Abstract: In this paper, a generalized nonautonomous stochastic competitive system with impulsive perturbations is studied. By the theories of impulsive differential equations and stochastic differential equations, we have established some asymptotic properties of the system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on. In order to show the correctness and feasibility of the theoretical results, several numerical examples are presented. Finally, the effects of different white noise perturbations and different impulsive perturbations are discussed and illustrated. PubDate: 2017-07-21

Abstract: The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model. PubDate: 2017-07-19

Abstract: In this paper, we investigate the problem of exponential stabilization criteria for a nonlinear system with mixed time-varying delays, including discrete interval and distributed time-varying delays. The time-varying delays are not necessarily differentiable. The exponential stabilization criteria of the nonlinear system are proposed via hybrid intermittent feedback control. Based on the improved Lyapunov-Krasovskii functionals with Leibniz-Newton’s formula, Jensen’s inequality and the reciprocal convex combination technique, the novel delay-dependent sufficient condition is derived in terms of linear matrix inequalities (LMIs). The obtained LMIs can be efficiently solved by standard convex optimization algorithms. A numerical example is given to demonstrate the effectiveness of the obtained result. Moreover, the results in this article generalize and improve the corresponding results of the recent works. PubDate: 2017-07-14

Abstract: In order to comprehend the effects of the duration of pesticide residual effectiveness on successful pest control, a stochastic integrated pest management (IPM) model with pesticides which have residual effects is proposed. Firstly, we show that our model has a global and positive solution and give its explicit expression when pest goes extinct. Then the sufficient conditions for pest extinction combined with the ones for the global attractivity of the pest solution only chemical control are established. Moreover, we also derive sufficient conditions for weak persistence which show that the solution of stochastic IPM models is stochastically ultimately bounded under some conditions. PubDate: 2017-07-11

Abstract: In this paper, we consider a class of nonlocal fractional stochastic differential equations driven by fractional Brownian motion with Hurst index \(H>\frac{1}{2}\) . Sufficient conditions for the existence and uniqueness of mild solutions are obtained. Finally, an example is presented to illustrate our obtained results. PubDate: 2017-07-11

Abstract: Atherosclerosis usually occurs within the large arteries. It is characterized by the inflammation of the intima, which involves dynamic interactions between the plasma molecules; namely, LDL (low density lipoproteins), monocytes or macrophages, cellular components and the extracellular matrix of the arterial wall. This process is referred to as plaque formation. If the accumulation of LDL cholesterol progresses unchecked, atherosclerotic plaques will form as a result of increased number of proliferating smooth muscle cells (SMCs) and extracellular lipid. This can thicken the artery wall and interfere further with blood flow. The growth of the plaques can become thrombotic and unstable, ending in rupture which gives rise to many life threatening illnesses, such as coronary heart disease, cardiovascular diseases, myocardial infarction, and stroke. A mathematical model of the essential chemical processes associated with atherosclerotic plaque development is analyzed, considering the concentrations of LDLs, oxidized LDLs, foam cells, oxidized LDL-derived chemoattractant and macrophage-derived chemoattractant, the density of macrophages, smooth muscle cells (SMCs), and extracellular matrix (ECM). The positive invariant set is found and local stability is established. Oscillatory behavior of the model solutions is also investigated. Numerical solutions show various dynamic behaviors that can occur under suitable conditions on the system parameters. PubDate: 2017-07-10

Abstract: In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent optimized finite difference Crank-Nicolson iterative (OFDCNI) scheme containing very few degrees of freedom but holding sufficiently high accuracy for the two-dimensional (2D) Sobolev equation by means of the proper orthogonal decomposition (POD) technique, analyzing the stability and convergence of the OFDCNI solutions and using the numerical simulations to verify the feasibility and effectiveness of the OFDCNI scheme. PubDate: 2017-07-10

Abstract: In this paper, we show that approximate derivations on Banach ∗-algebras are exactly derivations and also show that approximate quadratic ∗-derivations on Banach ∗-algebras are exactly quadratic ∗-derivations by the fixed point theorem. PubDate: 2017-07-05

Abstract: We prove that the system \(\dot{\theta}(t) =\Lambda(t)\theta(t)\) , \(t\in\mathbb{R}_{+}\) , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem \(\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)\) , \(t\in\mathbb{R}_{+}\) , \(\phi(0)=\theta_{0}\) as the approximate solutions of \(\dot{\theta}(t)=\Lambda(t)\theta(t)\) , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with \(\xi(0)=0\) , and \(\Lambda(t)\) is a 2-periodic square matrix of order l. PubDate: 2017-07-04

Abstract: The stability of coupled systems with time-varying coupling structure (CSTCS) is considered in this paper. The graph-theoretic method on a digraph with constant weight has been successfully generalized into a digraph with time-varying weight. In addition, we construct a global Lyapunov function for CSTCS. By using the graph theory and the Lyapunov method, a Lyapunov-type theorem and some sufficient criteria are obtained. Furthermore, the theoretical conclusions on CSTCS can successfully be applied to the predator-prey model with time-varying dispersal. Finally, a numerical example of CSTCS is given to illustrate the effectiveness and feasibility of our results. PubDate: 2017-07-03

Abstract: The existence theory for the vector-valued stochastic differential equations driven by G-Brownian motion and pure jump G-Lévy process (G-SDEs) of the type \(dY_{t}=f(t,Y_{t})\, dt+g_{j,k}(t,Y_{t})\, d\langle B^{j},B^{k}\rangle _{t}+\sigma_{i}(t,Y_{t}) \, dB^{i}_{t}+\int _{R_{0}^{d}}K(t,Y_{t},z)L(dt,dz)\) , \(t\in[0,T]\) , with first two and last discontinuous coefficients, is established. It is shown that the G-SDEs have more than one solution if the coefficients f, g, K are discontinuous functions. The upper and lower solution method is used. PubDate: 2017-06-30

Abstract: In this paper, the nonlinear multiple base points boundary value problems of the impulsive fractional differential equations are studied. By using the fixed point theorem and the Mittag-Leffler functions, the sufficient conditions for the existence of the solutions to the given problems are formulated. An example is presented to illustrate the result. PubDate: 2017-06-30

Abstract: In this paper, our aim is to develop a compensated split-step θ (CSSθ) method for nonlinear jump-diffusion systems. First, we prove the convergence of the proposed method under a one-sided Lipschitz condition on the drift coefficient, and global Lipschitz condition on the diffusion and jump coefficients. Then we further show that the optimal strong convergence rate of CSSθ can be recovered, if the drift coefficient satisfies a polynomial growth condition. At last, a nonlinear test equation is simulated to verify the results obtained from theory. The results show that the CSSθ method is efficient for simulating the nonlinear jump-diffusion systems. PubDate: 2017-06-30

Abstract: In this paper, we introduce a general \((p, q)\) -Sturm-Liouville difference equation whose solutions are \((p, q)\) -analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as \((p, q) \to(1,1)\) . In this direction, some basic characterization theorems for the introduced \((p, q)\) -Sturm-Liouville difference equation, such as Rodrigues representation for the solution of this equation, a general three-term recurrence relation, and a structure relation for the \((p, q)\) -classical polynomial solutions are given. PubDate: 2017-06-29

Abstract: In this article, we discuss some properties of positive solutions for a third-order differential equation with three-point boundary conditions and a positive parameter. By using recent fixed point theory, we establish the existence and uniqueness of positive solutions for any given parameter, and we show that the positive solution is continuous, increasing with respect to the parameter. Moreover, we give some properties of limits for positive solutions. An example is provided to demonstrate the main result. PubDate: 2017-06-29

Abstract: In this paper, we solve two-dimensional modified anomalous fractional sub-diffusion equation using modified implicit finite difference approximation. The stability and convergence of the proposed scheme are analyzed by the Fourier series method. We show that the scheme is unconditionally stable and the approximate solution converges to the exact solution. A numerical example is given to show the application and feasibility of the proposed scheme. PubDate: 2017-06-29

Abstract: We give two extensions of the classical formula for sums of powers on arithmetic progressions. This is achieved by using an identity involving binomial mixtures, which can be viewed as a generalization of the binomial transform. PubDate: 2017-06-27

Abstract: Using some recent results of fixed point of weakly contractive mappings on the partially ordered space, the existence and uniqueness of solution for interval fractional delay differential equations (IFDDEs) in the setting of the Caputo generalized Hukuhara fractional differentiability are studied. The dependence of the solution on the order and the initial condition of IFDDE is shown. A new technique is proposed to find the exact solutions of IFDDE by using the solutions of interval integer order delay differential equation. Finally, some examples are given to illustrate the applications of our results. PubDate: 2017-06-27

Abstract: This article deals with some existence and Ulam-Hyers-Rassias stability results for a class of functional differential equations involving the Hilfer-Hadamard fractional derivative. An application is made of a Schauder fixed point theorem for the existence of solutions. Next we prove that our problem is generalized Ulam-Hyers-Rassias stable. PubDate: 2017-06-27