In this paper, we investigate the pseudo Drazin invertibility of the sum and the product of elements in a Banach algebra $\mathscr{A}$. Given pseudo Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is pseudo Drazin invertible and $a+b$ is pseudo Drazin invertible if and only if so is $1+a^{\ddagger} b$, and the related formulae are provided.

In this article, we first establish the sharp distortion theorems of the Fréchet derivative for a subclass of starlike mappings on the unit ball of complex Banach spaces and the bounded starlike circular domain in $\mathbb{C}^n$. Meanwhile, we also obtain the sharp distortion theorems of the Jacobi determinant for a subclass of starlike mappings on the bounded starlike circular domain in $\mathbb{C}^n$. Our derived conclusions are the generalizations of some known results in several complex variables and the classical results in one complex variable.

In this paper, we deal with the existence of infinitely many solutions for a class of sublinear Schrödinger equation $$ \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbb{R}}^{N},\\ u\in H^{1}({\mathbb{R}}^{N}). \end{array} \right. $$ Under the assumptions that $\inf_{{\mathbb{R}}^{N}}V(x) \gt 0$ and $f(x, t)$ is indefinite sign and sublinear as $|t|\to +\infty$, we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrival solutions by using the genus properties in critical point theory.

In this paper, we study a class of differential inclusion problems driven by the $p(x)$-Kirchhoff with non-standard growth depending on a real parameter. Working within the framework of variable exponent spaces, a new existence result of at least three solutions for the considered problem is established by using the nonsmooth version three critical points theorem.

In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize all extremal connected graphs which attain the minimum value among the spectral radii of all connected graphs with order $n=k\alpha$ and the independence number $\alpha$. Moreover, all extremal graphs which attain the maximum value among the spectral radii of clique trees with order $n=k\alpha$ and the independence number $\alpha$ are characterized.

In this paper, we deal with the impulsive control systems of fractional order and their optimal controls in Banach spaces. We firstly show the existence and uniqueness of mild solutions for a broad class of impulsive fractional infinite dimensional control systems under suitable assumptions. Then by using generally mild conditions of cost functionals, we extend the existence result of optimal controls to the impulsive fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

In this paper, two kinds of nonlinear scalarization functions are applied to characterize $E$-efficient solutions and weak $E$-efficient solutions of vector optimization problems and some nonlinear scalarization characterizations are obtained. Some examples also are given to illustrate the main results.

We show new versions of reverse Young inequalities by virtue of the Kantorovich constant, and utilizing the new reverse Young inequalities we give the reverses of the weighted arithmetic-geometric and geometric-harmonic mean inequalities for two positive operators. Also, new versions of reverse Young and Heinz mean inequalities for unitarily invariant norms are established.

It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.

A signed graph $\Gamma=(G, \sigma)$ consists of an unsigned graph $G=(V, E)$ and a mapping $\sigma: E \rightarrow \{+, -\}$. Let $\Gamma$ be a connected signed graph and $L(\Gamma), {\cal L}(\Gamma)$ be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n\geq 0$ and $\lambda_1\geq \cdots \geq \lambda_{n-1}\geq \lambda_n\geq 0$ are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of $\Gamma$, respectively. In this paper, we give two new lower bounds on $\lambda_1$ which are both stronger than Li's bound [8] and obtain a new upper bound on $\lambda_n$ which is also stronger than Li's bound [8]. In addtion, Hou [6] proposed a conjecture for a connected signed graph $\Gamma: \sum\limits_{i=1}^k\mu _i\gt \sum\limits_{i=1}^k d _i (1\leq k\leq n-1)$. We investigate $\sum\limits_{i=1}^k\mu_i (1\leq k\leq n-1)$ and partly solve the conjecture.

The goal of this paper is to prove null 2-type hypersurfaces with at most three distinct principal curvatures in a Euclidean space have constant mean curvature.

Given a sequence $\{D_m\}$ of bounded and convex domains in a complex Banach space, we describe the limiting behavior of the corresponding sequence $\{k_{D_m}\}$ of Kobayashi distances when the given sequence $\{D_m\}$ is either monotonic or convergent in the Hausdorff metric.

A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions on the relaxation function and with certain initial data and by adopting the perturbed energy method, we establish uniform decay rate of the solution energy in terms of the behavior of the relaxation function, which are not necessarily of exponential or polynomial decay.

The aim of this paper is to extend the study of Riesz transforms associated to the Dunkl harmonic oscillator considered by A.Nowak and K. Stempak to higher order. The methods used to establish the $L^p$-boundedness of these transforms call the Calder$\acute{o}$n-Zygmund theory.

Let $X=(X_t)_{t\geq0}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma\gt 0$, and let $Y=(Y_t)_{t\geq0}$ be the associated maximum process of $X$. Under certain conditions, we prove a sharp maximal inequality for the geometric Brownian motion. The method of proof is essentially based on explicit forms of the following optimal stopping problem: Find a stopping time $\tau^*$, if it exists, such that \begin{eqnarray*} \Phi(x,y):=\sup_{\tau}\mathbf{E}^{x,y}\left[Y_\tau-c\int_0^\tau X^\theta_sds\right],\quad c,\theta\gt 0 \end{eqnarray*} where the supremum is taken over all stopping times $\tau$ for the process $X$. The present result complements and extends a similar result proved by Graversen and Peskir [1].

Let $X$ be a dimensional finite lattice (not necessary distributive) and let $F$ be a mapping from $X$ to $X$. Here we introduce a new notion of neighbours of an element of $X$ and prove that if all the neighbours of each element of $X$ are in $X$ and there is no negative circuit in the interaction graph of $F$, then $F$ has a fixed point.

We prove that every space of homogeneous type $(X,\rho,\mu)$ is either an LCH space and $\mu$ is a Radon measure, or $X$ may be identified as a dense subset, with inherited quasi-distance and measure, of another space of homogeneous type which is LCH. We also take an opportunity to present a metamathematical principle which is useful in proving results for general quasi-metric measure spaces by reducing arguments to the case of metric measure spaces.

In this paper, a class of holomorphic mappings is introduced in the unit ball of complex Banach space which is the extension of subclass of close-to-convex functions in $\mathbb{C}$, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space $X$ or the unit polydisc in $\mathbb{C}^n$ . As an application, a sharp growth theorem for this class is obtained. These results generalize many known results.

We present two new semilocal convergence analyses for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods include the secant, Newton's method and other popular methods as special cases. The convergence analysis is based on our idea of recurrent functions. Using more precise majorizing sequences than before we obtain weaker convergence criteria. These advantages are obtained because we use more precise estimates for the upper bounds on the norm of the inverse of the linear operators involved than in earlier studies. Numerical examples are given to illustrate the advantages of the new approaches.

In this paper, some new fixed point theorems for semi-closed 1-set-contractive operators of functional type are obtained, which generalize the famous Petryshyn's theorem and Altman's theorem by replacing the norm with some classes of functionals.