In this paper, the nonlocal Cauchy problem is discussed for the fractional evolution equations in an arbitrary Banach space and various criteria on the existence and uniqueness of mild solutions are obtained. An example to illustrate the applications of main results is also given. (C) 2010 Elsevier Ltd. All rights reserved.
This paper concerns the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the a-norm. A suitable alpha-mild solution of the semilinear fractional evolution equations is introduced. The existence and uniqueness of alpha-mild solutions are proved by means of fractional calculus, singular version Gronwall inequality and Leray-Schauder fixed point theorem. The existence of optimal pairs of system governed by fractional evolution equations is also presented. Finally, an example is given for demonstration. (C) 2010 Elsevier Ltd. All rights reserved.
Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work we present a new approach via variational methods and critical point theory to obtain the existence of solutions to impulsive problems. We consider a linear Dirichlet problem and the solutions are found as critical points of a functional. We also study the nonlinear Dirichlet impulsive problem. (C) 2007 Elsevier Ltd. All rights reserved.
Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study the integrability of the equation, including its bilinear representation, soliton solutions, periodic wave solutions, Backlund transformation and Lax pairs, respectively. Furthermore, by virtue of its Lax equations, the infinite conservation laws of the equation are also derived with the recursion formulas. Finally, the asymptotic behavior of periodic wave solutions is shown with a limiting procedure. (C) 2016 Elsevier Ltd. All rights reserved.
In this paper, we discuss similarity reductions for problems of magnetic field effects on free convection flow of a nanofluid past a semi-infinite vertical flat plate. The application of a one-parameter group reduces the number of independent variables by L and consequently the governing partial differential equation with the auxiliary conditions to an ordinary differential equation with the appropriate corresponding conditions. The differential equations obtained are solved numerically and the effects of the parameters governing the problem are discussed. Different kinds of nanoparticles were tested. (C) 2010 Elsevier Ltd. All rights reserved.
In this paper, we first investigate the existence of a unique equilibrium to general bidirectional associative memory neural networks with time-varying delays in the leakage terms by the fixed point theorem. Then, by constructing a Lyapunov functional, we establish some sufficient conditions on the global exponential stability of the equilibrium for such neural networks, which substantially extend and improve the main results of Gopalsamy [K. Gopalsamy, Leakage delays in BAM, J. Math. Anal. Appl. 325 (2007) 1117-1132]. (C) 2012 Elsevier Ltd. All rights reserved.
In the present paper, the following Schrodinger-Kirchhoff-type problem: -(a + b integral(RN) vertical bar del u vertical bar(2)dx) Delta u + V(x)u = f(x, u), in R-N (1.1) is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem. (C) 2010 Elsevier Ltd. All rights reserved.
SIR models With distributed delay and with discrete delay are studied. The global dynamics are fully determined for R-0 > 1 by using a Lyapunov functional. For each model it is shown that the endemic equilibrium is globally asymptotically stable whenever it exists. (C) 2008 Elsevier Ltd. All rights reserved.
This paper is concerned with the stability of delayed recurrent neural networks with impulse control and Markovian jump parameters. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. By applying the Lyapunov stability theory, Dynkin's formula and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the exponential stability of the equilibrium point. Moreover, three numerical examples and their simulations are given to show the less conservatism and effectiveness of the obtained results. In particular, the traditional assumptions on the differentiability of the time varying delays and the boundedness of their derivatives are removed since the time varying delays considered in this paper may not be differentiable, even not continuous. (C) 2012 Elsevier Ltd. All rights reserved.
This paper is devoted to studying the synchronization control of impulsive dynamical networks. A single impulsive controller is proved to be effective for the stabilization of dynamical networks with impulse-coupling. Some simple and easily verified criteria are given for the stabilization of impulsive dynamical networks under a single impulsive controller and/or a single negative state-feedback control. Moreover, the effects of a single impulsive controller, a single state-feedback controller and an isolated dynamical system on the synchronization process are respectively distilled and explicitly expressed in the derived criteria. The structure of the dynamical network can be directed and weakly connected with a rooted spanning tree. Moreover, the convergence rate of the dynamical network is also explicitly estimated, and there is no requirement on the lower and upper bounds of the impulsive intervals. A numerical example is presented to illustrate the efficiency of the designed controller and the validity of the analytical results. (C) 2012 Elsevier Ltd. All rights reserved.
The long-time asymptotics and bright N-soliton solutions of the Kundu-Eckhaus equation are studied by Riemann-Hilbert approach. Firstly, the initial value problem of the defocusing Kundu-Eckhaus equation is considered and its long-time asymptotics is derived based on the nonlinear steepest descent method of Deift-Zhou. Then the linear spectral problem of the focusing Kundu-Eckhaus equation is investigated via Riemann-Hilbert formulation and the bright N-soliton solutions of this equation are obtained explicitly. (C) 2017 Elsevier Ltd. All rights reserved.
This paper studies a nonlinear Langevin equation involving two fractional orders alpha is an element of (0, 1] and beta is an element of (1, 2] with three-point boundary conditions. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions for the problem. The existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results. Some illustrative examples are also discussed. (C) 2011 Elsevier Ltd. All rights reserved.
In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust-Karlin's fixed point theorem. At last, an example is given to illustrate the theory. (C) 2011 Elsevier Ltd. All rights reserved.
This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient. (C) 2011 Elsevier Ltd. All rights reserved.
The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number R-0 and establish that the global dynamics are completely determined by the values of R-0: if R-0 1, then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed. (C) 2011 Elsevier Ltd. All rights reserved.
We propose a reliable method for constructing a directed weighted complex network (DWCN) from a time series. Through investigating the DWCN for various time series, we find that time series with different dynamics exhibit distinct topological properties. We indicate this topological distinction results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Furthermore, we associate different aspects of dynamics with the topological indices of the DWCN, and illustrate how the DWCN can be exploited to detect unstable periodic orbits of different periods. Examples using time series from classical chaotic systems are provided to demonstrate the effectiveness of our approach. (C) 2011 Elsevier Ltd. All rights reserved.
In this paper, we propose a novel computer virus propagation model and study its dynamic behaviors; to our knowledge, this is the first time the effect of anti-virus ability has been taken into account in this way. In this context, we give the threshold for determining whether the virus dies out completely. Then, we study the existence of equilibria, and analyze their local and global asymptotic stability. Next, we find that, depending on the anti-virus ability, a backward bifurcation or a Hopf bifurcation may occur. Finally, we show that under appropriate conditions, bistable states may be around. Numerical results illustrate some typical phenomena that may occur in the virus propagation over computer network. Crown Copyright (C) 2011 Published by Elsevier Ltd. All rights reserved.
In this paper, we investigate the dynamical outcomes of a host parasite model incorporating both horizontal and vertical transmissions in a spatial heterogeneous environment analytically and numerically. Our study provides valuable insights in two aspects: Mathematically, we propose three threshold parameters, the demographic reproduction number R-0(v), the horizontal transmission reproduction number R-0(h) and the vertical transmission reproduction number N, to identify the conditions that lead to disease-free dynamics, or susceptible-free dynamics, or endemic dynamics. Epidemiologically, we find that both host population movements and spatial heterogeneity strongly affect the disease dynamics of our proposed epidemic model: (1) the larger random mobility can result in 100% infection prevalence; and (2) the heterogeneity tends to, enhance the persistence of the infected hosts with uninfected ones. As a consequence, our work suggests that, in order to control the invasion of the parasite, different preventive measures can be implemented in different regions. (C) 2017 Elsevier Ltd. All rights reserved.
In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle's invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results. (C) 2015 Elsevier Ltd. All rights reserved.
The paper deals with Kirchhoff type equations on the whole space RN, driven by the p-fractional Laplace operator, involving critical Hardy Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem. (C) 2016 Elsevier Ltd. All rights reserved.