On Impulsive Boundary Value Problem with Riemann-Liouville Fractional Order Derivative

Our manuscript is devoted to investigating a class of impulsive boundary value problems under the concept of the Riemann-Liouville fractional order derivative. The subject problem is of implicit type. We develop some adequate conditions for the existence and puniness of a solution to the proposed problem. For our required results, we utilize the classical ﬁ xed point theorems from Banach and Scheafer. It is to be noted that the impulsive boundary value problem under the fractional order derivative of the Riemann-Liouville type has been very rarely considered in literature. Finally, to demonstrate the obtained results, we provide some pertinent examples.


Introduction
The fractional order differential equations (abbreviated as FODEs) are the generalization of the ordinary differential equations of the integer order. In the 17th century (1665), the great mathematicians Newton, Leibnitz and L'Hospital introduced for the first time the idea of fractional order differential equations (FODEs). Later on, in 1823, another mathematician by the name of Lacroix, introduced the fractional derivative [1] of simple power function. Furthermore, this area has been studied by many researchers because it has significant applications in various fields of science and technology in mathematical modeling of different fields of Science and Technology. For instance, some phenomena including the diffusion process [2], some chemical processes of electrochemistry [3], infectious disease in biology [4], signal and image processing [5], dynamic processes [6], and systems control theory [7] can be excellently described by using FODEs instead of the ordinary derivative. For further applications of FODEs, we refer to [8][9][10][11][12][13] and the references therein.
On the other hand, an interesting and important branch recently got warm attention known as impulsive differential equations (IDEs). In recent times, the said area has been increasingly used to model many physical and social phenomena in social sciences in a very interesting way. Currently in the said area, significant contribution has been done by various researchers like Simeonov and Bainov [14], Benchohra et al. [15], Lakshmikantham et al. [16], and Samoilenko and Perestyuk [17]. Benchohra and Slimani [18] has initiated the study of FODEs under impulsive conditions by using fractional derivatives of the Caputo and Riemann-Liouville type with order α ∈ ð0, 1Þ. In addition, some applications of IDEs have been studied in various scientific disciplines such as biology, geography, engineering, dynamics, physics, geology, and management sciences. In terms of the important applications of IDEs, due to important applications of IDEs this field has a lot of significance and concentration (see [19][20][21][22]). For general research and significance, we refer some more important publications [23][24][25][26]. Many researchers have recently studied nonlinear FODEs with different kinds of boundary and initial conditions. Boundary value problems have significant applications in various fields of dynamics and fluid mechanics as well as engineering disciplines. Here, it is remarkable that problems under integral boundary conditions have some important applications in fluid mechanics, chemical engineering, thermoelasticity, flow of groundwater, population dynamics, and more (see [27][28][29]). Furthermore, we also refer some significance of FODEs under integral boundary conditions as discussed in [27][28][29][30][31].
Recently, due to increasing applications of FODEs to model real world problems more comprehensively, researchers are taking keen interest to investigate different areas of fractional calculus. In particular, the use of FODEs in mathematical modeling of infectious diseases and other biological phenomena have got more attention. Various researchers have studied the fractional predator-prey pathogen model, the n-predator-prey model with herd behavior, etc. (see [32][33][34]). Further, there is also modeling of the interaction between tumor growth and the immune system, edge-detecting techniques, an infectious diesease on a prepredator model, etc., (we refer to [35][36][37][38][39][40]).
As stated earlier, the area devoted to IDEs with a fractional order has many applications. These differential equations can be modeled to those evolutionary processes which are subjected to abrupt changes in their states. Recently, some authors have used IDEs for the mathematical modeling of certain biological events. It is remarkable that impulsive differential equations are using in mathematical models which give rise to some important duallayered impulsive systems. The said systems will open new doors in the future to develop a general mathematical theory for the said systems. For instance, the author of [41] has obtained very interesting results in this regard for various kinds of biological models of infectious diseases. Here, we remark that a very basic and important qualitative problem in the investigation of IDEs with a fractional order concerns the existence theory of solutions. For these purposes, researchers have used the classical fixed point theory and some tools of nonlinear analysis. For instance, in [42], the authors have applied fixed point results to develop the corresponding existence theory of solutions by using the Caputo derivative of the fractional order. In the same line, in [43], the authors have used the Picard-type analysis to investigate the stochastic-type IDEs of a fractional order by using the Caputo operator. In all these papers, the Caputo operator has been increasingly used. It is to be noted that the fractional order derivative of the Riemann-Liouville type has been very rarely used in IDEs.
Authors [44] have established existence theory for fractional order IDEs with initial conditions by using the fixed point theory. The authors in [45] investigated the following problem of IDEs under the fractional order derivative of the Riemann-Liouville type as Δr z m ð Þ= ψ m r z m ð Þ ð Þ, m = 1, 2, 3, ⋯, q, where β ∈ R and f : ½0, T × R ⟶ R is a continuous function. They developed sufficient conditions for the existence of at least one solution to the considered problem by using a fixed point approach. Motivated from the said work given in (1), we are interested in studying a class of nonlinear implicit fractional order IDEs under the Riemann-Liouville derivative with the Riemann-Liouville-type integral boundary conditions as where RL D is denoted as the Riemann-Liouville fractional order derivative, J = ½0, 1, f : J × R × R ⟶ R is a continuous function. Furthermore, ψ m , ψ * m : R ⟶ R are continuous functions for m = 1, 2, ⋯, q and And also, where I 1−α , I 2−α are denoted as the Riemann-Liouville integral of fractional order 1 − α < 0, 2 − α > 0 on J , respectively. To establish the required results, we utilize the Scheafer fixed point theorem to investigate sufficient conditions for the existence of at least one solution to the problem under consideration (2). Furthermore, the criterion of uniqueness is derived by using the Banach contraction theorem. For the demonstration of our results, we provide some concrete examples.

Preliminaries
In this section, we provide some important results, basic definitions, and lemmas from the literature of fractional calculus [1,3,10,11], which are needed in this manuscript. Let Therefore, the right side is point-wise defined on ð0, ∞Þ, where Γ is the symbol of gamma function defined as ΓðβÞ = Ð ∞ 0 e −s s β−1 ds.

Journal of Function Spaces
Definition 2. The derivative of the Riemann-Liouville fractional order β > 0, of a continuous function where n = ½β + 1, ½β represent the whole part of the real number β; therefore, the right side is point-wise defined on ð0, ∞Þ: If the function f : ðc, dÞ ⟶ R is at least n -times differentiable, then the Caputo fractional derivative of order β > 0 is defined as where n = ½β + 1.

Main Works
To convert our considered problem in to an impulsive fractional integral equation, the given Lemma is provided.
The solution of the given linear IDE of the fractional order with the Riemann-Liouville derivative is given by Proof. Suppose rðzÞ is a solution to Problem (9); then, taking the Riemann-Liouville integral on both sides to using Lemma 5, there exist some constants c 0 , c 1 ∈ R such that Again taking the Riemann-Liouville integral to using Lemma (9), for some constant d 0 , d 1 ∈ R, we have Now, by using the impulsive conditions, we have

Journal of Function Spaces
Thus, putting the values in (12), we have The above process can be repeated in this way until we obtain the solution rðzÞ for z ∈ ðz m , z m+1 as Now, applying boundary condition I 1−α rð0Þ = 0 and I 2−α rð1Þ = 0 to get the values of constant c 0 and c 1 , we have c 0 = 0, The values of c 0 , c 1 putting in (11) and (15), one can obtain (10). On the contrary, suppose rðzÞ is a solution of the impulsive fractional integral equation (10). Following the direct calculation, we see that (10) satisfies the problem (9).
For simplification, we use the following notations: For the existence and uniqueness of the solution, we use some fixed point theorems. To transform the considered Problem (2) to a fixed point problem, we need to define the operator by M : PCðJ , RÞ ⟶ PCðJ , RÞ as for any r, r ∈ PCðJ , RÞ, τ, τ ∈ R, and z ∈ J . (H 3 ) There exists a constant N * 1 > 0, such that for each r, r ∈ PCðJ , RÞ and m = 1, 2, 3, ⋯, q.
(H 4 ) There exists a constant N * 2 > 0, such that for every r, r ∈ PCðJ , RÞ and m = 1, 2, 3, ⋯, q. We use Banach fixed point theorem to prove that problem (2) has unique solution. 4 Journal of Function Spaces

Theorem 7. Under hypotheses (H 1 )-(H 4 ) and if the following condition holds
then, there exists a unique solution for Problem (2) on J .
Proof. Let r, r ∈ PCðJ , RÞ for some z ∈ J , we have which further gives where τ, τ ∈ CðJ , RÞ are given by By using hypothesis ðH 2 Þ, we have Repeating this process, we get Therefore, for every z ∈ J and from (24), using hypothesis ðH 3 Þ, ðH 4 Þ, and (27), one has Upon further simplification, (29) yields Hence, from (29), we have By (22), operator M is a contraction. Thus, according to Banach's contraction principle, operator M has a unique fixed point which is the unique solution to Problem (2).

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Next, we will prove that Problem (2) has at least one solution for this, and we use Schaefer's fixed point theorem. Let the given hypotheses hold true: (H 5 ) There exist x, y, h ∈ PCðJ , RÞ, with such that for z ∈ J , r ∈ PCðJ , RÞ, and τ ∈ R.

(Step 1) The operator M is continuous
Assume fr n g be a sequence such that r n ⟶ r on PC ðJ , RÞ.
For z ∈ J , we have where τ n ðzÞ, τðzÞ ∈ PCðJ , RÞ are given by Now, from assumption ðH 2 Þ, we have Repeating this process, we get Since r n ⟶ r, τ n ðzÞ ⟶ τðzÞ as n ⟶ ∞ for every z ∈ J . We know that every convergent sequence is bounded, so for this, let ζ > 0 such that for every z ∈ J , we have jτ n ðzÞ j ≤ ζ and |τðzÞ | ≤ζ. Then, we have for every z ∈ J ; the function s ⟶ 2ζðz − sÞ α−1 and s ⟶ 2ζ ðz m − sÞ α−1 are integrable on ½0, 1, upon the use of these facts and the Lebesque-dominated convergence theorem in (35). After using assumptions ðH 5 Þ-ðH 7 Þ, we see that and hence, we have Therefore, operator M is continuous. ( Step 2) The operator M assigns bounded sets to bounded sets on PCðJ , RÞ. Just prove it for any ξ * > 0, there exists a positive constant E * , such that for every r ∈ D = fr ∈ PCðJ , RÞ, krk ≤ ξ * g, we have kMrk ≤ E * . To derive this result for each z ∈ J , one has 6 Journal of Function Spaces where τðzÞ ∈ CðJ , RÞ is given by By hypothesis (H 5 ) and for every z ∈ J , we have Thus, we have Therefore, from (42) by using (45), one has Hence, one has Therefore, the operator M is bounded. ( Step 3) The operator M assigns bounded sets to equicontinuous sets of PCðJ , RÞ: Let z 1 , z 2 ∈ J , and z 1 < z 2 . D is a bounded set as in Step 2, and let r ∈ D; then, we have = ψ 1 r 1 ,
Thus, hypothesis (H 7 ) is satisfied with A * 2 = ð1/55Þ and B * 2 = ð1/30Þ. Therefore, all of the hypotheses of Theorem 8 are satisfied, and therefore, the considered problem (62) has at least one solution on J .

Conclusion
IDEs of the fractional order have received proper attention due to their important applications in various fields of applied sciences. In the past, most studies have been done using the Caputo-type fractional derivatives to handle IDEs. In very few papers, investigating IDEs of the fractional order was done using the Riemann-Liouville derivative. Therefore, we have established successfully some important results devoted to the existence theory of a solution to the considered nonlinear implicit IDE with the Riemann-Liouville-type integral boundary conditions under the Riemann-Liouville fractional order derivative. The corresponding results for the existence and unique-ness of the solution have been archived by utilizing the classical Schiefer and Banach contraction fixed point theorems. For the demonstration of our results, we have enriched the paper by providing two pertinent examples.

Data Availability
Data used to support the findings of this study are included within the article.

Conflicts of Interest
No conflict of interest exist.