The following viscoelastic wave equation with a delay term in internal feedback: \begin{equation*} \left\vert u_{t}\right\vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt} + \int_{0}^{t} g(t-s) \Delta u(s) ds + \mu _{1} u_{t}(x,t) + \mu _{2} u_{t}(x,t-\tau ) = 0, \end{equation*} is considered in a bounded domain. Under appropriate conditions on $\mu_{1}$, $\mu_{2}$ and on the kernel $g$, we prove the local existence result by Faedo-Galerkin method and establish the decay result by suitable Lyapunov functionals.

In this paper we give some interesting equation of $p$-adic $q$-integrals on $\mathbb Z_p$. From those $p$-adic $q$-integrals, we present a systemic study of some families of extended Carlitz $q$-Bernoulli numbers and polynomials with weight $\alpha$ in $p$-adic number field.

In this paper, we establish sharp inequalities of homogeneous expansions for starlike mappings and starlike mappings of order $\alpha$ defined on the unit ball of Banach complex spaces. As corollaries, we also obtain the sharp estimates of the third homogeneous expansions for the above mappings defined on the unit polydisk in $\mathbb{C}^n$ with some restricted conditions.

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times\cdots {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre [9]. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics. In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination of 2-ranks of compact connected simple Lie groups $G$ via the maximal antipodal sets $A_{2}G$ of $G$ introduced in [16, 17].

In this paper, a new class of roughly geodesic $B$-invex sets, quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are introduced and studied on Hadamard manifolds by relaxing the definitions of geodesic convex sets and functions. Some properties of quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are proved on Hadamard manifolds. As applications, some sufficient and necessary conditions for optimal solution of the nonlinear programming problems involving the quasi roughly geodesic $B$-invex functions and the pseudo roughly geodesic $B$-invex functions are given on Hadamard manifolds. The Mond-weir type dual problems for the nonlinear programming problems are also considered on Hadamard manifolds.

In this paper we are concernedwith a class of sublinear Schrödinger-Maxwell equations $$\begin{cases} -\triangle u + V(x)u + \phi u = f(x,u), &\textrm{in $\mathbb{R}^{3}$}, \\ -\triangle \phi = u^{2}, \lim\limits_{|x| \to +\infty} \phi(x) = 0, &\textrm{in $\mathbb{R}^{3}$}, \end{cases}$$ where $V: \mathbb R^3 \rightarrow \mathbb R$ and $f: \mathbb R^3 \times \mathbb R \rightarrow \mathbb R$. Under certain assumptions on $V$ and $f$, some new criteria on theexistence and multiplicity of negative energy solutions for theabove system are established via the genus properties in criticalpoint theory. Recent results from the literature are significantly improved.

In this paper, a new class of functions called semi-$G$-preinvex functions, which is generalization of semipreinvex functions and $G$-preinvex functions, is introduced. Some examples are given to show that there exist functions which are semi-$G$-preinvex functions, but are not $G$-invex functions, semipreinvex functions and $G$-preinvex functions. Furthermore, some interesting properties and characterizations of semi-$G$-preinvexity are established. Our results extend and improve the corresponding ones in the literature.

The spread of a matrix has been introduced by Mirsky in 1956. The classical theory provides an upper bound for the spread of the shape operator in terms of the second fundamental form of a hypersurface in the Euclidean space. In the present work, we are extending our understanding of the phenomenon by proving a lower bound, inspired from an idea developed recently by X.-Q. Chang. As we are exploring the very concept of curvature on hypersurfaces, we are introducing a new curvature invariant called amalgamatic curvature and we explore its geometric meaning by proving an inequality relating it to the absolute mean curvature of the hypersurface. In our study, a new class of geometric object is obtained: the absolutely umbilical hypersurfaces.

In this paper we introduce the notion of slant submanifolds of a Lorentzian almost contact manifold and of a Lorentzian almost para contact manifold.

In 2006, Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16(4) (2006), 1230-1241.] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality in a real Hilbert space via combining two well-known methods: hybrid and extragradient. In this paper, motivated by Nadezhkina and Takahashi's hybrid-extragradient method we propose and analyze a relaxed hybrid-extragradient method for finding a solution of a constrained convex minimization problem, which is also a common element of the solution set of a variational inclusion and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We obtain a strong convergence theorem for three sequences generated by this algorithm. Based on this result, we also construct an iterative algorithm for finding a solution of the constrained convex minimization problem, which is also a common fixed point of two mappings taken from the more general class of strictly pseudocontractive mappings.

As a generalization of slant Riemannian maps, semi-slant Riemannian maps, almost h-slant submersions, and almost h-semi-slant submersions, we introduce the notion of almost h-semi-slant Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate the integrability of distributions, the harmonicity of such maps, the geometry of fibers, etc. We also deal with the condition for such maps to be totally geodesic and study some decomposition theorems. Moreover, we give some examples.

Let $\alpha$ be a nonnegative number, and $\text{C}: \text{X} \to \text{X}$ a bounded linear operator on a Banach space $\text{X}$. In this paper, we shall deduce some basic properties of a nondegenerate local $\alpha$-times integrated $\text{C}$-cosine function on $\text{X}$ and some generation theorems of local $\alpha$-times integrated $\text{C}$-cosine functions on $\text{X}$ with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $\alpha$-times integrated $\text{C}$-cosine function on $\text{X}$ with generator $\text{A}$ and the unique existence of solutions of the abstract Cauchy problem: $$\textrm{ACP}(A,f,x,y) \qquad \begin{cases} u''(t) = Au(t) + f(t) \quad \textrm{for } t \in (0,T_0), \\ u(0) = x, u'(0) = y, \end{cases}$$ just as the case of $\alpha$-times integrated $\text{C}$-cosine function when $\text{C} :\text{X}\to\text{X}$ is injective and $\text{A}:\text{D}(\text{A})\subset\text{X}\to\text{X}$ a closed linear operator in $\text{X}$ such that $\text{C}\text{A}\subset \text{A}\text{C}$. Here $0 \lt T_0 \leq \infty$, $x,y \in X$, and $f$ is an $X$-valued function defined on a subset of $\mathbb{R}$ containing $(0,T_0)$.

Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x) = x^3 - mx^2 - (m+1)x - 1$, where $m$ is an integer with $m\geq 4$. In this paper, we will give a class number one criterion for $K_m$.

Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$ such that a graph $G$ has an $r$-equitable $k$-coloring is denoted by $\chi_{r=}(G)$, and the least $n$ such that a graph $G$ has an $r$-equitable $k$-coloring for all $k \geqslant n$ is denoted by $\chi^*_{r=}(G)$. In this paper, we propose a necessary and sufficient condition for a complete multipartite graph $G$ to have an $r$-equitable $k$-coloring, and also give exact values of $\chi_{r=}(G)$ and $\chi^*_{r=}(G)$.

The feature of the present work is to demonstrate that the method of regular variation can be effectively applied to fourth order quasilinear differential equations of the forms \begin{equation*} (|x''|^{\alpha-1}x'')'' + q(t)|x|^{\beta-1}x = 0, \end{equation*} under the assumptions that $\alpha \gt \beta$ and $q(t): [a,\infty) \to (0,\infty)$ is regularly varying function, providing full information about the existence and the precise asymptotic behavior of all possible positive solutions.

In the present note we deal with a bifurcation problem involving an asymptotically linear function. By employing a recent result of Ricceri we make much more precise the conclusion of a theorem of Mironescu and Radulescu.

In this paper, we prove two kinds of wavelet characterizations of the predual spaces of the Morrey spaces through considering some micro-local quantities of the predual spaces.

A robust estimating method to search for the bounded function minimizing a risk is proposed. The estimators obtained by the proposed method are the generalizations of several classical robust estimators. In addition, a variety of robust estimators can be obtained, including robust one and multidimensional estimators, robust regression and robust model selection criteria for univariate or multivariate data with independent or correlated errors, and robust Bayes estimators. Further, a simulation study is conducted to evaluate the proposed method.

We extend the existing concepts of secondary representation of a module, coregular sequence and attached prime ideals to the more general setting of any hereditary torsion theory. We prove that any $\tau$-artinian module is $\tau$-representable and that such a representation has some sort of unicity in terms of the set of $\tau$-attached prime ideals associated to it. Then we use $\tau$-coregular sequences to find a nice way to compute the relative width of a module. Finally we give some connections with the relative local homology.

We show that, for several function Banach spaces, multiplication considered as a bilinear continuous srjection, is an open mapping. In particular, we prove that multiplication from $L_p \times L_q$ to $L_1$ (for $p,q \in [1,\infty]$, $1/p + 1/q = 1$) is open.