分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE

We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.     

带有边界值问题的脉冲分数阶微分方程正解的存在性

本文研究了具有边界值条件的脉冲分数阶微分方程.利用Kuratowski非紧性测度理论和Sadovskii不动点定理,得到了脉冲分数阶微分方程正解的存在性的结果,推广了已有文献的结论.     

Non-instantaneous impulsive fractional-order implicit differential equations with random effects

In this article, we study existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects. First, we establish a framework to study impulsive fractional sample path associated with impulsive fractional L^p-problem, and present the relationship between them. We also derive the formula of the solution for inhomogeneous impulsive fractional L^p-problem and sample path. Second, we construct a sequence of Picard functions, which admits us to apply successive approximations method to seek the solution of impulsive fractional sample path. Further, we derive the existence of solutions to impulsive fractional L^p-problem. Third, the concepts of Ulam's type stability are introduced and sufficient conditions to guarantee Ulam–Hyers–Rassias stability are derived. Finally, an example is given to illustrate the theoretical results.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

A Uniform Method to Ulam–Hyers Stability for Some Linear Fractional Equations

In this paper, we first utilize fractional calculus, the properties of classical and generalized Mittag-Leffler functions to prove the Ulam–Hyers stability of linear fractional differential equations using Laplace transform method. Meanwhile, Ulam–Hyers–Rassias stability result is obtained as a direct corollary. Finally, we apply the same techniques to discuss the Ulam’s type stability of fractional evolution equations, impulsive fractional evolutions equations and Sobolev-type fractional evolution equations.     

On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator

We study a nonlocal problem for a fractional partial differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator. By separation of variables, we prove a theorem on the existence and uniqueness of a regular solution of this problem.     

具有脉冲的分数阶Bagley-Torvik 模型边值问题

将具有脉冲的分数阶Bagley-Torvik微分方程边值问题巧妙地转化为积分方程,定义加权Banach空间及全连续算子,运用不动点定理获得该边值问题解的存在性定理.举例说明了定理的应用.最后提出有趣的研究问题.     

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.     

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.     

含有积分、反周期和带P-Laplacian算子的分数阶微分方程边值问题解的存在性

利用格林函数的性质和Banach压缩映射原理讨论了含P-Laplacian算子反周期边值问题的解.首先,求出与该边值问题相关的格林函数并给出了格林函数的性质;然后将边值问题转化为与其等价的积分方程,利用格林函数的性质及Banach压缩映射原理得到边值问题解的唯一性;最后给出实例验证结果的合理性.     

Some results for fractional boundary value problem of differential inclusions

In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay.     

On positive solutions of a nonlocal fractional boundary value problem

In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.     

Periodic solutions of a fractional neutral equation with finite delay

In this paper, we prove the maximal regularity property of an abstract fractional differential equation with finite delay on periodic Besov and Triebel–Lizorkin spaces and use these results to guarantee the existence and uniqueness of periodic solution of a neutral fractional differential equation with finite delay. The main tool used to achieve our goal is an operator-valued version of Miklhin’s Fourier multiplier theorem and fixed-point argument.     

Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects

We prove the global solvability and weakly asymptotic stability for a semilinear fractional differential inclusion subject to impulsive effects by analyzing behavior of its solutions on the half-line. Our analysis is based on a fixed point principle for condensing multi-valued maps, which is employed for solution operator acting on the space of piecewise continuous functions. The obtained results will be applied to a lattice fractional differential system.     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations

In this paper, we study a class of integral boundary value problem for fractional order impulsive differential equations, where both the nonlinearity and the impulsive terms contain the fractional order derivatives. By using fixed‐point theorems, the existence results of solution for the boundary value problem are established. Finally, some examples are presented to illustrate the existence results. Copyright © 2015 John Wiley & Sons, Ltd.     

The Lambert Function on the Solution of a Delay Differential Equation

Recently, a new method for computing the analytical solution of a delay differential equation was developed considering a constant initial function. It is based on the existence of a specific class of polynomials in the delay. In this article, we extend this new method to the case of a continuous initial function. We also show the relationship between the new solution's method and the solution expressed in terms of the Lambert function.     

Existence of solutions for impulsive differential models on half lines involving Caputo fractional derivatives

In this article, we establish existence results of at least one solution for two classes of initial value problems of nonlinear impulsive fractional differential models on half lines involving Caputo fractional derivatives. Our analysis rely on the well known fixed point theorem of Schauder and nonlinear alternative of Leray–Schauder type.