Mostrando recursos 1 - 20 de 78

  1. Coderivatives Related to Parametric Extended Trust Region Subproblem and Their Applications

    Tran, Van Nghi
    This paper deals with the Fréchet and Mordukhovich coderivatives of the normal cone mapping related to the parametric extended trust region subproblems (eTRS), in which the trust region intersects a ball with a single linear inequality constraint. We use the obtained results to investigate the Lipschitzian stability of parametric eTRS. We also propose a necessary condition for the local (or global) solution of the eTRS by using the coderivative tool.

  2. The Spectral Method for Long-time Behavior of a Fractional Power Dissipative System

    Lu, Hong; Zhang, Mingji
    In this paper, we consider the fractional complex Ginzburg-Landau equation in two spatial dimensions with the dissipative effect given by a fractional Laplacian. The periodic initial value problem of the fractional complex Ginzburg-Landau equation is discretized fully by Galerkin-Fourier spectral method, and the dynamical behaviors of the discrete system are studied. The existence and convergence of global attractors of the discrete system are obtained by a priori estimates and error estimates of the discrete solution. The numerical stability and convergence of the discrete scheme are proved.

  3. Periodic Solutions of Sublinear Impulsive Differential Equations

    Niu, Yanmin; Li, Xiong
    In this paper, we consider sublinear second order differential equations with impulsive effects. Basing on the Poincaré-Bohl fixed point theorem, we first will prove the existence of harmonic solutions. The existence of subharmonic solutions is also obtained by a new twist fixed point theorem recently established by Qian etc in 2015 [18].

  4. Fixed Point Theorems via MNC in Ordered Banach Space with Application to Fractional Integro-differential Evolution Equations

    Nashine, Hemant Kumar; Yang, He; Agarwal, Ravi P.
    In this paper, we propose fixed point results through the notion of measure of noncompactness (MNC) in partially ordered Banach spaces. We also prove some new coupled fixed point results via MNC for more general class of function. To achieve this result, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. Further, we apply the obtained fixed point theorems to prove the existence of mild solutions for fractional integro-differential evolution equations with nonlocal conditions. At the end, an example is given to illustrate the rationality of the...

  5. Existence of Solutions to Quasilinear Schrödinger Equations Involving Critical Sobolev Exponent

    Wang, Youjun; Li, Zhouxin
    By using variational approaches, we study a class of quasilinear Schrödinger equations involving critical Sobolev exponents \[ -\Delta u + V(x)u + \frac{1}{2} \kappa [\Delta(u^2)]u = |u|^{p-2}u + |u|^{2^*-2}u, \quad x \in \mathbb{R}^N, \] where $V(x)$ is the potential function, $\kappa \gt 0$, $\max \{ (N+3)/(N-2),2 \} \lt p \lt 2^* := 2N/(N-2)$, $N \geq 4$. If $\kappa \in [0,\overline{\kappa})$ for some $\overline{\kappa} \gt 0$, we prove the existence of a positive solution $u(x)$ satisfying $\max_{x \in \mathbb{R}^N} |u(x)| \leq \sqrt{1/(2\kappa)}$.

  6. Toeplitz Operator for Dirichlet Space Through Sobolev Multiplier Algebra

    Luo, Shuaibing; Xiao, Jie
    This paper is mainly concerned with the Toeplitz operator $T_{\phi}$ over the Dirichlet space $\mathcal{D}$ with the symbol $\phi$ in the Sobolev multiplier algebra $M(W^{1,2}(\mathbb{D}))$, thereby extending several known ones in a very different manner.

  7. Exact Controllability for Wave Equations with Switching Controls

    He, Yong
    In this paper, we analyze the exact controllability problem for wave equations endowed with switching controls. The goal is to control the dynamics of the system by switching among different actuators such that, in each instant of time, there are as few active actuators as possible. We prove that the system is exactly controllable under suitable geometric control conditions.

  8. Blow-up Phenomena for a Porous Medium Equation with Time-dependent Coefficients and Inner Absorption Term Under Nonlinear Boundary Flux

    Xiao, Suping; Fang, Zhong Bo
    This paper deals with blow-up phenomena for an initial boundary value problem of a porous medium equation with time-dependent coefficients and inner absorption term in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, the upper and lower bounds for $t^{\ast}$ are derived in the higher dimensional space. Finally, some examples are presented to illustrate applications of our results.

  9. Devaney's Chaos for Maps on $G$-spaces

    Shah, Ekta
    We study the notion of sensitivity on $G$-spaces and through examples observe that $G$-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be $G$-sensitive. Next, we define the notion of Devaney's chaos on $G$-space and show that $G$-sensitivity is a redundant condition in the definition.

  10. Remark on Proper Holomorphic Maps Between Reducible Bounded Symmetric Domains

    Seo, Aeryeong
    In this paper we study proper holomorphic maps between bounded symmetric domains when the source domain is not irreducible. More precisely, we provide sufficient conditions for semi-product proper holomorphic maps to be product proper. As an application we characterize proper holomorphic maps between equidimensional bounded symmetric domains.

  11. $b$-generalized $(\alpha,\beta)$-derivations and $b$-generalized $(\alpha,\beta)$-biderivations of Prime Rings

    Filippis, Vincenzo De; Wei, Feng
    Let $R$ be a ring, $\alpha$ and $\beta$ two automorphisms of $R$. An additive mapping $d \colon R \to R$ is called an $(\alpha,\beta)$-derivation if $d(xy) = d(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$. An additive mapping $G \colon R \to R$ is called a generalized $(\alpha,\beta)$-derivation if $G(xy) = G(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$, where $d$ is an $(\alpha,\beta)$-derivation of $R$. In this paper we introduce the definitions of $b$-generalized $(\alpha,\beta)$-derivation and $b$-generalized $(\alpha,\beta)$-biderivation. More precisely, let $d \colon R \to R$ and $G \colon R \to R$ be two additive...

  12. Hecke Bound of Vector-valued Modular Forms and its Relationship with Cuspidality

    Jin, Seokho; Lim, Jongryul; Lim, Subong
    In this paper, we prove that if the Fourier coefficients of a vector-valued modular form satisfy the Hecke bound, then it is cuspidal. Furthermore, we obtain an analogous result with regard to Jacobi forms by applying an isomorphism between vector-valued modular forms and Jacobi forms. As an application, we prove a result on the growth of the number of representations of $m$ by a positive definite quadratic form $Q$.

  13. A Note on Modularity Lifting Theorems in Higher Weights

    Yu, Yih-Jeng
    We follow the ideas of Khare and Ramakrishna-Khare and prove the modularity lifting theorem in higher weights. This approach somehow differs from that using Taylor-Wiles systems.

  14. An Extending Result on Spectral Radius of Bipartite Graphs

    Cheng, Yen-Jen; Fan, Feng-lei; Weng, Chih-wen
    In this paper, we study the spectral radius of bipartite graphs. Let $G$ be a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that $G$ is not a complete bipartite graph and $(e-1,e+1)$ is not a pair of twin primes. We describe the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1,e+1)$ of integers to...

  15. Coalescence on Supercritical Bellman-Harris Branching Processes

    Athreya, Krishna B.; Hong, Jyy-I
    We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...

  16. Coalescence on Supercritical Bellman-Harris Branching Processes

    Athreya, Krishna B.; Hong, Jyy-I
    We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...

  17. Quantitative Recurrence Properties for Systems with Non-uniform Structure

    Zhao, Cao; Chen, Ercai
    Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...

  18. Quantitative Recurrence Properties for Systems with Non-uniform Structure

    Zhao, Cao; Chen, Ercai
    Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...

  19. General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$

    Feng, Baowei
    A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

  20. General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$

    Feng, Baowei
    A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

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