Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
Abstract
:1. Introduction
2. Preliminaries
- X: Banach Space;
- : piece-wise continuous function from ;
- : infinitesimal generator of a strongly continuous semigroup ;
- : fractional integral of order for a function f;
- : Riemann-Liouville (R-L) fractional derivative of order ;
- : Caputo fractional derivative of order ;
- : strong Caputo derivative of order .
- 1.
- R-L and Caputo are just two different operators that are related to each other in a quite simple way. Quite a few details about this are given in the book by Diethelm [2]. There we can also see the exact description of when they are equivalent. The most important difference between them is, of course, the structure of their kernels (i.e., the set of functions that is mapped to zero). Depending on what we want from our operator, one of them or the other one may be the right choice for us. We are using here Caputo because the derivative of a constant is zero for Caputo but not for R-L.
- 2.
- If a derivative of Caputo type is used instead of R-L type then initial conditions for the corresponding Caputo fractional differential equations can be formulated as for classical ordinary equations, namely
- 3.
- One has to make sure to use a constant function and the Heaviside unit step. They must be considered different so they must have different fractional derivatives. The Heaviside unit step is expected to have a non zero fractional derivative. If one given derivative gives zero it is useless due to the importance it enjoys in practice and its relation with the Dirac delta.
- 1.
- If then
- 2.
- The Caputo derivative of a constant is equal to zero.
- 3.
- If f is an abstract function with values in X, then integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense.
- 1.
- In this paper, we emphasize that we use the generalized Caputo derivative with the lower bound at zero for Equation (1). However, we have not chosen the classical Caputo derivative and have not changed it in each sub-interval for the Equation (1), where the impulses start at the lower bound Obviously, we mean keeping a different one, in each of the impulses the lower bound is at zero. Moreover, Definition 5 is more reasonable since the generalized Caputo derivative in Equation (1) should be fixed at the lower bound at zero once we set the initial time at zero. So we do not expect to change the lower bound again and again in the definition of Caputo derivative for the same equation.
- 2.
- We use Definition 4 (generalized Caputo derivative), where the integrable function f can be discontinuous. Definition 4 is more general with respect to Remark 2 (1) (relationship between strong and weak Caputo derivatives). So the result would be wrong if we have used a strong Caputo derivative.
- 3.
- Finally, we would like to mention the recently published paper written by Liu and Ahmed [34], where the formula of solutions for semi-linear impulsive fractional Cauchy problems (see (20) in Ref. [34]) coincided with ours (see Definition 5), if one imposes that the semi-linear term and the impulsive term have the same expression in the given interval.
- 1.
- , (here I is the identity operator on X);
- 2.
- for every (the semi-group property).
- 1.
- For any fixed and linear and bounded operator, i.e., for any ,
- 2.
- and are strongly continuous.
- 3.
- and are uniformly continuous, that is, for each fixed , and , there exists such that
3. Existence Results
- (H1)
- is continuous, and there exist functions such that
- (H2)
- is continuous and there exist such that
- (H3)
- is continuous and there exists a constant such that
- (H4)
- The function are continuous and there exists such that
- (H5)
- The function is defined by
- (H6)
- The constants and are defined by
- (H7)
- is continuous and there exist functions , such that
- (H8)
- is continuous and there exist functions , such that
- (H9)
- There exist , such that
- (H10)
- For all bounded subsets , the set
- (H11)
- For all bounded subsets , the set
4. Example
- Consider the following fractional partial functional mixed differential equations with impulsive conditions of the form
- (i)
- Let as the state space and as the state;
- (ii)
- is defined as with domain
- 2.
- Consider the following numerical fractional partial functional differential equations with impulsive conditions of the form
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Chalishajar, D.; Ravichandran, C.; Dhanalakshmi, S.; Murugesu, R. Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces. Appl. Syst. Innov. 2019, 2, 18. https://doi.org/10.3390/asi2020018
Chalishajar D, Ravichandran C, Dhanalakshmi S, Murugesu R. Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces. Applied System Innovation. 2019; 2(2):18. https://doi.org/10.3390/asi2020018
Chicago/Turabian StyleChalishajar, Dimplekumar, Chokkalingam Ravichandran, Shanmugam Dhanalakshmi, and Rangasamy Murugesu. 2019. "Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces" Applied System Innovation 2, no. 2: 18. https://doi.org/10.3390/asi2020018