Communications in Nonlinear Science and Numerical Simulation
Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations☆
Introduction
During the past decades, fractional differential equations have been paid more and more attentions due to their significant applications in various sciences, such as physics, mechanics, chemistry, engineering, etc. For more details on fractional calculus theory, one can see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the reference therein.
As an appropriate model for describing the progress in physics, biological, engineering, etc., impulsive fractional differential equations have much greater applications. The advantage of them is that they can describe the model which at certain moments change their state rapidly and which cannot be modeled by the classical differential problems. At present, more and more authors have paid great attention to the existence of solutions for impulsive fractional differential equations, for example, one can see [4], [9], [19], [20].
Recently Fečkan et al., proposed a good way to define the solutions for fractional impulsive differential equations (for more details, see [9]). They considered the Cauchy problems for the following impulsive fractional differential equations:where denotes the Caputo fractional derivative of order with the lower limit zero, is jointly continuous, .
In [4], Ahmad studied the existence of solutions for impulsive fractional differential equations with integral boundary conditions:where denotes the Caputo fractional derivative of order with the lower limit zero, is a given continuous function, the functions are continuous.
In [19], Wang considered the following anti-periodic boundary value problem for impulsive fractional differential equations:where denotes the Caputo fractional derivative of order with the lower limit zero, is a given continuous function, the functions are continuous.
Motivated by the above works, we consider the following impulsive fractional differential equations with integral boundary conditions:where denotes the Caputo fractional derivative of order with the lower limit zero, is a given continuous function, the functions are continuous, and , and denote the right and the left limits of at , respectively, and have the similar meaning for and .
It is easy to know that if and , then problem (1.1) reduces to an anti-periodic boundary problem. If and , then it reduces to an integral boundary problem.
The rest of this paper is organized as follows: In Section 2, we present some preliminaries. In Section 3, by applying some standard fixed point principles, we verify the existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. In Section 4, an example is given to illustrate our main results.
Section snippets
Preliminaries
Let us introduce and exist, with the norm , and and exist, with the norm . Obviously, and are Banach spaces.
For the convenience of the readers, we first present some useful definitions and fundamental facts of fractional calculus theory, which can be found in [12], [16]. Definition 2.1 The integral
Main results
In this section, we mainly investigate the existence and uniqueness of impulsive fractional differential equations by some standard fixed point theorems. Before stating and verifying the uniqueness and existence of solutions, we make the following hypotheses:
is continuous and there exists a constant such that.
: The functions are continuous and there exist some constants ,
Example
Consider the following nonlinear impulsive fractional differential equation:where .
Note that
References (21)
- et al.
Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with antiperiodic boundary conditions
Nonlinear Anal
(2008) - et al.
Existence of solutions for impulsive integral boundary value problems of fractional order
Nonlinear Anal Hybrid Syst
(2010) - et al.
Boundary value problems for differential equations with fractional order and nonlocal conditions
Nonlinear Anal
(2009) - et al.
Positive solutions of boundary value problems of nonlinear fractional differential equation
J Math Anal Appl
(2005) - et al.
Antiperiodic solutions for fully nonlinear first-order differential equations
Math Comput Model
(2007) - et al.
On initial and boundary value problems for fractional order mixed type functional differential inclusions
Comput Math Appl
(2010) - et al.
On the concept and existence of solution for impulsive fractional differential equations
Commun Nonlinear Sci Numer Simul
(2012) - et al.
Nonlinear boundary value problems of fractional differential systems
Comput Math Appl
(2012) Anti-periodic solutions for shuting inhibitory cellular neural networks with time-varying delays
Phys Lett A
(2008)- et al.
Existence and uniqueness for fractional neutral differential with infinite delay
Nonlinear Anal
(2009)
Cited by (63)
Impulsive effects on stability and passivity analysis of memristor-based fractional-order competitive neural networks
2020, NeurocomputingCitation Excerpt :The impulsive effects on the stability of the system should be considered, which often widely used in many fields of science and engineering [28,29]. Many results on the stability of impulse NNs have recently been proposed and studied [30–33]. On the other hand, in the analysis of circuits, the concept of passivity theory was first proposed.
A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators
2024, Fractional Calculus and Applied AnalysisEvent-based delayed impulsive control for fractional-order dynamic systems with application to synchronization of fractional-order neural networks
2023, Neural Computing and ApplicationsA class of piecewise fractional functional differential equations with impulsive
2023, International Journal of Nonlinear Sciences and Numerical SimulationOptimal control and feedback control for nonlinear impulsive evolutionary equations
2023, Quaestiones MathematicaeOptimal feedback control for a class of fractional evolution equations with history-dependent operators
2022, Fractional Calculus and Applied Analysis