On complete monotonicity of solution to the fractional relaxation equation with the $n$th level fractional derivative

Riemann-Liouville分数阶微分方程解的单调性与渐近行为

Monotonicity and asymptotic behavior of solutions for Riemann-Liouville fractional differential equation

本文利用不动点定理和分数阶积分函数的新性质，证明了Riemann-Liouville分数阶微分方程存在递减解，并通过函数单调性研究了其渐近行为，最后用实例验证了结果。

Monotonicity Results for Delta and Nabla Caputo and Riemann Fractional Differences via dual identities

Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the $Q-$operator dual identities to prove monotonicity results for the right fractional difference operators.

A monotonicity result for the $q-$fractional operator

In this article we prove that if the $q-$fractional operator $(~_{q}\nabla_{qa}^αy)(t)$ of order $0<α\leq 1$ , $0<q<1$ and starting at some $qa \in T_q=\{q^k: k \in \mathbb{Z}\}\cup \{0\},~~a>0$ is positive such that $y(a) \geq 0$, then $y(t)$ is $c_q(α)-$increasing, $c_q(α)=\frac{1-q^α}{1-q}q^{1-α}$. Conversely, if y(t) is increasing and $y(a)\geq 0$, then $(~_{q}\nabla_{qa}^αy)(t)\geq 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem.

A note on one-dimensional time fractional ODEs

In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work \cite{fllx17}. Here we use the definition of Caputo derivative proposed in \cite{liliu17frac1,liliu2017} based on a convolution group. In particular, we establish generalized comparison principles consistent with the new definition of Caputo derivatives. In addition, we establish the full asymptotic behaviors of the solutions for $D_c^γu=Au^p$. Lastly, we provide a simplified proof for the strict monotonicity and stability in initial values for the time fracti…

On a Fractional Oscillator Equation with Natural Boundary Conditions

We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with such approach, we first prove a result on the monotonicity of the right Caputo derivative.

Fractional extreme distributions

We consider three classes of linear differential equations on distribution functions, with a fractional order $α\in [0,1].$ The integer case $α=1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $α-$stable subordinator. From the analytical viewpoint, this law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Ro…