一类分数阶脉冲微分方程Dirichlet边值问题非平凡解的存在性

研究了一类带有脉冲的分数阶微分方程Dirichlet边值问题非平凡解的存在性.通过利用变分法和Morse理论证明了此分数阶脉冲微分方程至少存在一个非平凡解.     

分数阶Kirchhoff型微分方程Dirichlet边值问题的研究

本文应用不动点定理研究一类不带P.S.条件的分数阶Kirchhoff型微分方程Dirichlet边值问题弱解的存在性.     

半正的分数阶微分方程(n-1,1)-型积分边值问题的多个正解

采用Riemann-Liouville分数阶导数,研究了半正的分数阶微分方程(n-1,1)-型积分边值问题,获得了参数λ的一个区间,使得λ落在这个区间的时候,该半正的分数阶微分方程边值问题有多个正解.     

Hilfer-Katugampola序列分数阶微分方程边值问题Lyapunov型不等式

研究了Hilfer-Katugampola序列分数阶微分方程多点边值问题Lyapunov型不等式.首先,利用Hilfer-Katugampola分数阶微积分的定义和性质将HilferKatugampola序列分数阶微分方程边值问题等价转化为带有Green函数的积分方程问题.其次,定义相应的Banach空间并结合先验估计方法得到了Lyapunov型不等式.最后,通过给出一系列推论说明该文研究结果推广和丰富了已有文献相关工作.     

分数阶微分方程组边值问题解的存在性

研究分数阶微分方程组边值问题在一类新型的边界条件——分数阶分离边界条件下解的存在性.通过将微分方程组边值问题转化为与之等价的积分方程组,利用Banach不动点定理和Leray-Schauder非线性更替得到边值问题解存在的充分条件,并给出两个例子说明了主要结论.     

共振条件下序列型分数微分方程耦合系统解的存在性[英文]

本文研究一类序列型分数阶微分方程耦合系统的多点边值问题,在共振条件下,通过迭合度理论,给出系统解的存在性的判断标准.最后,通过实例验证了本文结论的可行性.     

一类非线性项带导数的p-Laplacian边值问题正解的存在性和多解性

本文研究了一类非线性项带导数的p-Laplacian算子的分数阶微分方程边值问题正解的存在性和多解性.首先,利用分数阶微分方程和边值条件给出了该边值问题的Green函数,然后利用Guo-Krasnosel’skii’s不动点定理和Leggett-Williams不动点定理得出该边值问题一个或者三个正解的存在性结论.作为应用,给出两个例子验证了结论的适用性,特别是,用迭代法进行了逼近模拟,给出解的图形.值得一提的是此文研究的微分方程的非线性项是带有Riemann-Liouville型分数阶微分.     

一类非线性共形分数阶微分方程的边值问题和脉冲初值问题（英文）

本文研究一类非线性共形分数阶微分方程的边值问题和脉冲初值问题,利用基于锥理论的和型算子不动点定理和混合单调算子不动点定理,获得共形分数阶微分方程边值问题和脉冲初值问题正解的存在性和唯一性定理,并且得到一组可以逼近唯一正解的单调迭代序列,最后给出一个实例用来验证结论的有效性.     

一类Caputo分数阶微分方程多点边值问题的多解性

本文讨论了一类Caputo分数阶微分方程多点边值问题的多解性,通过把分数阶微分方程的边值问题转化成与其等价的积分方程问题求出边值问题的Green函数并得到其格林函数的相关性质,最后利用锥上不动点指数定理研究分数阶微分方程多点边值问题正解和多个正解的存在性.     

分数阶微分方程积分边值问题多个正解的存在性

研究了带有积分边值条件的分数阶微分方程的边值问题.利用LeggettWilliams不动点定理,以及一些分析技巧得到了这类分数阶微分方程边值问题多个正解的存在性.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples.     

The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations

In this paper we establish the existence of a positive solution to a singular coupled system of nonlinear fractional differential equations. Our analysis rely on a nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

The new exact solutions of variant types of time fractional coupled Schr\"{o}dinger equations in plasma physics

In the present article, the new exact solutions of fractional coupled Schr\"{o}dinger type equations have been studied by using a new reliable analytical method. We applied a relatively new method for finding some new exact solutions of time fractional coupled equations viz. time fractional coupled Schr\"{o}dinger--KdV and coupled Schr\"{o}dinger--Boussinesq equations. The fractional complex transform have been used here along with the property of local fractional calculus for reduction of fractional partial differential equations (FPDE) to ordinary differential equations (ODE). The obtained results have been plotted here for demonstrating the nature of the solutions.     

On the reflecting function and the qualitative behavior of solutions of some non-autonomous differential equations

In this article, we use the Mironenko''s method to discuss the qualitative behavior of some non-autonomous differential equations. We study the structure of the reflecting functions of the simplest differential equations, and obtain some sufficient conditions under which these equations have the rational reflecting functions. We apply the obtained results to discuss the numbers of periodic solutions of the non-autonomous differential systems and derive some sufficient conditions for a critical point of theirs to be a center.     

Positive bound states of systems of nonlinear Schrödinger equations

We study the existence of positive bound states of non-autonomous systems of nonlinear Schrödinger equations. Both the singular case and the regular case are discussed. The proof is based on a nonlinear alternative principle of Leray–Schauder.     

Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.     

Analysis of fractional order differential coupled systems

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B‐vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence results of periodic solutions for non-autonomous differential delay equations with asymptotically linear properties

In this paper, we study a class of non-autonomous differential delay equations which can be changed to Hamiltonian systems. By estimating Maslov-type index of the related Hamiltonian systems at infinity and at origin, we establish the existence of periodic solutions of the differential delay equations.     

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two‐dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two‐dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro‐differential equations and then converted weakly singular fractional partial integro‐differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.