On the Nonlinear Impulsive $\Psi$--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $\Psi$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $\Psi$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.


Introduction
The fractional differential equations (FDEs) over the years have been the object of investigation by many researchers [1]- [6]. The fact is that certain natural phenomena by means of fractional differential equations are modeled and allows to better describe the real situation of the problem compared to the problem modeled by means of differential equations of whole order [7]- [11]. Recently, Sousa et al. [12] presented a fractional mathematical model by means of the time-fractional diffusion equation, which describes the concentration of nutrients in the blood and allows analyzing the solution of the model, better than the integer case. In addition, other mathematical models can be obtained in the literature involving fractional differential equations [13]- [16].
On the other hand, investigating the existence, uniqueness and stability of solutions of FDEs of the following types: functional, impulsive, evolution, with instantaneous and non-instantaneous impulses [17]- [24]. In this direction the subject has picked up strength and interest of the researchers, since the fractional derivatives allows the variation of the order of the differential equation that is straightforwardly associated with the solution of such FDEs.
The FDEs with impulsive effect play vital role in modeling real world physical phenomena involving in the study of population dynamics, biotechnology and chemical technology. Advancement in the theory of impulsive differential equations and its applications in the real world phenomena have been marvelously given in the monographs of Bainov and Simeonov [33], Benchohra et al [34] and Samoilenko and Perestyuk [35].
In [26] Feckan et al. with the help of the examples it is demonstrated that the formula for the solutions of fractional impulsive FDEs (1.1) considered in the few referred papers in [26] were incorrect. They have derived the valid formula for the solution of impulsive FDEs (1.1) involving Caputo derivative and investigated the existence results for (1.1) using Banach contraction principle and Leray-Schauder theorem.
In another interesting paper [5], Wang and coauthor presented the idea of piecewise continuous solutions for Caputo fractional impulsive Cauchy problems and impulsive fractioanl boundary value problem. They acquired existence and uniqueness of solution and furthermore determined data dependence and Ulam stabilities of solutions by means of generalized singular Gronwall inequalities.
It is noticed that numerous works with refined and important mathematical tools have been published and others that are yet to come [19,20,25,26,27,28]. In any case, it is advantageous to utilize more broad fractional derivatives in which they hold a wide class of fractional derivatives as particular cases, particularly the traditional ones of Caputo and Riemann-Liouville (RL). Another fundamental advantage is the fact that the properties of the general fractional derivative viz, Ψ-Hilfer is the preservation of the properties of the respective cases, in particular, in the investigated property of a fractional differential equation, in this case, the existence and uniqueness of solutions [21]- [23].
In the present paper, we consider the following impulsive Ψ-Hilfer fractional differential equation ( impulsive Ψ-HFDE ) with initial condition 3) is the Ψ-Hilfer fractional derivative of order µ and type ν, The main motivation for this work comes from the work highlighted above, with the purpose of investigating the existence and uniqueness of solution of impulsive Ψ-HFDEs and to provide new and more general results in the field of fractional differential equations.
We highlight here a rigorous analysis of Eq.(1.2)-Eq.(1.4) regarding the main results and advantages obtained in this paper: • With Ψ(t) = t and taking the limits β → 0 and β → 1 of the Eq.(1.2)-Eq.(1.4), we obtain the respective special cases for the differential equations, that is, the classical fractional derivatives of Riemann-Liouville and Caputo, respectively. In addition to the integer case, by choosing α= 1. These are two special cases of fractional derivatives. However, a wide class of fractional derivatives can be obtained from the choice of the parameters β and Ψ(t); • Since it is possible to obtain a wide class of derivatives from the choice of β and Ψ(t); consequently, it is also possible to obtain a class of fractional differential equations with their respective fractional derivatives, as particular cases; • A new class of solutions for impulsive Ψ-HFDEs; • We investigate the existence and uniqueness results for the impulsive Ψ-HFDEs and extend it to the non-local impulsive Ψ-HFDEs.
Organization of Paper: In section 2, some definitions and results that are important for the development of the paper have been provided via Lemmas and Theorems. In section 3, we present a representation formula for the solution, i.e., we show that the problem (1.2)-(1.4) is equivalent to the Volterra fractional integral equation. In section 4, we investigated the existence and uniqueness of the impulsive Ψ-HFDE. In Section 5, we will investigate the existence and uniqueness of a nonlocal impulse Ψ-HFDE. Concluding and remarks closing the paper.

Preliminaries
In this section, we introduce preliminary facts that are utilized all through this paper.
With suitable modification, the PC-type Arzela-Ascoli Theorem [33,41] can be extended to the weighted space PC 1−̺; Ψ (I, X ), where I is closed bounded interval.
. If the following conditions are satisfied: where W satisfy the conditions of Theorem 2.1 of [41]. Proceeding as in the proof of Theorem 2.1 of [41], there exist x ∈ P C(J , X ) such that x n → x in P C(J , X ) which in turn gives z n → z in PC 1−̺; Ψ (J , X ). This proves W 1−̺; Ψ is a relatively compact subset of PC 1−̺; Ψ (J , X ). ✷ Theorem 2.4 (Krasnoselskii, [6]) Let M be a closed, convex, and nonempty subset of a Banach space X , and A, B the operators such that

A is compact and continuous;
3. B is a contraction mapping.
Then there exists z ∈ M such that z = Az + Bz.

Representation formula for the solution
The following lemma play an important role in building an equivalent fractional integral equation of the impulsive Ψ-HFDE (1.2) -(1.4).
Then for any b ∈ J a function u ∈ C 1−̺,Ψ (J , R) defined by is the solution of the Ψ-Hilfer fractional differential equation Using the result ( [42], Page 10), and using the Theorem 2.2, we get This completes the proof of the Lemma. ✷ In the next result, utilizing the Lemma 3.1, we obtain the equivalent fractional integral of the problem (1.2)-(1.4).

5)
if and only if u is a solution of the following fractional integral equation Proof: Then the problem (3.7) is equivalent to the following fractional integral [42] By Lemma 3.1, we have Now, from (3.8), we have This gives Using (3.10) in (3.9), we obtain Again by Lemma 3.1, we have From (3.11), we have which gives Using (3.13) in (3.12), we get (3.14) Continuing the above process, we obtain Conversely, let u ∈ PC 1−̺; Ψ (J , R) satisfies the fractional integral equation (3.6). Then, for t ∈ [a, t 1 ], we have Applying the Ψ-Hilfer fractional derivative operator H D µ, ν; Ψ a + on both sides, we get Now, for t ∈ (t k , t k+1 ], (k = 1, 2, · · · , m), we have Applying the operator H D µ, ν; Ψ a + (·) on both sides and using (3.2) and the Theorem 2.2, we obtain We have proved that u satisfies (3.3). Next, we prove that u also satisfy the conditions (3.4) and (3.5).
Applying the Ψ-RL fractional operator I 1−̺; Ψ a + (·) on both sides of (3.8), we get and from which we obtain which is the condition (3.5).
Further, from equation (3.6), for t ∈ (t k , t k+1 ], we have and for t ∈ (t k−1 , t k ], we have Therefore, from (3.16) to (3.17), we obtain   We define the operators P and Q on B r by Then the fractional integral equation (4.1) can be written as operator equation Step 1: We prove that Pu + Qv ∈ B r for any u, v ∈ B r .
Let any u, v ∈ B r . Then using (A 1 ), for any t ∈ J , we have Further, from definition of the operator P and Q, one can verify that We have proved that, Pu + Qv ∈ B r .
Step 2 : Clearly P is a contraction with the contraction constant zero.
Step 3 : Q is compact and continuous.
The continuity of Q follows from the continuity of f . Next we prove that Q is uniformly bounded on B r .
Let any u ∈ B r . Then by (A 2 ), for any t ∈ J , we have Therefore This proves Q is uniformly bounded on B r . Next, we show that QB r is equicontinuous.
Let any u ∈ B r and t 1 , t 2 ∈ (t k , t k+1 ] for some k, (k = 0, 1, · · · , m) with t 1 < t 2 . Then, Note that This shows that Q is equicontinuous on (t k , t k+1 ]. Therefore Q is relatively compact on B r . By To prove u = T u has a fixed point, we show that T B r ⊂ B r . For that take any u ∈ B r . Then, by (A 2 ) for any t ∈ J , we have Thus, From the choices of constants r and L, it can be easily verified that This proves T B r ⊂ B r .
Now, we prove that the operator T is a contraction on B r . Let any u, v ∈ B r . Then by assumption (A 2 ) for any t ∈ J , From the choice of constant L, it follows that Thus, T is a contraction and by the Banach contraction principle it has a unique fixed point in B r ⊆ PC 1−̺; Ψ (J , R) which is the unique solution of impulsive Ψ-HFDE (1.2)-(1.4). ✷

Nonlocal Impulsive Ψ-HFDE
In this section we examine the existence and uniqueness results for impulsive Ψ-HFDE with non local initial conditions given by 2)  Define operator R and Q * on B r * by Then the fractional integral equation (5.4) is equivalent to the operator equation We apply the Krasnoselskii's fixed point theorem (Theorem 2.4) to prove that the operator equation (5.5) has fixed point. Firstly, we show that Ru+Q * v ∈ B r * for any u, v ∈ B r * . By assumption (A 2 ) and (A 3 ), for any u, v ∈ B r * and t ∈ J , From the choice of r * , L and L g , from the above inequality, we obtain Further, one can verify that This shows that Ru + Q * v ∈ B r * .
Next, we prove that R is a contraction mapping. Let any u, v ∈ B r * and t ∈ J .

Consider
From the choice of L g , we obtain This shows that R is a contraction. The operator Q * is compact and continuous as proved in the Theorem 4. Proof: The proof can be completed following the same steps as in the proof the Theorem 4.2. ✷
Theorem 6.1 Assume that the function f ∈ C(J , R) satisfies the Lipschitz condition Then, the Caputo impulsive FDE (6.1)-(6.3) has at least one solution in the space PC (J , R).

Examples
In this section, we give examples to illustrate the utility of the results we obtained.
Define the function f : Clearly, Thus f satisfies the Lipschitz condition with the constant L = 1 8 . If the function Ψ satisfies the condition then problem (7.1)-(7.3) has unique solution.
By direct substitution one can verify that u(t) = t 2 is the solution of the problem (7.5) -(7.7).

Concluding remarks
We close the present paper with the destinations we accomplished. We investigated the existence and uniqueness of solutions of nonlinear Ψ-HFDE and of also their respective extension to nonlocal case by means of strong analysis results. Some examples were illustrated in order to elucidate the results obtained. It is noted that since the Ψ-Hilfer fractional derivative is global and contains a wide class of fractional derivatives, the properties investigated herein are also valid for their respective particular cases.
Here we have not investigated the continuous dependence on the various data and Ulam-Hyers stabilities of solution of (1.2)-(1.4), which is the point of our next investigation and will be published a future work. Now, if we consider Ψ(t) = t in the problem (1.2)-(1.4) with A : D(A) ⊂ X → X generator of C 0 -semigroup (P t≥0 ) on a Banach space X, we have the following impulsive Ψ-HFDE with initial condition H D µ, ν a + u(t) = Au(t) + f (t, u(t)), t ∈ J = [a, T ] − {t 1 , t 2 , · · · , t m }, (7.11) ∆I 1−̺ a + u(t k ) = ζ k ∈ R, k = 1, 2, · · · , m, (7.12) with the same conditions of problem (1.2)-(1.4). The next step of the research is to analyze the problem (7.11)-(7.13). But the question may arise " Why not get the existence and uniqueness of mild solutions to the problem (7.11)-(7.13), with a formulation in the sense?". The reason for non-investigation with the Ψ-Hilfer fractional derivative comes from the fact of the non-existence of an integral transform, in particular, of Laplace with respect to another function, since it is an important condition in the investigation of the mild solution.
Research in this sense has been developed and, in the near future, results can be published.