Stability of fractional functional differential equations
J. Vanterler da C. Sousa, E. Capelas de Oliveira, F. G. Rodrigues
本文利用Banach不动点定理研究了分数阶泛函微分方程解的Ulam-Hyers和Ulam-Hyers-Rassias稳定性.
In this paper, we present a study on the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solution of the fractional functional differential equation using the Banach fixed point theorem.
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Solutions of Sequential Conformable Fractional Differential Equations around an Ordinary Point and Conformable Fractional Hermite Differential Equation
本文给出了2α阶齐次序列线性协调分数阶微分方程在变系数情形下在常点附近的幂级数解.在此基础上，介绍了共形分数阶Hermite微分方程、共形分数阶Hermite多项式及其基本性质。
Solution of Conformable Fractional Ordinary Differential Equations via Differential Transform Method
最近，在文献[1]中导数的基本极限定义的基础上，给出了一种新的分数阶导数，称为协调分数阶导数。然后，在[2]中发展了链式法则、指数函数、Gronwall不等式、分部积分法、Taylor幂级数展开式的分数阶形式。给出了共形分数阶微分变换方法及其在共形分数阶微分方程中的应用。
We establish partial semigroup property of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems, and prove existence and uniqueness of solution to multi-term Caputo fractional differential systems
Local existence and uniqueness of solutions for fractional differential problem with Hilfer-Hadamard fractional derivative
本文讨论了具有Hilfer-Hadamrd分数阶导数的分数阶微分方程解的局部存在唯一性。利用Picard逼近和推广非线性函数的约束条件，构造了一致逼近解的迭代格式。
On a Hilfer fractional differential equation with nonlocal Erdélyi-Kober fractional integral boundary conditions
考虑一类具有非局部Erdélyi-Kober分数次积分边界条件的Hilfer分数阶微分方程.利用Krasnoselskii的不动点定理和Banach的不动点定理研究了该方程的存在性、唯一性和Ulam-Hyers稳定性.最后给出一个例子来说明主要结果。
A new fractional derivative of variable order with non-singular kernel and fractional differential equations
在本文中，我们引入了两个新的非奇异核分数阶导数，并给出了一类由新公式导出的分数阶导数。我们提出了一些重要的结果一致收敛序列的连续函数，特别是比较的原则，和其他人，允许，研究的限制分数阶非线性微分方程。
Positive solutions of $p$-Laplacian fractional differential equations with fractional derivative boundary condition
本文给出了一类具有分数阶导数边界条件的$p$ -Laplacian分数阶微分方程正解的存在唯一性结果。我们的结果是基于Krasnosel'skii的不动点定理，Leray-Schauder型的非线性替换和压缩映射原理。最后给出了三个例子来说明我们的主要结果的适用性。
Existence results for a class of Caputo type fractional differential equations with Riemann-Liouville fractional integrals and Caputo fractional derivatives in boundary conditions
研究了一类具有非局部Riemann-Liouville分数次积分和Caputo分数阶导数边界条件的分数阶边值问题解的存在唯一性.我们的结果是基于一些已知的不动点理论的工具。最后，通过一些算例验证了所得结果的有效性.
Lyapunov type inequality for fractional differential equation with k-Prabhakar derivative
本文对含k-Prabhakar分数阶导数的分数阶边值问题建立了一类李雅普诺夫型不等式.
In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations.
General Solution to Sequential Linear Conformable Fractional Differential Equations With Constant Coefficients
本文给出了序列线性协调分数阶微分方程在常系数情形下α（\in）（0，1]的通解.在齐次情况下，我们使用一个分数阶指数函数，它推广了相应的普通函数。在非齐次情形下，给出了求序列线性一致分数阶微分方程特解的参数变分方法。
Lyapunov type inequality for hybrid fractional differential equation with Prabhakar Derivative
本文对含有Prabhakar分数阶导数的混合分数阶边值问题建立了一个李雅普诺夫型不等式。
Global Attractivity for Fractional Differential Equations in Weighted Spaces
研究分数阶Cauchy型问题。利用Schauder不动点定理得到了加权空间中非线性分数阶微分方程解的全局吸引性的充分条件。
Asymptotic behavior of solutions of linear multi-order fractional differential equation systems
本文研究了多阶分数阶微分方程系统的定性理论。首先，我们得到了多阶分数阶微分方程系统解的存在唯一性的一个基本结果。其次，给出了多阶齐次线性分数阶微分方程组解的级数表示。最后，我们给出了几类多阶线性分数阶微分方程组解的渐近性态的特征。
Existence and Uniqueness Theorems for Sequential Linear Conformable Fractional Differential Equations
最近，在[4]中给出了一种新的分数阶导数，称为协调分数阶导数。然后，在[5]中发展了链式法则、指数函数、Gronwalls不等式、分部积分法、Taylor幂级数展开式的分式形式。本文给出了序列线性协调分数阶微分方程解的存在唯一性定理。
Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation
利用$ψ-$ Hilfer分数阶导数，利用不动点方法研究了分数阶Volterra积分-微分方程的Hyers-Ulam-Rassias稳定性和Hyers-Ulam稳定性.
Analytic Solution of Linear Fractional Differential Equation with Jumarie Derivative in Term of Mittag-Leffler Function
分数阶微分方程的求解没有统一的方法。本文中所用的导数类型是Jumarie公式，对于所研究的几个微分方程。本文给出了用Mittag-Leffler函数求解由Jumarie分数阶导数构成的线性分数阶微分方程的一种算法，并证明了该算法与普通微积分的共轭性.在这些分数阶微分方程中，单参数Mittag-Leffler函数起着类似于常微分方程中指数函数的作用。
Solution of System of Linear Fractional Differential Equations with Modified derivative of Jumarie Type
分数阶微分方程的求解是当今研究的一个新兴领域，因为这类方程出现在各种应用领域中。本文发展了用Mittag-Leffler函数和广义Sine函数、广义Cosine函数求解分数阶微分方程组的解析方法，其中分数阶导数算子为Jumarie型。利用改进的Rieman-Liouvellie分数阶导数Jumarie型分数阶导数，简化了对此类分数阶系统的求解。这种类型的Jumarie分数导数的使用给出了一个共轭的线性整数阶微分方程系统的解决方案的经典方法，通过使用Mittag-Leffler和广义三角函数。这种方法的简便性及其与求解线性分数阶微分方程的经典方法的共轭性吸引了分数阶动力系统的研究者。在此基础上，将该算法应用于分数阶微分方程的物理系统。然后，以线性分数阶微分方程为例，用图解法给出了分析结果。
Non-existence of Global Solutions for a Generalized Fractional Differential Problem
研究了一类含广义Katugampola导数的分数阶微分问题整体解的不存在性。利用检验函数法和分数次分部积分法公式得到结果。文中还给出了一个说明性的例子。
In present paper, Hartman-Wintner-type inequality is established for a nonlocal fractional boundary value problem involving k-Prabhakar fractional derivative.
Fractional order pseudoparabolic partial differential equation: Ulam-Hyers stability
利用格朗沃尔不等式研究了一类分数阶拟抛物型偏微分方程解的Ulam-Hyers和广义Ulam-Hyers-Rassias稳定性.
In this paper, we will develop a definition of mild solution for impulsive fractional differential equation of order $α\in (1,2)$ with the help of solution operator and study the existence results of mild solution for impulsive fractional differential equations with state dependent delay by using fixed point theorems. Finally, we present an example of partial fractional derivative to illustrate the existence and uniqueness result.
Existence and Stability of Fractional Differential Equations Involving Generalized Katugampola Derivative
给出了一类含广义Katugampola导数的分数阶微分方程解的存在性和稳定性结果。利用了一些不动点定理，并给出了一些有启发性的例子。
Solutions around a Regular α Singular Point of a Sequential Conformable Fractional Differential Equation
本文给出了2α阶齐次序列线性协调分数阶微分方程在变系数情形下，在正则奇点附近的幂级数解.
Stability of the fractional Volterra integro-differential equation by means of $ψ-$Hilfer operator
利用Riemann-Liouville分数次积分和$ψ-$ Hilfer分数次导数，建立了分数阶Volterra积分方程和分数阶Volterra积分微分方程.在这个意义下，我们利用巴拿赫空间中的不动点定理，研究了这类新的分数阶Volterra积分微分方程的Ulam-Hyers稳定性以及分数阶Volterra积分方程.作为应用，我们给出了索伯列夫空间$W^{1,1}(\mathbb{R}_{+},\mathbb{C})$中$α$ -预解算子的Ulam-Hyers稳定性.
Existence and continuation of solutions of Hilfer fractional differential equations
本文研究了具有Hilfer分数阶导数的一般分数阶微分方程解的存在性和连续性。首先，我们建立了新的局部存在定理。然后我们得到了延拓定理。利用本文导出的延拓定理，构造了几个整体存在性结果。
Stability of scalar nonlinear fractional differential equations with linearly dominated delay
本文研究了一类分数阶时滞微分方程解在平衡点附近的渐近性态。更精确地说，我们给出了线性分数阶时滞方程渐近稳定的系数条件，并证明了分数阶时滞微分方程在非线性Lipschitz小扰动下平凡解的渐近稳定性.
Stability of $ψ$-Hilfer Impulsive Fractional Differential Equations
研究了一类含$ψ$ -Hilfer分数阶导数的脉冲分数阶微分方程解的存在唯一性和δ-Ulam-Hyers-Rassias稳定性的充分条件.主要结果是用不动点方法得到的。
In this article we study the asymptotic behavior of solutions of some fractional differential equations. We prove convergence to power type functions under some assumptions on the nonlinearities. Our results extend and generalize some existing well-known results on solutions of ordinary differential equations. Appropriate estimations and lemmas such as a fractional version of L'Hopital's rule are used.
Hartman-Wintner-type inequality for fractional differential equation with Prabhakar derivative
本文考虑一类带Prabhakar导数的非局部分数阶边值问题，得到了它的一个Hartman-Wintner型不等式.
In this work we wish to highlight some consequences of a recent result proved in [N. D. Cong and H. T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations, J. Integral Equations Appl. 29 (2017), no. 4, 585-608]. Particular emphasis will be given to its application on fractional variational problems of Herglotz type.
Existence and uniqueness of solution of Cauchy-type problem for Hilfer fractional differential equations
考虑一类含分数阶Hilfer导数的非线性微分方程的Cauchy型问题.利用逐次逼近和广义格朗沃尔不等式证明了Cauchy型问题解的存在唯一性和连续依赖性。
Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations
利用最近的$ψ$ -Hilfer分数阶导数和Banach不动点定理，研究了一类特殊的分数阶积分微分方程的Ulam-Hyers，Ulam-Hyers-Rassias和semi-Ulam-Hyers-Rassias在闭区间$[a,b]$和$[a,\infty)$上的稳定性.
Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals
本文给出了求解由Jumarie型分数阶导数构成的线性非齐次分数阶微分方程的一种方法，并描述了我们发展的求几种强迫函数的特积分的方法。用Mittag-Leffler函数、分数阶正弦函数和分数阶余弦函数的形式给出了解。我们已经使用我们早期开发的方法，通过Jumarie分数导数组成的均匀FDE的解决方案，并将其扩展到非均匀FDE。我们用几个FDE的例子证明了这些方法，并将其应用于分数阶阻尼强迫微分方程。我们提出的方法是有用的，因为它与求解非齐次线性微分方程的经典方法具有共轭性，并且对于理解FDE描述的物理系统也是有用的。
On existence and approximation of solution of nonlinear Hilfer fractional differential equation
本文给出了具有Hilfer导数的分数阶微分方程解的存在唯一性结果。利用一些新的技巧和推广所考虑函数的约束条件，建立了一致逼近解的迭代格式。
Recent developments of Lyapunov-type inequalities for fractional differential equations
综述了分数阶微分方程在各种边界条件下的Lyapunov型不等式的结果。这包括狄利克雷，混合，罗宾，分数，Sturm-Liouville，积分，非局部，多点，反周期，共轭，右焦点和脉冲条件。此外，我们还研究了Riemann-Liouville，Caputo，Hadamard，Prabhakar，Hilfer和协调型分数阶导数。分数$p$ -拉普拉斯算子，分数阶非奇异Mittag-Leffler核算子，$q$ -差分，离散和脉冲方程的边值问题的结果，也被考虑在内。
A new smoothness result for Caputo-type fractional ordinary differential equations
给出了Caputo型分数阶常微分方程的一个新的光滑性结果，该结果表明，对于某个正整数$ m $，减去一个可由已知信息得到的非光滑函数，非光滑解属于$ C^m $.
Analysis of Caputo impulsive fractional order differential equations with applications
利用Sadavoskii的不动点方法，以脉冲Logistic模型为例，研究了α阶介于0和1之间的Caputo脉冲分数阶微分方程解的存在唯一性，并给出了其它几个例子.我们还讨论了有限时滞Caputo脉冲分数阶微分方程.已证明的结果是新的，是对现有结果的补充。
Generalization of the final-value theorem and its application in fractional differential systems
本文推广了已知的拉普拉斯变换终值定理。从我们的结论可以推出文献[1，3，12]中已有的结果。利用终值定理给出了Caputo分数阶微分方程不存在非常数周期解的一个新的证明。
Existence, uniqueness and exponential boundedness of global solutions to delay fractional differential equations
在较弱的利普希茨条件下，证明了一类时滞分数阶微分方程整体解的存在唯一性定理。然后，我们建立了这些解的指数有界性的一个结果。
On Ulam type stability for nonlinear implicit fractional differential equations
本文建立了非线性隐式分数阶微分方程组解的存在性和稳定性的充分条件。主要技术是基于逐次逼近法。最后，通过一个算例说明了理论结果的适用性.
Non-existence for fractionally damped fractional differential problems
研究了一个右端含有一个多项式源项和一个低阶分数阶导数的分数阶微分不等式。在适当的空间中，利用测试函数法证明了非平凡整体解的不存在性。爆破的范围被发现只取决于低阶导数。这与内部弱阻尼波动方程的解将收敛到抛物部分的解的众所周知的事实是一致的。
On the stability of a hyperbolic fractional partial differential equation
本文利用格朗沃尔不等式，引入了Riemann-Liouville分数次积分和Hilfer分数次偏导数，并恢复了它们的一些特殊情况.利用这些结果，我们研究了巴拿赫空间（B，|.|),真实的或复杂的。
Discrete Fractional Solutions of a Physical Differential Equation via $ \nabla $-DFC Operator
离散数学，研究有限结构，是数学和优化中发展最快的领域之一。离散分数阶微积分（DFC）理论是分数阶微积分的一个重要分支，它包含了分数阶差分。本文讨论了径向薛定谔方程，它是一个物理奇异的微分方程。利用nabla（$ \nabla $）离散分数阶算子，得到了该方程的特解.该算子对奇异方程给出了成功的结果，并且解具有分数形式，包括离散移位算子$ E $。
In this work, we study the regularity of positive solutions for nonlinear fractional differential equation with a singular weight. We define the new Banach space and use this space to show the regularity. We also give an example with a singular weight which may not be in $L^1.$
Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations
研究了分数阶中立型积分微分方程$r$类的伪$S$ -渐近$ω$ -周期温和解的存在唯一性.最后给出一个例子说明抽象结果的应用。
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula to a general clas…
In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of Fractional Calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann-Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikus…
On the existence and uniqueness of solution for fractional differential equations with nonlocal multi-point boundary conditions
给出了一类分数阶微分方程在Caputo分数阶导数和积分边界条件下的非局部多点边值问题解存在的充分条件.我们的分析依赖于Banach压缩原理，Boyd和Wong不动点定理，Leray-Schauder非线性替代。最后，给出了一些例子来说明我们的主要结果.
It is shown that the attractor of an autonomous Caputo fractional differential equation of order $α\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of "N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dyn…
Fractional Differential Equations with Periodic Boundary Conditions of Constant Ratio
本文讨论了一类分数阶边值问题解的存在唯一性。我们的结果是基于一些不动点定理。对于我们的结果的适用性，我们提供了一个例子。
Existence of smooth solutions of multi-term Caputo-type fractional differential equations
研究了多项分数阶微分方程的初值问题。分数阶导数是在Caputo意义下定义的。首先在适当的假设下将初值问题转化为等价的Volterra型积分方程。然后利用Schauder不动点定理建立了光滑解的新的存在性结果.
Global attractivity for some classes of Riemann--Liouville fractional differential systems
本文给出了一类含Riemann-Liouville导数的多维分数阶微分方程整体解的存在性和吸引性的一些结果。首先，利用Bielecki型范数和Banach不动点定理，证明了关于解的存在唯一性的Picard-Lindelöf型定理.然后，利用Mittag-Leffler函数的性质，我们描述了几类Riemann-Liouville线性分数阶微分系统解的吸引性。
Stability Properties of Multi-Order Fractional Differential Systems in 3D
本文研究具有Caputo导数的三维非公度分数阶微分方程系统，获得了其渐近稳定性的充分必要条件。
Fractional Differential Equations Involving Caputo Fractional Derivative with Mittag-Leffler Non-Singular Kernel: Comparison Principles and Applications
本文研究了具有$0<α<1.$阶Mittag-Leffler非奇异核的含有Caputo分数阶导数的线性和非线性分数阶微分方程.首先得到了函数在极值点处的分数阶导数的一个新的估计，并得到了线性分数阶微分方程解存在的一个必要条件.所得到的充分条件确定了相应的分数阶初值问题的初始条件。然后，我们推导出比较原则的线性分式方程。应用这些原理得到了线性方程解的范数估计和非线性方程解的唯一性结果。我们还得到了非线性方程的解的上下界上界。通过几个例子说明了新结果的适用性。
Non-differentiable solutions for local fractional nonlinear Riccati differential equations
通过将局部分数阶非线性里卡蒂微分方程（LFNRDE）转化为局部分数阶线性常微分方程，研究了它们的解.考虑了常系数LFNRDE的情形，得到了特殊情形的不可微解。
Existence and Uniqueness results for a class of Generalized Fractional Differential Equations
作者（公牛.数学.肛门. App.6（4）（2014）：1-15），引入了一种新的分数导数，\[{}^ρ\mathcal{D}_a^αf (x) = \frac{ρ^{α-n+1}}{Γ({n-α})} \, \bigg(x^{1-ρ} \,\frac{d}{dx}\bigg)^n \int^x_a \frac{τ^{ρ-1} f(τ)}{(x^ρ- τ^ρ)^{α-n+1}}\, dτ\]，其将两种熟悉的分数导数，即Riemann-Liouville和Hadamard分数导数推广为单一形式。本文给出了一个由分数阶导数控制的广义分数阶微分方程解的存在唯一性结果。
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new …
In this paper, we focus on the existence and uniqueness of solutions of boundary value problems for a coupled system of fractional differential equations with four-point boundary conditions involving $ψ$-Caputo fractional derivatives. Our investigation is based on Leray-Schauder alternative and Banach's fixed point theorem. Two examples are presented to illustrate the applicability of the results developed.
Viability for stochastic differential equations driven by fractional Brownian motion
本文利用路径方法证明了Hurst参数为1/2 < H < 1的分数阶布朗运动驱动的多维不定常随机微分方程解的一个生存性结果.该充分条件也是带状态约束的分数阶微分方程解的另一个整体存在性结果。
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