Given a Vilenkin group G, a scalar matrix $\Lambda = [\lambda_{ij}]_{i \in \mathbb{N}, j \in \mathbb{N}_0}$, a function $f \in L^1(G)$, and a point $x \in G$ we introduce, for each $\alpha \in \mathbb{R}$, the $(\Lambda,\alpha)$ − derivative $f$ at $x$ denoted by $f^{(\Lambda,\alpha)}(x)$. We also introduce the sets: $$ M_\alpha = M(G,\Lambda,\alpha,x) := \big\{ f \in L^1(G)：\exists f^{(\Lambda,\alpha)}(x) \big\}, $$ $$ M = M(G,\Lambda,x) := \{f \in L^1(G)：\exists f^\Lambda(x)\}; $$ where $f^\Lambda(x)$ derivative in [8], which is a generalization of Onneweer’s derivative $f^{[1]}(x)$ in [6]. We proved:

(a) Five theorems which express essential characteristics of $(\Lambda,\alpha)$− derivative,

(b) $M = M_0$,

(c) $(\forall \alpha,\beta \in \mathbb{R}) \wedge (\alpha \lt \beta) \Rightarrow (M_\alpha \subseteq M_\beta) \wedge (M_\beta \setminus M_\alpha \not= \theta)$.

Statement b) states that the method $(\Lambda,\alpha)$ − differentiation, for $\alpha = 0$, is equal to $\Lambda$ − differentiation and statement c) says that $(\Lambda,\alpha)$− differentiation increases with increasing $\alpha \in \mathbb{R}$.

We discussed the upper and lower bounds of packing constants in Orlicz-Lorentz sequence spaces equipped with both the Luxemburg norm and the Orlicz norm. Provided $\Phi \in \Delta_2(0)$, we showed that the Kottman constant of $\lambda_{\Phi,\omega}$ and $\lambda^o_{\Phi,\omega}$ satisfies $$\max \left\{ \frac{1}{\alpha_\Phi(0)}, \frac{1}{\alpha'_{\Phi,\omega}} \right\} \leq K\left(\lambda_{\Phi,\omega}\right) \leq \frac{1}{\tilde{\alpha}_{\Phi,\omega}},$$ $$ \max \left\{ \frac{1}{\alpha_\Phi(0)}, \frac{1}{\alpha''_{\Phi,\omega}} \right\} \leq K\left(\lambda^o_{\Phi,\omega}\right) \leq \frac{1}{\alpha^{\ast}_{\Phi}}. $$ As a corollary, the packing constant of Lorentz space $\lambda_{p,\omega}$ is $1/(1+2^{1-\frac{1}{p}})$.

The q-Hausdorff matrices are defined in terms of symbols from q-mathematics. The matrices become ordinary Hausdorrf matrices as $q \rightarrow 1$. In this paper, we consider the q-analogues of the Cesàro matrix of order one, both for $0 \lt q \lt 1$ and $q \gt 1$, and obtain the lower bounds for these matrices for any $1 \lt p \lt \infty$.

Given a graph $G,$ an $L(3,2,1)$-labeling of $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(u)-f(v)|\geqslant 1$ if $d(u,v) = 3,$ $|f(u)-f(v)| \geqslant 2$ if $d(u,v) = 2$ and $|f(u)-f(v)| \geqslant 3$ if $d(u,v) = 1$. For a nonnegative integer $k$, a $k$-$L(3,2,1)$-labeling is an $L(3,2,1)$-labeling such that no label is greater than $k$. The $L(3,2,1)$-labeling number of $G$, denoted by $\lambda_{3,2,1}(G)$, is the smallest number $k$ such that $G$ has a $k$-$L(3,2,1)$-labeling. We study the $L(3,2,1)$-labelings of graphs in this paper. We give upper bounds for the $L(3,2,1)$-labeling numbers of general graphs and trees, and consider the $L(3,2,1)$-labeling numbers of several classes of graphs, such as the Cartesian product of paths and cycles, and the powers of paths.

In this paper, let $X$ be a reflexive Banach space which either is uniformly smooth or has a weakly continuous duality map. We prove, under the convergence of no parameter sequences to zero, the strong convergence of their iterative scheme to a zero of $m$-accretive operator $A$ in $X$, which solves a variational inequality on the set $A^{-1}(0)$ of zeros of $A$. Such a result includes their main result as a special case. Furthermore, we also give a weak convergence theorem for hybrid viscosity iterative approximation method involving a maximal monotone operator in a Hilbert space.

The notion of Lagrangian $H$-umbilical submanifolds of Kähler manifolds introduced in [3, 4] is closely related with several problems in Lagrangian geometry (cf. [7]). The classification of such submanifolds was done in a series of author's papers [3, 4, 5]. On the other hand, the study of Lagrangian submanifolds of para-Kähler manifolds was initiated very recently in [10]. In this paper we study Lagrangian $H$-submanifolds of para-Kähler manifolds. As results we prove several fundemental properties of such submanifolds. Moreover, we are able to classify Lagrangian $H$-umbilical submanifolds of the para-Kähler $n$-plane $({\mathbb E}^{2n}_n,g_0,P)$ for $n \geq 3$.

The standard Pompeiu results for complex balls in the setting of the Heisenberg group $\bf H^n$ are also shown to carry over to Heisenberg balls. This extension is important because it allows for integration over sets of the same dimension as the ambient space $\bf H^n$. Several different concepts of the Heisenberg ball are considered, using differing definitions for the metric on $\bf H^n$. For purposes of this work, none of these is more natural than the others. The results for each of the spaces $L^2$, $L^p$ for $1 \leq p \lt \infty$, and for $L^\infty$, are directly comparable to the Pompeiu results for complex balls in $\bf H^n$, as in [2, 3, 4]. The natural expression for the Pompeiu problem in $\bf H^n$ is integration over complex balls, and the extension to the Heisenberg ball builds upon the methods for this case. The extra dimension primarily leads to extra complexity in the integrals. At the level of $L^\infty$, where the results require balls of two radii satisfying appropriate conditions, the additional dimension adds to the complexity in the functions defining the conditions for the radii. The different concepts of the Heisenberg ball lead to different forms for these arithmetic conditions defining the radii. The differences between these balls can also be seen when they are rotated with $\bf H^n$, an issue to be further considered in a later work. The standard Pompeiu results have been extended to all of the cases considered here.

In this paper, by virtue of coderivatives, we investigate sensitivity analysis of parametric vector set-valued optimization problems in Banach spaces. Several examples are provided to illustrate the main results.

This paper is concerned with the first initial boundary value problem for hyperbolic systems in infinite cylinders with the nonsmooth base. Some results on the regularity of generalized solutions of this problem are given.

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies Lipschitz condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

In this paper, we first obtain a distortion theorem of Jacobian matrix $J_f(z)$ for biholomorphic starlike mapping $f$ along a unit direction on the unit polydisc. This result extends the classical distortion theorem of starlike functions to higher dimensions. We then give an upper bound estimate of distortion theorem for biholomorphic starlike mappings along a unit direction in a complex Banach space. Finally, we propose two conjectures on starlike mappings in several complex variables.

Given a general monic sextic polynomial with six real coefficients, necessary and sufficient conditions are found such that the polynomial does not have any positive roots. This `nonlinear eigenvalue problem' is a relatively difficult one since we have $6$ real parameters. Fortunately, we succeed in applying the Cheng-Lin envelope method in [1] together with several new ideas and techniques to express our criteria in terms of roots of quartic polynomials and explicit parametric curves and therefore our problem is completely solved. Several specific examples are also included to illustrate various applications including the seeking of periodic solutions of the logistic equation studied in chaos theory.

Let $\varphi_{1}$ and $\varphi_{2}$ be holomorphic self-maps of the open unit ball $\mathbb B$ in $\mathbb C^N$, $u_{1}$ and $u_{2}$ be holomorphic functions on $\mathbb B$ and let weighted composition operators $W_{\varphi_{1},u_{1}}$; $W_{\varphi_{2},u_{2}}: A^{p}_{\alpha} \to H^{\infty}_{v}$ be bounded. This paper characterizes the compactness of the difference of these operators from the weighted Bergman space $A^p_\alpha$, $0-1$, to the weighted-type space $H^\infty_v$ of holomorphic functions on $\mathbb B$ in terms of inducing symbols $\varphi_{1}$, $\varphi_{2}$, $u_{1}$ and $u_{2}$. For the case $p \gt 1$ we find an asymptotically equivalent expression to the essential norm of the operator.

In this paper, the derivations between ideals of the Banach algebra $\mathfrak{E}_\infty(I)$ are characterized. Necessary and sufficient conditions for weak amenability of Banach algebras $\mathfrak{E}_p(I)$, $1 \leq p \leq \infty$, are found. Also, some applications to compact groups and hypergroups are given.

In this paper the authors give a sufficient condition such that the commutator generated by the weighted Lipschitz function and the fractional integral operator with homogeneous kernel satisfying certain Dini condition is bounded on weighted Lebesgue spaces.

Let $H$ be a weak Hopf algebra in the sense of Böhm and Szlachányi [3] and $A$ a weak $H$-bimodule algebra. Then in this paper we first introduce the notion of a weak twisted smash product $A \star H$ and then find some sufficient and necessary conditions making it into a weak bialgebra. Furthermore, we give a Maschke-type theorem for the weak twisted smash product over semisimple weak Hopf algebra $H$, which generalizes the well-known Maschke-type theorem in [5, 15, 17]. Finally, we obtain an analogue of the duality theorem for the weak twisted smash products.

In this paper, we give the sufficient as well as necessary condition of the Schur-convexity and Schur-harmonic-convexity of the generalized Heronian means with two positive numbers. Our main results provide the perfected versions of the results given in 2008 by Shi et al. [9].

Recently Bauschke et al. introduced a very interesting and new notion of proximal average in the context of convex analysis, and studied this subject systemically in [3-7] from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix diagonalization [8]. Inspired by their works and the well-known fact that the second-order cone is the other typical example of symmetric cone, we study the resolvent average on second-order cone, and derive corresponding results in a different manner from Bauschke et al. [8].

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by $H_A$, $H_B$ and $H_C$. Each of these three triple hypergeometric functions $H_A$, $H_B$ and $H_C$ has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for each of Srivastava's triple hypergeometric functions $H_A$, $H_B$ and $H_C$.