Analysis of Impulsive $φ$--Hilfer Fractional Differential Equations

On the Nonlinear $Ψ$-Hilfer Fractional Differential Equations

We consider the nonlinear Cauchy problem for $ Ψ$- Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $ Ψ$- Hilfer fractional derivative and an estimation have been obtained for the error bound. Further, by Picard's successive approximation, we derive the repre…

Stability of Cobweb economic model involving Hilfer fractional derivative

This paper evaluates the solution of cobweb models when there is a Hilfer fractional derivative in the demand and supply function. Particular cases when Hilfer derivative reducing to Caputo and Riemann-Liouville derivative are discussed. Subsequently, the solution of the cobweb model with Riemann-Liouville derivative is derived instantly. Two numerical examples interpreting the crafted theory is deliberated with illustrations and comparisons. Conclusions based on the graphical illustrations is outlined in detail. These illustrations highlight the advantage of the Hilfer fractional derivative o…

Global Existence and Ulam--Hyers Stability of $Ψ$--Hilfer Fractional Differential Equations

In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving $Ψ$-Hilfer fractional derivative and prove the existence and uniqueness of solutions in the weighted space of functions. The Ulam--Hyers and Ulam--Hyers--Rassias stability of Cauchy--type problem is investigated via successive approximation method. Further, we investigate the dependence of solutions on the initial conditions and uniqueness via $ε$-approximated solution. An example is provided to illustrate the results we obtained.

Properties of relaxed trajectories of non-linear fractional impulsive control systems

A non-convex control system governed by a nonlinear impulsive evolution equation of Hilfer fractional order in a Banach space is considered. The existence of admissible state-control pair is established. Then the introduction of suitable measure-valued control convexifies the system, and the relaxed system is obtained. Further, the relaxation theorem for the described class is proved along with the existence of optimal relaxed control.

Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system $D_{0,t}^{μ,ν}z\left( t\right) +Az\left( t\right) +Ωz\left( t-h\right) =f\left( t\right) $ of order $1<μ<2$ and type $0\leqν\leq1,$ with nonpermutable matrices $A$ and $Ω$. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear tim…

非线性 conformable 分数阶时滞脉冲系统的有限时间稳定性：脉冲控制与扰动视角

Finite-time stability of nonlinear conformable fractional-order delayed impulsive systems: Impulsive control and perturbation perspectives

本文基于 conformable 分数阶导数框架，研究了非线性时滞脉冲系统的有限时间稳定性，建立了考虑脉冲控制与扰动的 Lyapunov 判据并估计了稳定时间，最后应用于忆阻神经网络验证有效性。

Stochastic impulsive fractional differential evolution equations with infinite delay

In this paper, we investigate a class of stochastic impulsive fractional differential evolution equations with infinite delay in Banach space. Firstly sufficient conditions of the existence and uniqueness of the mild solution for this type of equations are derived by means of the successive approximation. Then we use the Bihari's inequality to get the stability in mean square of the mild solution. Finally an example is presented to illustrate the results.

线性分数脉冲演化方程在Hilbert空间中的近似可控性

Approximate Controllability of Linear Fractional Impulsive Evolution Equations in Hilbert Spaces

本文研究了Hilbert空间中线性分数脉冲演化方程的近似可控性，通过分数解算子与脉冲算子结合推导mild解，并基于残差算子的收敛性条件提出近似可控性判据。

New notion of mild solutions for nonlinear differential systems involving Riemann-Liouville derivatives of higher order with non-instantaneous impulses

The artefact is dedicated towards the inspection of nonlinear fractional differential systems involving Riemann-Liouville derivative with higher order and fixed lower limit, including non-instantaneous impulses for existence and uniqueness results in Banach spaces. The motive of the paper is to set sufficient conditions to guarantee the existence of mild solution in Banach spaces. Firstly, appropriate integral type initial conditions depending on the impulsive functions are chosen at suitable points. A mild solution of the concerned system is constructed using fractional resolvent. Subsequentl…