Abstract: In this paper, we apply the fractional calculus and a suitable fixed point theorem with the measure of noncompactness to give the sufficient conditions of the controllability for a new class of fractional neutral integro-differential evolution systems with infinite delay and nonlocal conditions. The results are obtained here under some weakly noncompactness conditions. Thus they improve and generalize many well-known results. At the end of this paper, two examples are given to explain our abstract conclusions. PubDate: 2017-05-18

Abstract: The solvability of the following system of difference equations $$z_{n+1}=\alpha z_{n}^{a}w_{n}^{b},\qquad w_{n+1}=\beta w_{n-2}^{c}z_{n-2}^{d},\quad n\in {\mathbb {N}}_{0}, $$ where \(a,b,c,d\in {\mathbb {Z}}\) , \(\alpha ,\beta , w_{-2}, w_{-1}, w_{0}, z_{-2}, z_{-1}, z_{0}\in {\mathbb {C}}\setminus\{0\}\) , is studied in detail by using several methods. The system has the most complex structure of solutions of all the related systems studied so far, and some of the forms of solutions appear for the first time. PubDate: 2017-05-18

Abstract: In this paper, mathematical analysis is proposed on the synchronization problem for stochastic reaction-diffusion Cohen-Grossberg neural networks with Neumann boundary conditions. By introducing several important inequalities and using Lyapunov functional technique, some new synchronization criteria in terms of p-norm are derived under periodically intermittent control. Some previous known results in the literature are improved, and some restrictions on the mixed time-varying delays are removed. The influence of diffusion coefficients, diffusion space, stochastic perturbation and control width on synchronization is analyzed by the obtained synchronization criteria. Numerical simulations are presented to show the feasibility of the theoretical results. PubDate: 2017-05-18

Abstract: A mathematical model of the infection of CD4+ T-cells by HIV that includes the effects of treatment by a reverse transcriptase inhibitor (RTI) and a protease inhibitor (PI) is studied. The model includes three populations of CD4+ T-cells (healthy cells, latently-infected cells which cannot produce virus, and productively-infected cells which can produce virus) and two populations of free virus in the blood (infectious virus and non-infectious virus). The model includes a time delay between a T-cell becoming latently infected and productively infected. The model has a virus-free and a chronic infection equilibrium. It is shown that the model has Andronov-Hopf bifurcations leading to limit cycle behavior in the chronic infection region at critical values of the time delays. For three data sets obtained from the work of previous authors, numerical simulations have given critical delay values ranging from approximately 15 days to more than 200 days. This range includes the period of approximately 50 days for intermittent viral blips reported by Rong and Perelson (Plos Comp. Biol. 5(10), 1-18 (2009)). Simple formulas are derived for the sensitivity indices of the equilibrium populations and the basic reproductive number with respect to all parameters in the model. Numerical simulations are carried out to support the analytical results. The numerical results suggest that the most effective methods of reducing both the basic reproductive number and the chronic infection CD4+ T-cell and virus populations are the following: (1) to increase the efficacy of the antiretroviral treatments and (2) to increase virus clearance rate, decrease infection rate, or decrease viral reproduction rate. PubDate: 2017-05-16

Abstract: The existence of positive solutions is considered for a fractional differential equation with p-Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained for the boundary value problems. An example is also presented to illustrate the effectiveness of the main result. PubDate: 2017-05-12

Abstract: An HIV/AIDS epidemic model with general nonlinear incidence rate and treatment is formulated. The basic reproductive number \(\Re_{0}\) is obtained by use of the method of the next generating matrix. By carrying out an analysis of the model, we study the stability of the disease-free equilibrium and the unique endemic equilibrium by using the geometric approach for ordinary differential equations. Numerical simulations are given to show the effectiveness of the main results. PubDate: 2017-05-12

Abstract: We consider a nonlinear Schrödinger equation with Dirac interaction defect. Moreover, non-standard boundary conditions are introduced in connection to the behavior of the solutions. First, we prove that this kind of Schrödinger equation can be characterized by an autonomous dynamical system. Then, based on this result, we show that such an equation possesses a maximal compact attractor in the weak topology of \(\mathbf{H}^{\mathbf{1}}\) . PubDate: 2017-05-12

Abstract: In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter \(1/2<\hbar <1\) on the Hilbert space. By combining some stochastic analysis theory and new integral inequality techniques, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the pth moment to the stochastic systems are obtained. Last, an example is presented to illustrate our theory in the work. PubDate: 2017-05-10

Abstract: In this study, a mathematical model of bacterial resistance considering the immune system response and antibiotic therapy is examined under random conditions. A random model consisting of random differential equations is obtained by using the existing deterministic model. Similarly, stochastic effect terms are added to the deterministic model to form a stochastic model consisting of stochastic differential equations. The results from the random and stochastic models are also compared with the results of the deterministic model to investigate the behavior of the model components under random conditions. PubDate: 2017-05-09

Abstract: In this paper, we study the convergence of iterative learning control for some fractional equation. Firstly, by using the Laplace transform and the M-L function, we show the concept of mild solutions. Secondly, by using the Gronwall inequality, we show the sufficient conditions of convergence for the open P-type and the close P-type iterative learning control. At last, we give some examples to illustrate our main results. PubDate: 2017-05-08

Abstract: In this paper, the input-to-state stability for coupled control systems is investigated. A systematic method of constructing a global Lyapunov function for the coupled control systems is provided by combining graph theory and the Lyapunov method. Consequently, some novel global input-to-state stability principles are given. As an application to this result, a coupled Lurie system is also discussed. By constructing an appropriate Lyapunov function, a sufficient condition ensuring input-to-state stability of this coupled Lurie system is established. Two examples are provided to demonstrate the effectiveness of the theoretical results. PubDate: 2017-05-05

Abstract: In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions: $$ \textstyle\begin{cases} L(D)u(t) = f(t,u(t)),\quad t\in[0,T], T>0, \\ u(0) = u(T)>0,\qquad u'(0)=u'(T)>0, \end{cases} $$ where \(L(D)=\sum^{n}_{i=0}a_{i}D^{S_{i}}\) , \(1\leq S_{0}<\cdots<S_{n-1}<S_{n}<2\) , \(a_{i}\in\mathbb{R}\) , \(a_{n}\neq0\) , and \(f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}\) is a continuous operation. We get the Green’s function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case \(f(t,u(t))= u(t) ^{p}\) , \(a_{i}\geq0\) , and give an upper bound on the blow-up time \(T_{\mathrm{max}}\) . PubDate: 2017-05-05

Abstract: In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees. Firstly, we give two alternative formulas of the solutions for a delay linear differential equation. Secondly, we obtain a norm estimation of the delayed matrix sine and cosine of polynomial degrees, which are used to establish sufficient conditions to guarantee our finite time stability results. Meanwhile, a numerical example is presented demonstrating the validity of our theoretical results. Finally, we extend our study to the same issue of a delay differential equation with nonlinearity by virtue of the Gronwall inequality approach. PubDate: 2017-05-05

Abstract: In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive \((q,p)\) -Laplacian dynamical systems. The key contributions of this paper lie in (i) Exploiting the least action principle, we deduce that the system we are interested in has at least one weak solution if the potential function has sub- \((q,p)\) growth or \((q,p)\) growth; (ii) Employing a critical point theorem due to Ding (Nonlinear Anal. 25(11):1095-1113, 1995), we derive that the system involved has infinitely many weak solutions provided that the potential function is even. PubDate: 2017-05-03

Abstract: The oscillatory and asymptotic behavior results for a class of third-order nonlinear neutral dynamic equations on time scales are presented. The results obtained can be extended to more general third-order neutral dynamic equations of the type considered here. Examples are provided to illustrate the applicability of the results. PubDate: 2017-05-03

Abstract: In this paper, we consider a class of infinite-point boundary value problems of fractional differential equations on the infinite interval \([0,+\infty)\) with a disturbance parameter. By using the method of upper and lower solutions, fixed point index theory and some fixed point theorems, the existence, multiplicity and nonexistence for the positive solution of the boundary value problem are obtained, respectively. The impact of the disturbance parameters on the existence of positive solutions is also given. Finally, some examples are presented to illustrate the wide range of potential applications of our main results. PubDate: 2017-05-03

Abstract: In the present paper, a stochastic mutualism model subject to white noises is established. We first investigate the existence and uniqueness of globally positive solution of the stochastic model. Then we study its asymptotic behavior, such as stochastic permanence and extinction, and estimate the limit of the average in time of the sample paths of every component. We also show that the stochastic system is globally attractive under some appropriate conditions. Finally, numerical simulations are presented to justify the analytical results. PubDate: 2017-05-01

Abstract: This paper investigates modified function projective synchronization (MFPS) for complex dynamical networks with mixed time-varying and hybrid asymmetric coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix needs not be symmetric or irreducible. The MFPS of delayed complex dynamical networks is considered via either hybrid control or hybrid pinning control with nonlinear and adaptive linear feedback control, which contains error linear term, time-varying delay error linear term and distributed time-varying delay error linear term. Based on Lyapunov stability theory, adaptive control technique, the parameter update law and the technique of dealing with some integral terms, we will show that control may be used to manipulate the scaling functions matrix such that the drive system and response networks could be synchronized up to the desired scaling function matrix. Numerical examples are given to demonstrate the effectiveness of the proposed method. The results in this article generalize and improve the corresponding results of the recent works. PubDate: 2017-04-28

Abstract: This paper is concerned with stochastic differential equations of fractional-order \(q \in(m-1, m)\) (where \(m \in \mathbb{Z}\) and \(m \geq 2\) ) with finite delay at a space \(BC ( [ - \tau, 0];R^{d} )\) . Some sufficient conditions are obtained for the existence and uniqueness of solutions for these stochastic fractional differential systems by applying the Picard iterations method and the generalized Gronwall inequality. PubDate: 2017-04-27

Abstract: In this paper, we define the 2k-step Jordan-Fibonacci sequence, and then we study the 2k-step Jordan-Fibonacci sequence modulo m. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step Jordan-Fibonacci sequence when read modulo m, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulo m. Furthermore, we extend the 2k-step Jordan-Fibonacci sequence to groups, and then we examine this sequence in the finite groups. Finally, we obtain the period of the 2k-step Jordan-Fibonacci sequence in the generalized quaternion group \(Q_{2^{n}}\) as applications of the results produced. PubDate: 2017-04-26