s, this is equivalent to the case when, for any given $\varepsilon >0$, there exist $\delta =\delta \left(\varepsilon \right)>0$ and $N=N\left(\varepsilon \right)$ such that if $n\ge N\left(\varepsilon \right)$ and $n\le m\le \left(1+\delta \right)n$. A new type compactness is also defined and some new results related to compactness are obtained.

We consider an integral variational control system on a Banach space $X$ and we study the connections between its uniform exponential stability and the $\left(I\left({ℝ}_{+},X\right),O\left({ℝ}_{+},X\right)\right)$ stability, where $I$ and $O$ are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.

We obtain the general solution and the stability of the functional equation $f\left(x+y+z,u+v+w\right)+f\left(x+y-z,u+v+w\right)+2f\left(x,u-w\right)+2f\left(y,v-w\right)=f\left(x+y,u+w\right)+f\left(x+y,v+w\right)+f\left(x+z,u+w\right)+f\left(x-z,u+v-w\right)+f\left(y+z,v+w\right)+f\left(y-z,u+v-w\right)$. The function $f\left(x,y\right)=x^3+ax+b-y^2$ having level curves as elliptic curves is a solution of the above functional equation.

We need to take account of the superstability for generalized left derivations (resp., generalized derivations) associated with a Jensen-type functional equation, and we also deal with problems for the Jacobson radical ranges of left derivations (resp., derivations).

We will introduce and study the pairwise weakly regular-Lindelöf bitopological spaces and obtain some results. Furthermore, we study the pairwise weakly regular-Lindelöf subspaces and subsets, and investigate some of their characterizations. We also show that a pairwise weakly regular-Lindelöf property is not a hereditary property. Some counterexamples will be considered in order to establish some of their relations.

Commutative symmetric operator families of the so-called $D_{\kappa }^{+}$-class are considered in Krein spaces. It is proved that the restriction of a family of this type on a special kind of invariant subspace is similar to a family of operators adjoint to multiplication operators by scalar functions acting on a suitable function space.

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

For $s\in ℂ$, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ${\zeta }_E\left(s\right)=2{\sum }_{n=1}^{\infty }\left({\left(-1\right)}^n/n^s\right)$, and ${\zeta }_E\left(s,x\right)=2{\sum }_{n=0}^{\infty }\left({\left(-1\right)}^n/{\left(n+x\right)}^s\right)$. Thus, we note that the Euler zeta functions are entire functions in whole complex $s$-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, ${\zeta }_E\left(-k\right)=E_k^{\ast }$, and ${\zeta }_E\left(-k,x\right)=E_k^{\ast }\left(x\right)$. We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

We introduce a $q$-analogues of Wright function and its auxiliary functions as Barnes integral representations and series expansion. The relations between $q$-analogues of Wright function and some other functions are investigated.

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionic $p$-adic invariant integral on ${\mathbb{Z}}_p$, we construct $p$-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula: $G_{n+k,q}^{\left(k\right)}\left(x\right)=2^{k}k!\left(\begin{array}{c}n+k\\ k\end{array}\right){\sum }_{l=0}^{\infty }{\sum }_{d_0+d_1+\cdots +d_k=k-1,d_i\in \mathbb{N}}{\left(-1\right)}^l{\left(l+x\right)}^n$, where $G_{n+k,q}^{\left(k\right)}\left(x\right)$ are the $q$-Genocchi polynomials of order $k$.

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

A molecular motor utilizes chemical free energy to generate a unidirectional motion through the viscous fluid. In many experimental settings and biological settings, a molecular motor is elastically linked to a cargo. The stochastic motion of a molecular motor-cargo system is governed by a set of Langevin equations, each corresponding to an individual chemical occupancy state. The change of chemical occupancy state is modeled by a continuous time discrete space Markov process. The probability density of a motor-cargo system is governed by a two-dimensional Fokker-Planck equation. The operation of a molecular motor is dominated by high viscous friction and large thermal fluctuations from surrounding fluid. The instantaneous velocity of a molecular motor is highly stochastic: the past velocity is quickly damped by the viscous friction and the new velocity is quickly excited by bombardments of surrounding fluid molecules. Thus, the theory for macroscopic motors should not be applied directly to molecular motors without close examination. In particular, a molecular motor behaves differently working against a viscous drag than working against a conservative force. The Stokes efficiency was introduced to measure how efficiently a motor uses chemical free energy to drive against viscous drag. For a motor without cargo, it was proved that the Stokes efficiency is bounded by 100% [H. Wang and G. Oster, (2002)]. Here, we present a proof for the general motor-cargo system.

We derive two sets of explicit homogeneous algebraic constraint-preserving boundary conditions for the second-order in time reduction of the linearized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system. Our second-order reduction involves components of the linearized extrinsic curvature only. An initial-boundary value problem for the original linearized BSSN system is formulated and the existence of the solution is proved using the properties of the reduced system. A treatment is proposed for the full nonlinear BSSN system to construct constraint-preserving boundary conditions without invoking the second order in time reduction. Energy estimates on the principal part of the BSSN system (which is first order in temporal and second order in spatial derivatives) are obtained. Generalizations to the case of nonhomogeneous boundary data are proposed.

The authors introduce new classes of analytic functions in the open unit disc which are defined by using multiplier transformations. The properties of these classes will be studied by using techniques involving the Briot-Bouquet differential subordinations. Also an integral transform is established.

We prove that the semilinear elliptic equation $-\Delta u=f\left(u\right)$, in $\Omega$, $u=0$, on $\partial \Omega$ has a positive solution when the nonlinearity $f$ belongs to a class which satisfies $\mu t^q\le f\left(t\right)\le C{t}^p$ at infinity and behaves like $t^q$ near the origin, where if $N\ge 3$ and if $N=1,2$. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of $p$.

Let ${\delta }_{\text{X}}\left(ϵ\right)$ and $R\left(1,X\right)$ be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space $X$ has normal structure if $2{\delta }_{\mathrm{X}}(1+ϵ)>\text{max}\left\{\left(\mathrm{R}\left(1,\mathrm{x}\right)-1\right)ϵ,1-\left(1-ϵ/R\left(1,X\right)-1\right)\right\}$ for some $ϵ\in \left[0,1\right]$ which generalizes the known result by Gao and Prus.

In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz's type $q$-Bernoulli polynomials and obtained the distribution relation for them. They also raised the following problem: “are there analytic multiple twisted Carlitz's type $q$-zeta functions which interpolate multiple twisted Carlitz's type $q$-Euler (Bernoulli) polynomials?” The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twisted $q$-extension of Euler polynomials and multiple twisted Carlitz's type $q$-Euler polynomials.

We prove that a two-variable $p$-adic $l_q$-function has the series expansion $l_{p,q}(s,t,\chi )={\left(\right[2]}_q/{\left[2\right]}_F\left){\sum }_{a=1,(p,a)=1}^F{(-1)}^a\right(\chi \left(a\right)q^a/{\langle a+pt\rangle }^s){\sum }_{m=0}^{\infty }\left(\begin{array}{c}-s\\ m\end{array}\right){(F/\langle a+pt\rangle )}^m{E}_{m,q^F}^{*}$ which interpolates the values $l_{p,q}\left(-n,t,\chi \right)=E_{n,{\chi }_n,q}^{\ast }\left(pt\right)-p^n{\chi }_n\left(p\right)({\left[2\right]}_q/{\left[2\right]}_{q^p})E_{n,{\chi }_n,q^p}^{\ast }\left(t\right)$, whenever $n$ is a nonpositive integer. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given in this note is due to Washington.

Let $B_n$ be the unit ball of $C^n$, $H({\mathbb{B}}_n)$ the space of all holomorphic functions in $B_n$. Let $u\in H(B_n)$ and $\alpha$ be a holomorphic self-map of $B_n$. For $f\in H({\mathbb{B}}_n)$, the weigthed composition operator $u{C}_{\alpha }$ is defined by $(u{C}_{\alpha }f)(z)=u(z)f(\alpha (z)),z\in B_n\mathrm{.}$ The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.

We study the $h$-stability of dynamic equations on time scales, without the regressivity condition on the right-hand side of dynamic equations. This means that we can include noninvertible difference equations into our results.

Let ${ℂ}_{\ge 0}:=\{s\in ℂ\mid \text{Re}(s)\ge 0\}$, and let ${𝒲}^{+}$ denote the ring of all functions $f:{ℂ}_{\ge 0}\to ℂ$ such that $f(s)=f_a(s)+{\displaystyle {\sum }_{k=0}^{\infty }{f}_k{e}^{-s{t}_k}}\hspace{0.17em}(s\in {ℂ}_{\ge 0})$, where $f_a\in L^1\left(0,\infty \right),\hspace{0.17em}{\left(f_k\right)}_{k\ge 0}\in {\ell }^1$, and equipped with pointwise operations. (Here $\widehat{\cdot }$ denotes the Laplace transform.) It is shown that the ring ${𝒲}^{+}$ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain ${𝒲}^{+}$ whose intersection is not finitely generated.

This paper gives some sufficient conditions for an analytic function to belong to the space consisting of all analytic functions $f$ on the unit disk such ${\lim }_{\left|a\right|\to 1}{\int }_{\hspace{0.17em}D}^{}{|f^{\prime }(z)|}^p{(1-{\left|z\right|}^2)}^{q}K\left(g\left(z,a\right)\right)dA\left(z\right)=0.$

The functional inequality $\Vert f\left(x\right)+2f\left(y\right)+2f\left(z\right)\Vert \le \Vert 2f\left(x/2+y+z\right)\Vert +\varphi \left(x,y,z\right)\left(x,y,z\in G\right)$ is investigated, where $G$ is a group divisible by $2,f:G\to X$ and $\varphi :G^3\to \left[0,\infty \right)$ are mappings, and $X$ is a Banach space. The main result of the paper states that the assumptions above together with (1) $\varphi \left(2x,-x,0\right)=0=\varphi \left(0,x,-x\right)$ $\left(x\in G\right)$ and (2) ${\lim }_{n\to \infty }\left(1/2^n\right)\varphi \left(2^{n+1}x,2^{n}y,2^{n}z\right)=0$, or ${\lim }_{n\to \infty }{2}^n\varphi \left(x/2^{n-1},y/2^n,z/2^n\right)=0\left(x,y,z\in G\right)$, imply that $f$ is additive. In addition, some stability theorems are proved.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of $C^{\ast }$-algebra homomorphisms and of generalized derivations on $C^{\ast }$-algebras for the following functional equation of Apollonius type .

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

We give the existence of multiple twisted $p$-adic $q$-Euler $\zeta$-functions and $l$-functions, which are generalization of the twisted $p$-adic $\left(h,q\right)$-zeta functions and twisted $p$-adic $\left(h,q\right)$-Euler $l$-functions in the work of Ozden and Simsek (2008).

Kupershmidt and Tuenter have introduced reflection symmetries for the $q$-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of the $q$-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the $q$-Bernoulli polynomials, we can obtain some interesting relationships between $q$-Bernoulli numbers and polynomials.

We will study a new $q$-analogue of Bernoulli polynomials associated with $p$-adic $q$-integrals. Furthermore, we examine the Hurwitz-type $q$-zeta functions, replacing $p$-adic rational integers $x$ with a $q$-analogue ${\left[x\right]}_q$ for a $p$-adic number $q$ with , which interpolate $q$-analogue of Bernoulli polynomials.

We establish two inequalities concerning the weakly convergent sequence coefficient and other parameters, which enable us to obtain some sufficient conditions for normal structure.

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from $\alpha$-Bloch spaces to the weighted-type space $H_{\mu }^{\infty }$ on the unit ball for the case $\alpha \ge 1$.

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

This paper shows that if $S$ is a bounded linear operator acting on the weighted Bergman spaces $A_{\alpha }^2$ on the unit ball in ${ℂ}^n$ such that $S{T}_{z_i}=T_{{\overline{z}}_i}S\left(i=1,\dots ,n\right)$, where $T_{z_i}=z_{i}f$ and $T_{{\overline{z}}_i}=P\left({\overline{z}}_{i}f\right)$; and where $P$ is the weighted Bergman projection, then $S$ must be a Hankel operator.

Recently, Choi et al. (2008) have studied the $q$-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order $n$ and multiple Hurwitz zeta function. In this paper, we define Apostol's type $q$-Euler numbers $E_{n,q,\xi }$ and $q$-Euler polynomials $E_{n,q,\xi }\left(x\right)$. We obtain the generating functions of $E_{n,q,\xi }$ and $E_{n,q,\xi }\left(x\right)$, respectively. We also have the distribution relation for Apostol's type $q$-Euler polynomials. Finally, we obtain $q$-zeta function associated with Apostol's type $q$-Euler numbers and Hurwitz's type $q$-zeta function associated with Apostol's type $q$-Euler polynomials for negative integers.

This paper gives some estimates of the essential norm for the difference of composition operators induced by $\phi$ and $\psi$ acting on the space, $H^{\infty }\left({\mathbb{D}}^n\right)$, of bounded analytic functions on the unit polydisc ${\mathbb{D}}^n$, where $\phi$ and $\psi$ are holomorphic self-maps of ${\mathbb{D}}^n$. As a consequence, one obtains conditions in terms of the Carathéodory distance on ${\mathbb{D}}^n$ that characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on $H^{\infty }\left({\mathbb{D}}^n\right)$ is compact.

We first define upper sign continuity for a set-valued mapping and then we consider two types of generalized vector equilibrium problems in topological vector spaces and provide sufficient conditions under which the solution sets are nonempty and compact. Finally, we give an application of our main results. The paper generalizes and improves results obtained by Fang and Huang in (2005).

Impulsive Cohen-Grossberg neural networks with bounded and unbounded delays (i.e., mixed delays) are investigated. By using the Leray-Schauder fixed point theorem, differential inequality techniques, and constructing suitable Lyapunov functional, several new sufficient conditions on the existence and global exponential stability of periodic solution for the system are obtained, which improves some of the known results. An example and its numerical simulations are employed to illustrate our feasible results.

We investigate Jordan $\ast$-derivations on $C^{\ast }$-algebras and Jordan $\ast$-derivations on $J{C}^{\ast }$-algebras associated with the following functional inequality $\Vert f\left(x\right)+f\left(y\right)+kf\left(z\right)\Vert \le \Vert kf\left(\left(x+y\right)/k+z\right)\Vert$ for some integer $k$ greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan $\ast$-derivations on $C^{\ast }$-algebras and of Jordan $\ast$-derivations on $J{C}^{\ast }$-algebras associated with the following functional equation $f\left(\left(x+y\right)/k+z\right)=\left(f\left(x\right)+f\left(y\right)\right)/k+f\left(z\right)$ for some integer $k$ greater than 1.

We consider a mathematical model to describe the growth of a vascular tumor including tumor cells, macrophages, and blood vessels. The resulting system of equations is reduced to a strongly $2\times 2$ coupled nonlinear parabolic system of degenerate type. Assuming the initial data are far enough from 0, we prove existence of a global weak solution with finite entropy to the problem by using an approximation procedure and a time discrete scheme.

We obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation $4\left(f\left(3x+y\right)+f\left(3x-y\right)\right)=-12\left(f\left(x+y\right)+f\left(x-y\right)\right)+12\left(f\left(2x+y\right)+f\left(2x-y\right)\right)-8f\left(y\right)-192f\left(x\right)+f\left(2y\right)+30f\left(2x\right)$.

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the $T_0$-periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. By virtue of the generalized new Gronwall lemma with impulse and $B$-norm, the estimate on the PC-mild solutions is derived. Showing the continuity and compactness of the Poincaré operator, we utilize Horn's fixed point theorem to prove the existence of $T_0$-periodic PC-mild solutions when the PC-mild solutions are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

We work on RD-spaces $𝒳$, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $𝒳$. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spaces $H^p\left(𝒳\right)$ and local Hardy spaces $h^p\left(𝒳\right)$ on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces when $p\ge 1$. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains in ${ℂ}^N$. Our theory applies in a wide range of settings.

The nonmonotonic differentiable solutions of equations involving iterated functional series are investigated. Conditions for the existence, uniqueness, and stability of such solutions are given. These extend earlier results due to Murugan and Subrahmanyam.