Abstract
Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.
1. Introduction
We consider the following impulsive fractional differential equations involving Hilfer fractional derivative and almost sectorial operatorswhere Hilfer fractional derivative of order and type and is an almost sectorial operator in having norm and denotes the jump of at , i.e., , where and . and . is a function which is defined later.
In fact, rapid changes in the dynamics of evolutionary processes might occur due to shocks, harvesting, or natural disasters, among other things. These short-term disturbances are frequently handled as impulses. Hilfer derivative fractional differential equations have recently gained a lot of attention in the literature. [1–12] appears to be the case.
In [3], Anjali Jaiswal and Bahuguna studied the equations of Hilfer fractional deritaive with almost sectorial operator in the abstract sense.
We also refer to Hamdy M. Ahmed et.al studied in [13], which looked at the existence of nonlinear Hilfer fractional derivative differential equations with control. Sufficient circumstances have been found where the Hilfer derivative is the time fractional derivative. In [14], Yong Zhoy et.al studied the factional cauchy problems with almost sectorial operators of the formwhere is Riemann–Liouville derivative of order §, is Riemann–Liouville integral of order , . is an almost sectorial operator on a complex Banach space, and is a given function.
The paper is structured as follows: we have presented some information in section 2 about Hilfer derivative, almost sectorial operators, measure of non-compactness, mild solutions of equations along with some basic definitions, results and lemmas. We discuss the main results for mild solutions for the equations in section 3. In section 4 5, we have discussed the two cases if associated semigroup is compact and noncompact respectively. Finally, an example is discussed to illustrate the main result.
The following sections describes the supporting results of the given problem and also generalizes the results in [14–16].
2. Preliminaries
Definition 1. (see [17]). For , the fractional integral of order § of a function is defined by
Definition 2. (see [17]). For , the Riemann-Liouville (R-L) fractional derivative with order § of a function is defined by
Definition 3. (see [17]). For , the Caputo fractional derivative with order § of a function is defined by
Definition 4. (see [11]). Let and . The Hilfer fractional derivative derivative of order § and type is defined by
2.1. Measure of Non-compactness
Let also bounded. The Hausdorff measure of non-compactness is considered as follows
The Kurtawoski measure of noncompactness on a bounded set is considered as followswith the following properties1. gives where are bounded subsets of 2. iff is relatively compact in 3. for all 4.5.6. for .
Let and . We define
2.2. Almost Sectorial Operators
Let and . We define and its closure by , that is .
Definition 5. (see [18]). For , we define as a family of all closed and linear operators this implies1. is contained in .2.For all , there exists implieswhere is the resolvent operator and is said to be an almost sectorial operator on .
We assume the following Wright-type function [17]For ,(A1);(A2);(A3).We have define , by
Theorem 1 (Theorem 4.6.1 in [17]). For each fixed and are bounded and linear operators on . Alsowhere and are constants.
We introduce the following hypotheses to obtain our main results.(H1)For is continuous function and for each is strongly measurable.(H2) a function satisfying(H3) for and , where .(H4) constants such that for each .
Lemma 1. The fractional Cauchy problem is equivalent to an integral equation given by
Lemma 2. Let £ is a solution of the integral equation given in (2.3), then £ satisfieswhere .
Definition 6. By a mild solution of the Cauchy problem we mean a function that satisfiesNow, we define an operator as
Lemma 3 (see [3]). and are bounded linear operators on , for every fixed . Also for .
Proposition 1 (see [3]). and are strongly continuous, for .
3. Main results
Theorem 2. Let for and . Assuming are satisfied, the operators is equicontinuous provided with .
Proof. For and , we gave, as .
Now, let ,here, using the triangle inequality, we haveSince the strong continuous of , we get as . AlsoThen as , by using and the dominated convergence theorem. Sinceand exists, i.e, as .
For , we haveSince is uniformly continuous and , then as i.e. independent of .
Clearly, since the strongly continuous of of , we getHence independently of as therefore is equicontinuous.
Theorem 3. Let and and . Then under consideration the operator is continuous and bounded provided with .
Proof. We verify that maps . Taking and define , we have . Let ,From , we getHence , for any .
Now, to verify is continuous in , let , with . That is and and , on .
impliesas .
From to obtain the inequality ,
i.e.,Let . Now,Applying Theorem 1, we have as , i.e, pointwise on . Also Theorem 2 implies that uniformly on as . That is is continuous.
4. is Compact
Theorem 4. Let and . If is compact and hold then a mild solution of in for every with .
Proof. Since we have assumed is compact, then the equicontinuity of . Moreover, by Theorem 2 and 3, is continuous and bounded and is bounded, continuous and equicontinuous. We can write bywhereFor and , we define an operator on bySince is compact is precompact in and . Moreover, for any where, .
Therefore, are arbitrarily close to , for . Hence , for , is precompact in .
For , , we can present an operator on by Hence is precompact in and due to the compactness of . For every , we getTherefore, are arbitrarily close to . That is, is relatively compact by using the Arzela-Ascoli Theorem. Then is a mild solution of (1.1)-(1.3).
5. is Noncompact
We consider as follows, a constant satisfies the followingfor every bounded subset .
Theorem 5. Let , and . Suppose hold. Then, has a mild solution in for every with .
Proof. By Theorem 2 and 3, we get is continuous, bounded, and is equicontinuous. Also, we prove that there is a subset of such that is campact in it.
For any bounded set , setFor any , we can obtain from Propositions , a subsequence satisfyingFrom is arbitrary,Again for any , we can get from Propositions a subsequence implies thatNow,Let . We can find big enough such that and for . That isReplace by . Then R.H.S of inequality given above becomesTherefore, a constant that isNowFrom is bounded and equicontinuous, applying Proposition (2.2), we getHence,where . By applying the Schauder fixed point theorem, we obtain a fixed point in of . Let . Then, is a mild solution of .
6. Illustrate an application for main results
We consider the following impulsive Hilfer derivative fractional system:in Banach space of all Holder continuous functions, where . Here, we can convert the above problem in abstract form as
Here with . It follows from the work in [18] constants , implies . To verify the compactness of semigroup , it is enough to prove that is compact for every . By and is compactly embedded in , the compactness of the resolvent operator for every . We take .
Then, hypotheses are satisfied. According to Theorem 4, the problem (6.1) has mild solution in .
7. Conclusion
The main objective of in this paper, we discussed the mild solutions for Hilfer fractional derivative differential equations involving jump conditions and almost sectorial operator when associated semigroup is compact and noncompact using Schauder fixed point theorem. Our theorems guarantee the effectiveness of existence results It is the results of the equations concerned. we discussed an example to verify the existence results. We will find to to investigate stability of similar problem in our future research work.
Data Availability
There is no data used for this manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors have equal contribution and have finalized the manuscript.
Acknowledgments
This research was funded by National Science, Research and Innovation Fund (NSRF), and Suan Dusit University with Contract no. 65-FF-010.