Impulsive Fractional Evolution Equations Research Articles

The e-positive mild solutions for fractional impulsive evolution equations with 1<α<2

The e-positive mild solutions for fractional impulsive evolution equations with 1<α<2

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

In this paper, we investigate the controllability of a class of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive ψ-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results.

A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results.

Controllability of Impulsive Fractional Integro-Differential Evolution Equations

Controllability of Impulsive Fractional Integro-Differential Evolution Equations

A study on controllability of impulsive fractional evolution equations via resolvent operators

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the (alpha ,beta )-resolvent operator, we concern with the term u'(cdot ) and finding a control v such that the mild solution satisfies u(b)=u_{b} and u'(b)=u'_{b}. Finally, we present an application to support the validity study.

A New Study on the Mild Solution for Impulsive Fractional Evolution Equations

In this article, we consider mild solutions to a class of impulsive fractional evolution equations of order $0<\alpha<1$. After analyzing analytic results reported in the literature using Mittag-Leffer function, $\alpha$-resolvent operator theory, we propose a more appropriate new definition of mild solutions for impulsive fractional evolution equations by replacing the impulse term operator $S_\alpha(t-t_i)$ with $S_\alpha(t)S_\alpha^{-1}(t_i)$, where $S_\alpha^{-1}(t_i)$ denotes the inverse of the fractional solution operator $S_\alpha(t)$ at $t=t_i, (i=1,2,\cdots m)$.

Well‐posedness and Hyers‐Ulam results for a class of impulsive fractional evolution equations

In this work, we establish a new class of nonlinear implicit fractional evolution equation with integrable impulses. We investigate the qualitative properties of ‐mild solution of the proposed problem. The results are obtained using the theory of probability density functions, operators semigroup, and fixed‐point criteria. The main theoretical results are well demonstrated with the help of an example.

A study on impulsive fractional hybrid evolution equations using sequence method

In this paper, we introduce a new concept called $$\alpha $$ -order cosine-resolvent family, by using the theory of fractional calculus, the concepts of measure of noncompactness and Hybrid fixed point theorem, we consider the existence of PC-mild solutions for a class of impulsive fractional hybrid evolution equation in a Banach space . Furthermore, we obtain some sufficient conditions for approximate controllability of our concern problem. At the end, an example is given to illustrate the feasibility of our results.

Global attractiveness and exponential stability for impulsive fractional neutral stochastic evolution equations driven by fBm

This paper is concerned with a class of fractional neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion (fBm). First, by means of the resolvent operator technique and contraction mapping principle, we can directly show the existence and uniqueness result of mild solution for the aforementioned system. Then we develop a new impulsive-integral inequality to obtain the global attracting set and pth moment exponential stability for this type of equation. Worthy of note is that this powerful inequality after little modification is applicable to the case with delayed impulses. Moreover, sufficient conditions which guarantee the pth moment exponential stability for some pertinent systems are stated without proof. In the end, an example is worked out to illustrate the theoretical results.

Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.

The method of lower and upper solutions for impulsive fractional evolution equations

In this paper, we investigate the existence of the extremal solutions for a class of impulsive fractional integro-differential equation with order $$1< \alpha < 2$$, combine upper and lower solution method with the theory of $$(\alpha ,\beta )$$-resolvent family, a series of results about the solutions to such differential equations is obtained. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Finally, an example is given to illustrate the feasibility of our results.

Periodic mild solutions of impulsive fractional evolution equations

This paper studies the periodic mild solutions of impulsive fractional evolution equations. Firstly, the existence and stability of periodic solutions of impulsive fractional differential equations with varying lower limits for general impulses and small shifted impulses are considered. Secondly, the existence of periodic solutions of impulsive fractional differential equations with fixed lower limits is proved. Lastly, an example is given to demonstrate the result.

Controllability of impulsive fractional functional evolution equations with infinite state‐dependent delay in Banach spaces

Many evolutionary processes from various fields of physical and engineering sciences undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. These processes are more suitably modeled by impulsive differential equations. In this paper, we study the controllability of an impulsive fractional differential equation with infinite state‐dependent delay in an arbitrary Banach space. We apply semigroup theory and Schaefer fixed point theorem. As an application, we include an example to illustrate the theory.

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.

Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions

In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main 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.

Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions

Abstract In this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the fixed point theorems and fractional cosine family combined with solutions operator theorems. Our results improve and generalize some classical results. Finally, an example is given to illustrate the main results.

Existence Results of Mild Solutions for Impulsive Fractional Evolution Equations with Periodic Boundary Condition

Abstract This paper discusses the existence of mild solutions for a class of fractional impulsive evolution equation with periodic boundary condition and noncompact semigroup. By using some fixed-point theorems, the existence theorems of mild solutions are obtained, our results are also more general than known results. Furthermore, as an application that illustrates the abstract results, two examples are given.

Existence of solutions for impulsive fractional evolution equations with periodic boundary condition

This paper discusses the existence of mild solutions for a class of fractional impulsive evolution equation with periodic boundary condition and noncompact semigroup. By using some fixed point theorems, the existence theorems of mild solutions are obtained, our results are also more general than some well-known results. Furthermore, as applications that illustrate the abstract results, two examples are given.

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.