In this paper, we consider the generalized translations associated with the Dunkl and the Jacobi-Dunkl differential-difference operators on the real line which provide the structure of signed hypergroups on $\mathbf{R}$. Especially, we study the representation of the generalized translations of the product of two functions for these signed hypergroups.

In this paper, the authors prove that the multilinear fractional integral operator $T_{\Omega,\alpha}^{A_1,A_2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A_1,A_2}$ with rough kernel are both bounded from $L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−\alpha),\infty}$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A_1,A_2}(f)(x) =\int_{\mathbf{R}^n}\frac{R_{m_1}(A_1;x,y)R_{m_2}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2-2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A_1,A_2}(f)(x)= \sup_{r > 0}\frac{1}{r^{n−\alpha+m_1+m_2−2}}\int_{|x−y|<r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0<\alpha<n$, $\Omega \in L^s(S^{n−1})$ $(s \geq 1)$ is a homogeneous function of degree zero in $\mathbf{R}^n$, $A_i$ is a function defined on $\mathbf{R}^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor series of $A_i$ at $x$ about $y$. More precisely, $R_{m_i} (A_i;x,y) = A_i(x)− \sum\limits_{|\gamma|< m_i}\frac{1}{\gamma!}D^{\gamma}A_i(y)(x−y)^r,$ where $D^{\gamma} (A_i) \in BMO(\mathbf{R}^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.

A new generalization of Stancu polynomials based on the $q$-integers and a nonnegative integer $s$ is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated.

In this paper we extend Enestrom-Kakeya theorem to a large class of polynomials with complex coefficients by putting less restrictions on the coefficients. Our results generalise and extend many known results in this direction.

In this paper we develop the Banach contraction principle and Kannan fixed point theorem on generalized cone metric spaces. We prove a version of Suzuki and Kannan type generalizations of fixed point theorems in generalized cone metric spaces.

This paper concerns with the fractional integrals, which are also known as the Riesz potentials. A characterization for the boundedness of the fractional integral operators on generalized Morrey spaces will be presented. Our results can be viewed as a refinement of Nakai’s^[7].

Iterated function systems (IFS) were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle. In 2010, D. R. Sahu and A. Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings. In this paper, following Hutchinson, D. R. Sahu and A. Chakraborty, we present some new iterated function systems by using the so-called generalized contractive mappings, which will also cover a large range of mappings. Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.

In this paper, some weighted estimates for the multivariate Hausdorff operators are obtained. It is proved that the multivariate Hausdorff operators are bounded on $L^p$ spaces with power weights, which is based on the boundedness of multivariate Hausdorff operators on Herz spaces, and are bounded on weighted $L^p$ spaces with the weights satisfying the homogeneity of degree zero.

In this paper, we will discuss the behavior of a class of rough fractional integral operators on variable exponent Lebesgue spaces, and establish their boundedness from $L^{p_1(\cdot)}(\mathbf{R}^n)$ to $L^{p_2(\cdot)}(\mathbf{R}^n)$.

In this paper, we consider the following subadditive set-valued map $F : X \to P_0(Y):$ $$F(\sum_{i=1}^rx_i+\sum_{j=1}^sx_{r+j})\subseteq rF(\frac{\sum\limits_{i=1}^rx_i}{r})+sF(\frac{\sum\limits_{j=1}^sx_{r+j}}{s}), \forall x_i\in X, i=1,2,\cdots,r+s,$$ where $r$ and $s$ are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set-valued map.