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The purpose of this paper is to study the following system of nonlinear operator equations for N-variable pseudo-contractive mappings $$T_1, T_2,\ldots ,T_N$$ in a Banach space: $$\begin{aligned} {\left\{ \begin{array}{ll} T_1(x_{1},x_{2},\ldots ,x_{N})=x_1,\\ T_2(x_{1},x_{2},\ldots ,x_{N})=x_2,\\ T_3(x_{1},x_{2},\ldots ,x_{N})=x_3,\\ \ \ \ \ \ \ \ \cdots \\ T_N(x_{1},x_{2},\ldots ,x_{N})=x_N.\\ \end{array}\right. } \end{aligned}$$ The existence theorems of solutions are proved using a comprehensive analytical method. To get the expected results, the reflexivity of the product of Banach space and Opial’s condition of the product of Banach spaces are discussed. Meanwhile, the weak topology of the product of Banach spaces and normal structure of the product of Banach spaces are also discussed. The results of this article improve and extend the previous classical results given in the literature.

We prove some existence theorems for solutions of a certain system of multivariate nonexpansive operator equations and calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. The results of this paper improve and extend the previously known ones in the literature.

NaGd(WO4)2:RE3+ (RE = Eu, Tb, Tm) phosphors were synthesized by a hydrothermal process. The XRD and FTIR results show that the hydrothermal time has obvious influence on the crystallization of samples. The longer hydrothermal time benefits to the higher crystallization of NaGd(WO4)2:RE3+ samples. All of the excitation spectra of NaGd(WO4)2:RE3+ samples show the charge transfer bands, which suggests the effective energy transfer between the WO4 2− group and RE3+ ions. Under the excitation of 288 nm, NaGd(WO4)2:RE3+ (RE = Eu, Tb, Tm) phosphors show emissions in red, green, and blue regions, respectively. The results of this work indicate that NaGd(WO4)2:RE3+ (RE = Eu, Tb, Tm) phosphors have potentials to be used as red, green, and blue emitting phosphors.

Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN be N-variables monotone demi-continuous mappings from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN into X. Then: (1) the system of multivariate variational inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$\end{document}{〈A1(x1,x2,…,xN),y1−x1〉≥0,∀y1∈K,〈A2(x1,x2,…,xN),y2−x2〉≥0,∀y2∈K,⋯〈AN(x1,x2,…,xN),yN−xN〉≥0,∀yN∈K, has a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$\end{document}(x1∗,x2∗,…,xN∗)∈KN; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN; (3) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN are also strictly monotone, this system of multivariate variational inequalities has a unique solution.

Sr3Al2O6:Ce3+ powders were synthesized using a solid state reaction method in air with addition of H3BO3. The effects of Ce3+ dopant concentrations and the weight ratio (wt) of H3BO3 to total raw materials weight on the formation of Sr3Al2O6:Ce3+ samples were investigated. Single-phase and well-crystallized Sr3Al2O6:Ce3+ samples can be obtained when the addition of H3BO3 is lower than 7 wt%. The excitation spectrum indicates that Sr3Al2O6:Ce3+ phosphor powders exhibit two excitation bands centered at 345 and 462 nm. The emission spectrum shows that Sr3Al2O6:Ce3+ phosphor powders exhibit an emission band peaked at 536 nm when excited at 462 nm. Sr3Al2O6:Ce3+ sample at the Ce3+ concentration of 5 mol% shows the strongest emission intensity. The addition of H3BO3 has obvious influence on phase and emission intensity of Sr3Al2O6:Ce3+ phosphors. The results show that the Sr2.95Ce0.05Al2O6 sample with the 7 wt% addition of H3BO3 has the highest emission intensity and the longest lifetime.

The purpose of this paper is to present a new class of probabilistic normed spaces and to study fixed point problems in this class of probabilistic normed spaces. This paper includes the following two contents: (1) The definition of a new class of probabilistic normed spaces, the so-called S-probabilistic normed spaces, is given. In order to study the fixed point problems, some relevant properties of S-probabilistic normed spaces are discussed and some basic useful results are obtained; (2) The notion of probabilistic weak convergence is firstly presented in this paper. Therefore the probabilistic weak and strong convergence theorems of fixed points for nonexpansive mappings, asymptotically nonexpansive mappings and strongly pseudocontractive mappings are also proved by using the new methods and techniques.