We introduce the definition of $q$-Stancu operator and investigate its approximation and shape-preserving property. With the help of the sign changes of $f (x)$ and $L_n = f ( f ,q;x)$ the shape-preserving property of $q$-Stancu operator is obtained.

In this paper, we extend Hu and Zhang’s results in [2] to different cases.

In this paper we apply Bishop-Phelps property to show that if $X$ is a Banach space and $G \subseteq X$ is the maximal subspace so that $G^\bot = \{x^* \in X^*|x^*(y) = 0; \forall y \in G\}$ is an $L$-summand in $X^*$, then $L^1(\Omega,G)$ is contained in a maximal proximinal subspace of $L^1(\Omega,X)$.

The main purpose of this paper is to prove a collection of new fixed point theorems and existence theorems for the nonlinear operator equation $F(x) = \alpha x $ $(\alpha \geq 1)$ for so-called 1-set weakly contractive operators on unbounded domains in Banach spaces. We also introduce the concept of weakly semi-closed operator at the origin and obtain a series of new fixed point theorems and the existence theorems for the nonlinear operator equation $F(x) = \alpha x $ $(\alpha \geq 1)$  for such class of operators. As consequences, the main results generalize and improve the relevant results, which are obtained by O’Regan and A. Ben Amar and M. Mnif in 1998 and 2009 respectively. In addition, we get the famous fixed point theorems of Leray-Schauder, Altman, Petryshyn and Rothe type in the case of weakly sequentially continuous, 1-set weakly contractive ($\mu$-nonexpansive) and weakly semi-closed operators at the origin and their generalizations. The main condition in our results is formulated in terms of axiomatic measures of weak compactness.

In this paper, we give error estimates for the weighted approximation of $r$-monotone functions on the real line with Freud weights by Bernstein-type operators.

Let $0 <p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang^[11] established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.

We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0,  & x \in R, y \in R, 0 < z \leq 1,\\u(x,y,0) = g(x,y), & x \in R, y \in R,\\u_z(x,y,0) = 0,  &  x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 < z < 1$. The problem is ill-posed, the solution (if it exists) doesn’t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.

In this paper, we consider a class of Banach space valued singular integrals. The $L^p$ boundedness of these operators has already been obtained. We shall discuss their boundedness from BMO to BMO. As applications, we get BMO boundedness for the classic $g$-function and the Marcinkiewicz integral. Some known results are improved.

The present paper deals with the new type of Gamma operators, here we estimate the rate of pointwise convergence of these new Gamma type operators $M_{n,k}$ for functions of bounded variation, by using some techniques of probability theory.