Forced oscillation of conformable fractional partial delay differential equations with impulses

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems

Ulam Stabilities for Partial Impulsive Fractional Differential Equations

summary:In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem

Impulsive Partial Hyperbolic Functional Differential Equations of Fractional Order with State-Dependent Delay

MSC 2010: 26A33, 34A37, 34K37, 34K40, 35R11This paper deals with the existence and uniqueness of solutions of two classes of partial impulsive hyperbolic differential equations with fixed time impulses and state-dependent delay involving the Caputo fractional derivative. Our results are obtained upon suitable fixed point theorems

Controllability Problem of Fractional Neutral Systems: A Survey

The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces, controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, controllability for a class of fractional neutral integrodifferential equations with unbounded delay, controllability of neutral fractional functional equations with impulses and infinite delay, and controllability for a class of fractional order neutral evolution control systems

Direct methods for singular integral equations and non-homogeneous parabolic PDEs

In this article, the author presented some applications of the Laplace, $L^2$, and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient

On Asymptotic Stability of Stochastic Differential Equations with Delay in Infinite Dimensional Spaces

In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion