Abstract: Abstract In this paper, we investigate the existence of a reversed S-shaped component in the positive solutions set of the fourth-order boundary value problem $$\textstyle\begin{cases} u''''(x)=\lambda h(x)f(u(x)),\quad x\in(0,1),\\ u(0)=u(1)=u''(0)=u''(1)=0, \end{cases} $$ where \(\lambda>0\) is a parameter, \(h\in C[0,1]\) and \(f\in C[0,\infty )\) , \(f(0)=0\) , \(f(s)>0\) for all \(s>0\) . By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to parameter λ, and especially, we obtain the existence of three distinct positive solutions for λ being in a certain interval. PubDate: 2017-04-18

Abstract: Abstract By noting the fact that the intrinsic growth rate are not positive everywhere, we revisit Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. The corresponding results about permanence and extinction for the species given in (Chen and Chen in Int. J. Biomath. 8(1):1550012, 2015) are extended. Furthermore, a very important fact is found in our results, that is, the feedback controls and toxic substances have no effect on the permanence and extinction of species. Moreover, we also derive sufficient conditions for the global stability of positive solutions. Finally, some numerical simulations show the feasibility of our main results. PubDate: 2017-04-17

Abstract: Abstract In this paper, we study the third-order functional dynamic equation $$ \bigl\{ r_{2}(t)\phi_{\alpha_{2}} \bigl( \bigl[ r_{1}(t) \phi _{\alpha _{1}} \bigl( x^{\Delta}(t) \bigr) \bigr] ^{\Delta} \bigr) \bigr\} ^{\Delta}+q(t)\phi_{\alpha} \bigl( x\bigl(g(t)\bigr) \bigr) =0, $$ on an upper-unbounded time scale \(\mathbb{T}\) . We will extend the so-called Hille and Nehari type criteria to third-order dynamic equations on time scales. This work extends and improves some known results in the literature on third-order nonlinear dynamic equations and the results are established for a time scale \(\mathbb{T}\) without assuming certain restrictive conditions on \(\mathbb{T}\) . Some examples are given to illustrate the main results. PubDate: 2017-04-13

Abstract: Abstract A Crank-Nicolson scheme catering to solving initial-boundary value problems of a class of variable-coefficient tempered fractional diffusion equations is proposed. It is shown through theoretical analysis that the scheme is unconditionally stable and the convergence rate with respect to the space and time step is \(\mathcal{O}(h^{2} +\tau^{2})\) under a certain condition. Numerical experiments are provided to verify the effectiveness and accuracy of the scheme. PubDate: 2017-04-11

Abstract: Abstract For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant \(\mathcal{R}_{\sigma}\) is below unity. Here \(\mathcal{R}_{\sigma}< \mathcal{R}\) , the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the disease-free state if both components of the perturbation are non-trivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem. PubDate: 2017-04-11

Abstract: Abstract In this paper, the problem of the existence of periodic solutions is studied for the second-order differential equations with a singularity of repulsive type, $$ x''(t)+f\bigl(x'(t)\bigr)+\varphi(t)x(t)- \frac{1}{x^{r}(t)}=h(t), $$ where φ and h are T-periodic functions. By using topological degree theory, a new result on the existence of positive periodic solutions is obtained. The interesting thing is that the sign of the function \(\varphi(t)\) is allowed to be changed for \(t\in[0,T]\) . PubDate: 2017-04-08

Abstract: Abstract In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=\Delta_{\alpha}u+\frac{\partial ^{2}B}{\partial t\,\partial x}, $$ where \(\frac{\partial^{2}B}{\partial t\,\partial x}\) is a fractional Brownian sheet with Hurst indices \(H_{1},H_{2}\in(\frac{1}{2},1)\) and \(\Delta _{\alpha}=-(-\Delta)^{\alpha/2}\) is a fractional Laplacian operator with \(1<\alpha\leq2\) . In particular, when \(H_{2}=\frac{1}{2}\) we show that the temporal process \(\{u(t,\cdot),0\leq t\leq T\}\) admits a nontrivial p-variation with \(p=\frac{2\alpha}{2\alpha H_{1}-1}\) and study its local nondeterminism and existence of the local time. PubDate: 2017-04-08

Abstract: Abstract We introduce some weighted hypergeometric functions and the suitable generalization of the Caputo fractional derivation. For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator. Then some of the considered hypergeometric functions are determined in terms of the generalized Mittag-Leffler function \(E_{(\rho_{j}),\lambda}^{(\gamma_{j}),(l_{j})}[z_{1},\ldots,z_{r}]\) (Mainardi in Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010) and the generalized polynomials \(S_{n}^{m}[x]\) (Srivastava in Indian J. Math. 14:1-6, 1972). The boundary behavior of some other class of weighted hypergeometric functions is described in terms of Frostman’s α-capacity. Finally, an application is given using our fractional operator in the problem of fractional calculus of variations. PubDate: 2017-04-05

Abstract: Abstract This paper studies a new class of boundary value problems of sequential fractional differential equations of order \(q \in(2, 3]\) supplemented with nonlocal non-separated boundary conditions involving lower-order fractional derivatives. Existence and uniqueness results for the given problem are obtained by applying standard fixed-point theorems and are illustrated with the aid of examples. Some interesting observations are presented. PubDate: 2017-04-05

Abstract: Abstract This paper is devoted to generalizing the notion of almost periodic functions on time scales. We introduce a new class of almost periodic time scales called Hausdorff almost periodic time scales by using the Hausdorff distance and propose a more general notion of almost periodic functions on these new time scales. Then we explore some properties of Hausdorff almost periodic time scales and prove that the family of almost periodic functions on Hausdorff almost periodic time scales is a Banach space. Especially, our analysis also indicates that a function on a Hausdorff almost periodic time scale is almost periodic if and only if its affine extension is Bohr almost periodic on the real numbers \({\mathbb{R}}\) . As an application, we establish the existence of almost periodic solutions for a single species model on Hausdorff almost periodic time scales. PubDate: 2017-04-05

Abstract: Abstract The purpose of this paper is to study the solvability of a resonant boundary value problem for the fractional p-Laplacian equation. By using the continuation theorem of coincidence degree theory, we obtain a new result on the existence of solutions for the considered problem. PubDate: 2017-04-04

Abstract: Abstract The multilevel augmentation method with the anti-derivatives of the Daubechies wavelets is presented for solving nonlinear two-point boundary value problems. The anti-derivatives of the Daubechies wavelets are applied as the multilevel bases for the subspaces of approximate solutions. This process results in a full nonlinear system that can be solved by the multilevel augmentation method for reducing computational cost. The convergence rate of the present method is shown. It is the order of \(2^{s}\) , \(0\leq s\leq p\) when p is the order of the Daubechies wavelets. Various examples of the Dirichlet boundary conditions are shown to confirm the theoretical results. PubDate: 2017-04-04

Abstract: Abstract This paper focuses on a three-species food chain system which is formulated as stochastic differential equations with regime switching represented by a hidden Markov chain. Firstly, using the Wonham filter, we estimate the hidden Markov chain through the observable solution of the Markov chain in Gaussian white noise. Then two kinds of special dissipative control strategy are proposed to study the given model. That is, under \(H_{\infty}\) control and passive control, the sufficient conditions for global asymptotic stability are established, respectively. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results. PubDate: 2017-04-04

Abstract: Abstract We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations. PubDate: 2017-03-31

Abstract: Abstract This paper is concerned with near-optimality for stochastic control problems of linear delay systems with convex control domain and controlled diffusion. Necessary and sufficient conditions for a control to be near-optimal are established by Pontryagin’s maximum principle together with Ekeland’s variational principle. PubDate: 2017-03-31

Abstract: Abstract In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4+ cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4+ T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results. PubDate: 2017-03-29

Abstract: Abstract In this paper, we propose a new three-level implicit method based on a half-step spline in compression method of order two in time and order four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form \(u_{tt} =A(x,t,u)u_{xx} +f(x,t,u,u_{x},u_{t})\) . We describe spline in compression approximations and their properties using two half-step grid points. The new method for one-dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function \(u(x,t)\) and two half-step points for the known variable ‘x’ in x-direction. The proposed method, when applied to a linear test equation, is shown to be unconditionally stable. We have also established the stability condition to solve a linear fourth-order hyperbolic partial differential equation. Our method is directly applicable to solve hyperbolic equations irrespective of the coordinate system, which is the main advantage of our work. The proposed method for a scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method. PubDate: 2017-03-29

Abstract: Abstract The rodent is one important component of an ecosystem; however, abundance of rodents may have negative effects to humans. Based on a seasonally breeding model and the logistic model, the effect of impulse lethal control and impulse contraception control on rodent population dynamics is investigated. The existence and stability of the periodic solution were analyzed. The condition of rodent population dying out is the same for lethal and contraception control. However, the process of rodent population tending to a stable periodic solution is different obviously. Under lethal control, the rodent population tends to a stable periodic solution more quickly, whereas under contraception control, the rodent population develops slowly. PubDate: 2017-03-29

Abstract: Abstract In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: $$ \textstyle\begin{cases} (\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \quad 0 < t < 1, \\ \lambda_{1}\phi_{p} ( y ( 0 ) ) - \beta_{1} \phi_{p} ( y' ( 0 ) ) = 0, \\ \lambda_{2}\phi_{p} ( y ( 1 ) ) + \beta_{2} \phi_{p} ( y' ( 1 ) ) = 0,\qquad y'' ( 0 ) = 0, \end{cases} $$ where \(\phi_{p} ( s ) = s ^{ p-2 } s\) , \(p >1 \) . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term \(b ( t ) \) may be singular at \(t=0\) and/or \(t=1\) . Moreover, the nonlinear term \(g(t, y)\) is also allowed to have singularity at \(y=0\) . In particular, our results extend and unify some known results. PubDate: 2017-03-29