两类分数阶微分方程的边值问题

本文研究了两类非线性项含有未知函数导数的分数阶微分方程的边值问题.利用分数阶微积分的性质及Banach不动点定理,获得了解的存在唯一性等有关结果,推广了已有文献的结论.     

非线性随机分数阶微分方程Euler方法的弱收敛性

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方...     

一类非线性分数阶泛函微分方程的正解的存在性

本文研究了一类非线性分数阶泛函微分方程.利用锥拉伸与锥压缩不动点定理,得到这类分数阶泛函微分方程的正解的存在性,推广了马如云的结果.     

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.     

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

Existence of Solutions for Fractional Integro-differential Equations with Impulsive and Integral Conditions

This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differ- ential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.     

On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations

In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results.     

Mixed monotone iterative technique for semilinear impulsive fractional evolution equations

In this paper, we deals with the existence of mild $L$-quasi-solutions to the boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space $E$. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive fractional evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. As some application that illustrate our results, An example is also given.     

Boundary problems for fractional differential equations

In this paper, the existence of solutions of fractional differential equations with nonlinear boundary conditions is investigated. The monotone iterative method combined with lower and upper solutions is applied. Fractional differential inequalities are also discussed. Two examples are added to illustrate the results.     

分数阶中立型微分方程的有限时间稳定(英文)

本文研究了一类分数阶中立型微分方程的有限时间稳定性问题.利用分步法及微分不等式理论,获得了该方程解的存在唯一性及有限时间稳定性结果.推广了文献[8,9]的相关结果.     

一类广义B-T方程边值问题解的存在性

研究一类含有两个分数阶导数的非线性分数阶微分方程在边值问题以及在边值非零情况下mild解的存在性,进而利用函数的初始值非零时Riemann-Liouville分数阶导数的奇异性,在加权连续函数空间上,运用Schauder不动点定理,得到关于这类问题的mild解存在的充分条件,完善和推广了某些已有结论.     

Note on periodic solutions of fractional differential equations

The existence and nonexistence of periodic solutions are discussed for fractional differential equations by varying the lower limits of Caputo derivatives. The developed approach is illustrated on several examples.     

EXISTENCE OF MILD SOLUTIONS TO SEMILINEAR FRACTIONAL ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper,we use the analytic semigroup theory of linear operators and fixed point method to prove the existence of mild solutions to a semilinear fractional order functional differential equations in a Banach space.     

Mild solutions for abstract fractional differential equations

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a,?k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.     

Fixed‐point theorem for a generalized almost Hardy‐Rogers‐type F contraction on metric‐like spaces

In this paper, we introduce some fixed‐point theorems for a generalized almost Hardy‐Rogers‐type F contraction in a metric‐like space and give an example to illustrate these main results. Moreover, we show the applications of electric circuit equations, second‐order differential equations, and fractional differential equations. Our results improve, generalize, and extend the corresponding results in literature.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

奇异耦合分数阶微分方程组Dirichlet型边值问题的正解

研究了一非线性奇异非自治耦合半正分数阶微分方程组Dirichlet型边值问题.利用Schauder不动点定理,获得了该非自治耦合分数阶微分方程组Dirichlet型边值问题的正解.     

Positive Solutions of Fractional Differential Inclusions at Resonance

This paper deals with the fractional differential inclusions at resonance. By the recent Leggett-Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima in [19], we present a new result on the existence of positive solutions for a class of differential inclusion of fractional order with boundary conditions at resonance. And our results improve and generalize the existing results.     

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

Abstract

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.     

On the fractional differential equations with uncertainty

This paper is based on the concept of fuzzy differential equations of fractional order introduced by Agarwal et al. [R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862]. Using this concept, we prove some results on the existence and uniqueness of solutions of fuzzy fractional differential equations.