On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

A time fractional functional differential equation driven by the fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $$ \begin{cases} ^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0], \end{cases} $$ where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition.     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Operators associated with a stochastic differential equation driven by fractional Brownian motions

In this paper, by using a Taylor type development, we show how it is possible to associate differential operators with stochastic differential equations driven by fractional Brownian motions. As an application, we deduce that invariant measures for such SDE’s must satisfy an infinite dimensional system of partial differential equations.     

Existence results of positive solutions to boundary value problem for fractional differential equation

In this paper, we consider the existence, multiplicity, and nonexistence of positive solutions to some class of boundary vale problem for fractional differential equation of high order. Our analysis relies on the fixed point index.      

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.     

Existence and uniqueness of solutions for a fractional differential equation with multi-point boundary value problems

In this paper, we study the existence and uniqueness solutions of a fractional differential equation with multi-point boundary value problems. By using the fixed point theorems, some new results are established and two examples are given to demonstrate the application of main results.     

Positive solutions for boundary value problem of nonlinear fractional differential equation

In this paper, we investigate the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem:

     

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.     

Positive solutions of fractional differential equation boundary value problems at resonance

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.     

On the concept of solution for fractional differential equations with uncertainty

We consider a differential equation of fractional order with uncertainty and present the concept of solution. It extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty. Some examples are presented.     

Existence of solutions for fractional differential equation with p-Laplacian through variational method

In this paper, a class of fractional differential equation with p-Laplacian operator is studied based on the variational approach. Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for our equation is obtained. Additionally, we demonstrate the application of our main result through an example.     

Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H>0.5$H>0.5$. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition.     

Oscillatory behavior of a fractional partial differential equation

In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.     

Three positive solutions for boundary value problem for differential equation with Riemann-Liouville fractional derivative

In this paper, by using the Avery-Peterson fixed point theorem, we establish the existence result of at least three positive solutions of boundary value problem of nonlinear differential equation with Riemann-Liouville''s fractional order derivative. An example illustrating our main result is given. Our results complements and extends previous work in the area of boundary value problems of nonlinear fractional differential equations.     

Solvability for impulsive fractional Langevin equation

We investigate impulsive fractional Langevin equation involving two fractional Caputo derivatives with boundary value conditions. By Banach contraction mapping principle and Krasnoselskii"s fixed point theorem, some results on the existence and uniqueness of solution are obtained.