Abstract: In this paper, by employing fixed point theory, we investigate the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space. PubDate: 2019-04-08

Abstract: In the paper, we study a plasma fluid physical model, namely the Zakharov–Kuznetsov (ZK, for simplicity) equation with fractional power nonlinear terms by the complete discrimination system for polynomial method, and give a detailed construction of all its single traveling wave solutions. The results show abundant traveling wave patterns of the ZK equation. PubDate: 2019-04-08

Abstract: The aim of this paper is to use an analytic method and the properties of the classical Gauss sums to research the computational problem of one kind of character sum of polynomials modulo an odd prime p and obtain several meaningful third- and fourth-order linear recurrence formulae for them. PubDate: 2019-04-05

Abstract: In this paper, we establish some Lyapunov-type inequalities for a class of linear and nonlinear fractional q-difference boundary value problems under Cauchy boundary conditions. As applications, we use the inequality to obtain an interval, where Mittag-Leffler function has no real zeros. In addition, we also derive nonexistence results for fractional q-difference boundary value problem. PubDate: 2019-04-04

Abstract: This paper is devoted to the study of Gronwall–Bellman-type inequalities on an arbitrary time scale \(\mathbb{T}\) . We investigate some new explicit bounds of a certain class of nonlinear retarded dynamic inequalities of Gronwall–Bellman type on time scales. These inequalities extend some known dynamic inequalities on time scales. We also generalize and unify some continuous inequalities and their corresponding discrete analogues. To illustrate the benefits of our work, we present some applications of these results. The main results will be proved by using some analysis techniques and a simple consequence of the Keller’s chain rule on time scales. PubDate: 2019-04-02

Abstract: In this paper, we give an affirmative answer to the following question: Is the solvability of some nonlinear dynamic equations on a time scale \(\mathbb{T}\) not only sufficient but in a certain sense also necessary for the validity of some dynamic Hardy-type inequalities with two different weights' In fact, this answer will give a new characterization of the weights in a weighted Hardy-type inequality on time scales. The results contain the results when \(\mathbb{T}=\mathbb{R}\) , \(\mathbb{T}=\mathbb{N}\) , and when \(\mathbb{T}=q^{\mathbb{N}_{0}}\) as special cases. Some applications are given for illustrations. PubDate: 2019-04-01

Abstract: In this paper, we present a semi-analytic method called the local fractional homotopy analysis method (LFHAM) for solving differential equations involving local fractional derivatives based on the local fractional calculus and the homotopy analysis method. The suggested analytical technique always provides a simple way of constructing a series of solutions from the higher-order deformation equation. The LFHAM guarantees the convergence of the series solutions using the nonzero convergence-control parameter. Three examples are provided to illustrate the efficiency and high accuracy of the method. PubDate: 2019-03-29

Abstract: In this article, we study a coupled system of singular fractional difference equations with fractional sum boundary conditions. A sufficient condition of the existence of positive solutions is established by employing the upper and lower solutions of the system and using Schauder’s fixed point theorem. Finally, we provide an example to illustrate our results. PubDate: 2019-03-29

Abstract: In this paper, the bifurcations of a modified Degasperis–Procesi equation are studied under different parametric conditions, which have not been investigated by the bifurcation theory of dynamical systems before. The existence of loop, periodic wave and smooth solitary wave solutions for the modified Degasperis–Procesi equation is proved, and exact expressions of corresponding traveling wave solutions are obtained. PubDate: 2019-03-28

Abstract: In this paper, we study the existence of solutions for hybrid fractional differential equations involving fractional Caputo derivative of order \(1 < \alpha\leq 2\) . Our results rely on a hybrid fixed point theorem for a sum of three operators due to Dhage. An example is provided to illustrate the theory. PubDate: 2019-03-27

Abstract: In this paper, bifurcation analysis of a discrete Hindmarsh–Rose model is carried out in the plane. This paper shows that the model undergoes a flip bifurcation, a Neimark–Sacker bifurcation, and \(1:2\) resonance which includes a pitchfork bifurcation, a Neimark–Sacker bifurcation, and a heteroclinic bifurcation. The sufficient conditions of existence of the fixed points and their stability are first derived. The flip bifurcation and Neimark–Sacker bifurcation are analyzed by using the inner product method and normal form theory. The conditions for the occurrence of \(1:2\) resonance are also presented. Furthermore, the sufficient conditions of pitchfork, Neimark–Sacker, and heteroclinic bifurcations are derived and expressed by implicit functions. The numerical analysis shows us consistence with the theoretical results and exhibits interesting dynamics, especially symmetric and invariant closed orbits. The dynamics observed in this paper can be used to mimic the dynamical behaviors of one single neuron and design a humanoid locomotion model for applications in bio-engineering and so on. PubDate: 2019-03-25

Abstract: In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map. PubDate: 2019-03-22

Abstract: This paper is concerned with the stability criteria for a discrete-time linear system with interval time-varying delays. By using the time delay division we construct an augmented Lyapunov–Krasovskii functional for two delay subintervals. Moreover, we use a new summation inequality to estimate the derivatives of LKFs more accurately and derive less conservative criteria. Finally, we present two numerical examples to demonstrate that the obtained results are less conservative. PubDate: 2019-03-22

Abstract: In this paper the classification of single traveling wave solutions of \((1+1)\) dimensional Gardner equation with variable coefficients is obtained by applying the complete discrimination system to the polynomial and trial equation methods. In particular, the corresponding solutions for the concrete parameters are constructed to show that each solution in the classification can be realized. Moreover, numerical simulations shown in the paper could help us better understand the nature of each solution. PubDate: 2019-03-22

Abstract: This paper investigates the problem of a state bounding estimation for a linear continuous-time singular system with time-varying delay. By employing the maximal Lyapunov–Krasovskii functional and applying the new free-matrix-based integral inequality, some proper conditions are derived in terms of LMIs and a bounding estimation lemma and set are obtained for the studied singular system. PubDate: 2019-03-22

Abstract: This paper is concerned with a class of anti-periodic boundary value problems for fractional differential equations with the Riesz–Caputo derivative, which can reflected both the past and the future nonlocal memory effects. By means of new fractional Gronwall inequalities and some fixed point theorems, we obtain some existence results of solutions under the Lipschitz condition, the sublinear growth condition, the nonlinear growth condition and the comparison condition. Three examples are given to illustrate the results. PubDate: 2019-03-21

Abstract: In the paper, fourth-order delay differential equations of the form $$ \bigl(r_{3} \bigl(r_{2} \bigl(r_{1}y' \bigr)' \bigr)' \bigr)'(t) + q(t) y \bigl( \tau (t) \bigr) = 0 $$ under the assumption $$ \int _{t_{0}}^{\infty }\frac{\mathrm {d}t}{r_{i}(t)} < \infty , \quad i = 1,2,3, $$ are investigated. Our newly proposed approach allows us to greatly reduce a number of conditions ensuring that all solutions of the studied equation oscillate. An example is also presented to test the strength and applicability of the results obtained. PubDate: 2019-03-20

Abstract: In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of Li (Comput. Methods Funct. Theory 12:307–328, 2012 and partially answers Liu’s (J. Math. Anal. Appl. 359:384–393, 2009) question. PubDate: 2019-03-19

Abstract: Based on the weighted and shifted Grünwald formula, a fully discrete finite element scheme is derived for the variable coefficient time-fractional subdiffusion equation. Firstly, the unconditional stable and convergent of the fully discrete scheme in \(L^{1}(H^{1})\) -norm is proved. Secondly, through a new estimate approach, the superclose properties are obtained. The global superconvergence order \(\mathcal{O}(\tau ^{2}+h^{m+1})\) is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis. PubDate: 2019-03-19

Abstract: Pesticides often cause residual and delayed effects on pests. Considering these effects, we use a pollution emission model to simulate the process of spraying pesticides. Many pests reproduce only at a fixed time in a year. So a pest control model with birth pulse and spraying pesticides is proposed. Using the limit system of the developed model, we analyze the dynamics of the system. The stability of the trivial equilibrium and the positive equilibrium of the model is analyzed, and the threshold conditions of pest eradication and permanence of the system are given. We obtain the optimal frequency of spraying pesticides by numerical simulations. The important parameters related to the pest eradication or permanence of the system are given by analyzing the sensitivity of the parameters. Finally, biological explanations are provided. PubDate: 2019-03-19