Abstract: The Fibonacci sequence was firstly used in the theory of sequence spaces by Kara and Başarir (Casp. J. Math. Sci. 1(1):43–47, 2012). Afterward, Kara (J. Inequal. Appl. 2013(1):38, 2013) defined the Fibonacci difference matrix F̂ by using the Fibonacci sequence \((f_{n})\) for \(n\in{\{0, 1, \ldots\}}\) and introduced new sequence spaces related to the matrix domain of F̂. In this paper, by using the Fibonacci difference matrix F̂ defined by the Fibonacci sequence and the notion of ideal convergence, we introduce the Fibonacci difference sequence spaces \(c^{I}_{0}(\hat {F})\) , \(c^{I}(\hat{F})\) , and \(\ell^{I}_{\infty}(\hat{F})\) . Further, we study some inclusion relations concerning these spaces. In addition, we discuss some properties on these spaces such as monotonicity and solidity. PubDate: 2018-05-25

Abstract: The objective of this paper is to study the dynamics of the stochastic ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting. For the autonomous system, sufficient conditions for globally positive solution and stochastic permanence are established. Then, applying comparison theorem for stochastic differential equation, sufficient conditions for extinction and persistence in the mean are obtained. In addition, we prove that there exists a unique stationary distribution and it has ergodicity under certain parametric restrictions. For the periodic system, we obtain conditions for the existence of a nontrivial positive periodic solution. Finally, numerical simulations are carried out to substantiate the analytical results. PubDate: 2018-05-24

Abstract: A Lotka–Volterra commensal symbiosis model with first species subject to the Allee effect is proposed and studied in this paper. Local and global stability property of the equilibria are investigated. An amazing finding is that with increasing Allee effect, the final density of the species subject to the Allee effect is also increased. Such a phenomenon is different from the known results, and it is the first time to be observed. Numeric simulations are carried out to show the feasibility of the main results. PubDate: 2018-05-22

Abstract: In this paper, we prove the following result: Let f be a nonconstant meromorphic function of finite order, p be a nonconstant polynomial, and c be a nonzero constant. If f, \(\Delta _{c}f\) , and \(\Delta_{c}^{n}f\) ( \(n\ge 2\) ) share ∞ and p CM, then \(f\equiv \Delta_{c}f\) . Our result provides a difference analogue of the result of Chang and Fang in 2004 (Complex Var. Theory Appl. 49(12):871–895, 2004). PubDate: 2018-05-22

Abstract: In this paper, an attempt is being made to investigate a class of fractional Fourier integral operators on classes of function spaces known as ultraBoehmians. We introduce a convolution product and establish a convolution theorem as a product of different functions. By employing the convolution theorem and making use of an appropriate class of approximating identities, we provide necessary axioms and define function spaces where the fractional Fourier integral operator is an isomorphism connecting the different spaces. Further, we provide an inversion formula and obtain various properties of the cited integral in the generalized sense. PubDate: 2018-05-22

Abstract: In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence. PubDate: 2018-05-22

Abstract: In this paper, a class of first-order neutral differential equations with time-varying delays and coefficients is considered. Some results on the existence of positive almost periodic solutions for the equations are obtained by using the contracting mapping principle and the differential inequality technique. In addition, an example is given to illustrate our results. PubDate: 2018-05-21

Abstract: In this paper, we investigate oscillatory and asymptotic properties for a class of fractional order dynamic equations on time scales, where the fractional derivative is defined in the sense of the conformable fractional derivative. Based on the properties of conformable fractional differential and integral, some new oscillatory and asymptotic criteria are established. Applications of the established results show that they can be used to research oscillation for fractional order equations in various time scales such as fractional order differential equations, fractional order difference equations, and so on. PubDate: 2018-05-21

Abstract: A single species logistic model with Allee effect and feedback control $$\begin{aligned}& \frac{dx}{dt} = rx(1-x)\frac{x}{\beta+x}-axu, \\& \frac{du}{dt} = -bu+cx, \end{aligned}$$ where β, r, a, b, and c are all positive constants, is for the first time proposed and studied in this paper. We show that, for the system without Allee effect, the system admits a unique positive equilibrium which is globally attractive. However, for the system with Allee effect, if the Allee effect is limited ( \(\beta<\frac{b^{2}r^{2}}{ac(ac+br)}\) ), then the system could admit a unique positive equilibrium which is locally asymptotically stable; if the Allee effect is too large ( \(\beta>\frac{br}{ac}\) ), the system has no positive equilibrium, which means the extinction of the species. The Allee effect reduces the population density of the species, which increases the extinction property of the species. The Allee effect makes the system “unstable” in the sense that the system could collapse under large perturbation. Numeric simulations are carried out to show the feasibility of the main results. PubDate: 2018-05-18

Abstract: A nonlinear amensalism model of the form $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}= r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned}$$ where \(r_{i}, P_{i}, u, i=1, 2, \alpha_{1}, \alpha_{2}, \alpha_{3}\) are all positive constants, is proposed and studied in this paper. The dynamic behaviors of the system are determined by the sign of the term \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} \) . If \(1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}>0\) , then the unique positive equilibrium \(D(N_{1}^{*},N_{2}^{*})\) is globally attractive, if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\) , then the boundary equilibrium \(C(0, P_{2})\) is globally attractive. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Advances in Applied Mathematics 5(2):255–261, 2016). PubDate: 2018-05-18

Abstract: This paper investigates the design of disturbance attenuating controller for memristive recurrent neural networks (MRNNs) with mixed time-varying delays. By applying the combination of differential inclusions, set-valued maps and Lyapunov–Razumikhin, a feedback control law is obtained in the simple form of linear matrix inequality (LMI) to ensure disturbance attenuation of memristor-based neural networks. Finally, a numerical example is given to show the effectiveness of the proposed criteria. PubDate: 2018-05-18

Abstract: In this paper, we study a kind of difference equations with Riemann–Liouville-like fractional difference. The results on existence and attractivity are obtained by using the Picard iteration method and Schauder’s fixed point theorem. Examples are provided to illustrate the main results. PubDate: 2018-05-18

Abstract: In this paper, an adaptive neural network (NN) synchronization controller is designed for two identical strict-feedback chaotic systems (SFCSs) subject to dead-zone input. The dead-zone models together with the system uncertainties are approximated by NNs. The dynamic surface control (DSC) approach is applied in the synchronization controller design, and the traditional problem of “explosion of complexity” that usually occurs in the backstepping design can be avoided. The proposed synchronization method guarantees the synchronization errors tend to an arbitrarily small region. Finally, this paper presents two simulation examples to confirm the effectiveness and the robustness of the proposed control method. PubDate: 2018-05-18

Abstract: In this article, the following boundary value problem of fractional differential equation with Riemann–Stieltjes integral boundary condition $$\textstyle\begin{cases} D_{0+}^{\alpha}u(t)+\lambda f(t, u(t),u(t))=0, \quad0< t< 1, n-1< \alpha \leq n, \\ u^{(k)}(0)=0,\quad 0\leq k\leq n-2, \qquad u(1)= \int_{0}^{1}u(s)\,dA(s) \end{cases} $$ is studied, where \(n-1 < \alpha\le n\) , \(\lambda>0\) , \(D_{0+}^{\alpha}\) is the Riemann–Liouville fractional derivative, A is a function of bounded variation, \(\int_{0}^{1}u(s)\,dA(s)\) denotes the Riemann–Stieltjes integral of u with respect to A. By the use of fixed point theorem and the properties of mixed monotone operator theory, the existence and uniqueness of positive solutions for the problem are acquired. Some examples are presented to illustrate the main result. PubDate: 2018-05-16

Abstract: In this paper, we consider a spectrally negative Markov additive risk process. Using the theory of Jordan chain, a compact formula of Parisian ruin probability is given. The formula depends only on the scale matrix of spectrally negative Markov additive risk processes and the transition rate matrix \(\Lambda^{q}\) . PubDate: 2018-05-16

Abstract: In this work, we deal with two-point Riemann–Liouville fractional boundary value problems. Firstly, we establish a new comparison principle. Then, we show the existence of extremal solutions for the two-point Riemann–Liouville fractional boundary value problems, using the method of upper and lower solutions. The performance of the approach is tested through a numerical example. PubDate: 2018-05-16

Abstract: In this paper we consider the existence of positive solutions of nth-order Sturm–Liouville boundary value problems with fully nonlinear terms, in which the nonlinear term f involves all of the derivatives \(u',\ldots, u^{(n-1)}\) of the unknown function u. Such cases are seldom investigated in the literature. We present some inequality conditions guaranteeing the existence of positive solutions. Our inequality conditions allow that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) is superlinear or sublinear growth on \(x_{0}, x_{1},\ldots, x_{n-1}\) . Our discussion is based on the fixed point index theory in cones. PubDate: 2018-05-16

Abstract: This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter \(k_{1}\) and sufficiently large \(k_{2}\) in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large \(k_{1}\) . The results show that if \(k_{1}\) is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem. PubDate: 2018-05-16

Abstract: This paper is concerned with the two-point boundary value problems of a nonlinear fractional q-difference equation with dependence on the first order q-derivative. We discuss some new properties of the Green function by using q-difference calculus. Furthermore, by means of Schauder’s fixed point theorem and an extension of Krasnoselskii’s fixed point theorem in a cone, the existence of one positive solution and of at least one positive solution for the boundary value problem is established. PubDate: 2018-05-16

Abstract: Incorporating two delays ( \(\tau_{1}\) represents the maturity of predator, \(\tau_{2}\) represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov–LaSalle’s principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if \(\tau_{1}\in (m_{1},m_{2})\) and \(\tau_{2}\in(m_{4}, +\infty)\) , then the boundary equilibrium \(E_{2}(x^{0}, y_{1}^{0}, y_{2}^{0}, 0, 0)\) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if \(\tau_{1}\in(m_{2}, +\infty)\) , then the axial equilibrium \(E_{1}(k, 0, 0, 0, 0)\) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results. PubDate: 2018-05-16