Mathematical Analysis of Nonlocal Implicit Impulsive Problem under Caputo Fractional Boundary Conditions

This paper is related to frame a mathematical analysis of impulsive fractional order diﬀerential equations (IFODEs) under nonlocal Caputo fractional boundary conditions (NCFBCs). By using ﬁxed point theorems of Schaefer and Banach, we analyze the existence and uniqueness results for the considered problem. Furthermore, we utilize the theory of stability for presenting Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam-Rassias stability results of the proposed scheme. Finally, some applications are oﬀered to demonstrate the concept and results. The whole analysis is carried out by using Caputo fractional derivatives (CFDs).


Introduction
It has been observed that the focus of investigation has shifted from classical integer-order models to fractional-order models. It is because of the fact that many practical systems are excellently described by using fractional-order differential equations (FODEs) instead of classical differential equations. For basic theory and some important applications of fractionalorder derivatives, we refer the readers to see [1][2][3][4] and the references therein. Many researchers are devoted to work in this area and made significant contribution in this regard; we refer the readers to the recent work in [5][6][7][8][9][10].
e study of implicit systems of FODEs with impulsive conditions is quite important as such systems appear in a variety of problems of applied nature, especially in biosciences, economics, engineering, etc. Such problems arise due to abrupt changes in the state of systems like earth quack, fluctuation of pendulum, etc. Here, we refer to some recent papers on impulsive problems [11][12][13][14][15][16]. e important class of FODEs known as IFODEs has been given much devotion by researchers. One of the most important aspects is investigation of problems under boundary conditions. Such problems mostly occur in engineering. Boundary and initial conditions may be local or nonlocal and both are important, and increasingly many problems have been investigated under these conditions. Replacing the local conditions by nonlocal ones produces a significant effect. is is due to the fact that the measurement computed from a nonlocal condition is usually more precise than the only one measurement given by a local condition. erefore, the area of nonlocal boundary value problems has also attracted enough attention. In the last two decades, the area of IFODEs has been investigated from various directions including qualitative theory of existence of solution/solutions, stability, and numerical analysis. erefore, IFODEs have also been investigated under nonlocal boundary conditions. For instance, Gupta and Dabas [17] studied the existence and uniqueness results for a class of IFODEs with nonlocal boundary conditions. c 0 D ϱ t w(t) + f t, w(t), w ′ (t) � 0, ϱ ∈ (1,2], t ∈ [0, 1], w(0) � 0, c 0 D ϱ t w(1) � δ c 0 D ϱ t w(ρ), 0 < ξ < 1, 0 < ρ ≤ 1.

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(1) By employing the fixed point technique, the authors obtained the existence and uniqueness results.
is paper can be considered as generalization of the aforesaid work, in which we discuss existence, uniqueness, and various stability results for the following implicit IFODEs with three point NCFBCs of order ϱ ∈ (1, 2]: In the proposed problem, the notation and c 0 D ξ t represent Caputo fractional derivatives of orders ϱ, ], and ξ, respectively, where the points 0 and t in the subscript of the differential operator D are actually the limits of the definite integral involved in the definition of CFD. e function f: where R is the set of real numbers. e impulsive functions I q and I q in C(R, R) are bounded. For the sequence . e speciality of this proposed problem is that the nonlinear term depends not only on the unknown function but also on its fractional derivative compared with the available results in the literature. is type of study has rarely been discussed in the literature because of the complexity of fractional impulsive surfaces. e further organization of this manuscript is divided into four parts as follows: e second part of the paper demonstrates the preliminary portion in which we recall to readers the basics of used theory, notations, and definitions. e third part presents an existence result by employing Schaefer's fixed point theorem. e fourth section is introduced to analyze and study several stability results of the considered problem, and the last section is provided to illustrate the applications of the obtained results.

Preliminaries
We take S � [0, T], S 0 � [0, t 1 ], and S q � (t q , t q+1 ]. We introduce the following space of piecewise continuous functions by where B is the Banach space corresponding to the norm ‖w‖ B � max t∈S |w(t)|.
Definition 1 (see [18]). e fractional order integral of function g ∈ L 1 ([a, b], R + ) of order ϱ ∈ R + is defined by where Γ is the gamma function.

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Definition 4 (see [19]). If for ϵ > 0 and set of positive real numbers R + there exists ψ ∈ C(R + , R + ), such that for any solution h ∈ B of inequality (6), there is a unique solution w ∈ B of problem (2) which satisfies then problem (2) is called generalized Hyers-Ulam stable.
Definition 5 (see [19]). If for ϵ > 0 there exists a real number C > 0, such that for any solution h ∈ B of inequality (8), there is a unique solution w ∈ B of problem (2) which satisfies then problem (2) is called Hyers-Ulam-Rassias stable with respect to (Φ, ψ).
Definition 6 (see [19]). If there exists constant C > 0, such that for any solution h ∈ B of inequality (7), there is a unique solution w ∈ B of problem (2) which satisfies then problem (2) is called generalized Hyers-Ulam-Rassias stable with respect to (Φ, ψ).
Here, it is to be noted that Definitions 3-6 have been adopted from the paper [19].

Remark 1.
e function h ∈ B is called a solution for inequality (6) if there exists a function ϕ ∈ B together with a sequence ϕ q , where q � 1, 2, . . . , r (which depends on h) such that Lemma 1 (see [20]). For ϱ > 0, the given result holds: To investigate the nonlinear IFODE2, we first consider the associated linear problem and obtain its solution.
if and only if w(t) is a solution of the following BVP: Proof. Let for t ∈ [0, t 1 ), w(t) be the solution of (15), then by Lemma 1, we have Using the condition w(0) � 0, we get Substituting c 0 in (16), we get For t ∈ (t 1 , t 2 ], we get Applying the impulsive condition Δw(t 1 ) � I 1 (w(t − 1 )), we get Substituting c 2 in (19), we get From equations (18) and (22), we get Substituting c 3 in (22), we get For t ∈ (t 2 , t 3 ], we get Applying the impulsive condition Δw(t 2 ) � I 2 (w(t − 2 )) in (26) and (25), we get Substituting c 4 in (26), we get By equations (25) and (28), we get Now, using the impulsive condition Δ( c Mathematical Problems in Engineering Substituting c 5 in (28), we get Similarly for t ∈ S q , we get Using the boundary conditions, c By substituting the value of c 1 and summarizing, we get the required result.
Conversely, assume that u satisfies the impulsive fractional integral equation (8); then by direct computation, it can be seen that the solution given by (14) satisfies (15).

Existence and Uniqueness Results
In this section, we shall prove our main results. For which, we assume the following assumptions: (H 1 ) Let there exist positive constants L 1 , L 2 , L 3 , and L 4 such that for t ∈ [0, T] and all w 1 , w 2 , h 1 , h 2 ∈ R, the following inequalities hold: for t ∈ S, w ∈ B. (37) such that a * 3 � sup t∈[0,T] |a 3 (t)| < 1. (H 3 ) If f, I ı , and J ı are continuous functions such that for all w, h ∈ R, the following inequalities hold: (H 4 ) Let there exist constants β > 0 and ϵ > 0 and a nondecreasing function Φ ∈ C(S, R), such that the following inequalities hold: We transform problem (2) into a fixed point problem.
Considering an operator N: B ⟶ B, defined by , t ∈ t q , t q+1 , q � 1, 2, . . . , r, . In (40), we see that all the terms of solution w in the interval [0, t 1 ] are contained in the solution w in interval (t q , t q+1 ]; therefore, for simplicity purpose, we will study the solution in interval (t q , t q+1 ] only. Now, we shall prove some theorems. Our first result is based on Schaefer's fixed theorem.

Theorem 1. If the assumptions (H 1 ) − (H 3 ) are satisfied, then problem (2) has at least one solution.
Proof. We use Schaefer's fixed point theorem. e proof is given in the following four steps.
Step 1: to show that N is continuous, take a sequence w n such that w n ⟶ w ∈ B. en, for t ∈ S, we have where Y w n (t) � f(t, w n (t), Y w n (t)) and Y(t) � f(t, w(t), Y w (t)). Using (H 1 ), we have which implies erefore, by the continuity of f, I, andJ and the Lebesgue dominated convergent theorem, we conclude from (41) that |N(w n (t)) − N(w(t))| ⟶ 0 as n ⟶ ∞ which implies ‖N(w n ) − N(w)‖ B ⟶ 0 n ⟶ ∞. is proves the continuity of N.
Step 2: in this step, we will show that for each Mathematical Problems in Engineering (45) Using (H 2 ) and a * 1 � sup t∈X a 1 (t), a * 2 � sup t∈X a 2 (t), Taking the maximum value over the interval S and simplifying, we get (47) Using this result, (45) implies Further simplification implies is shows that the operator N maps bounded sets into bounded sets.
Step 3: in this step, we will show that N is equicontinuous. Let w ∈ W⊆B and t 1 , t 2 ∈ S such that t 1 < t 2 and consider

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Using assumptions (H 2 ) and (H 3 ), we obtain We see that as t 1 ⟶ t 2 , the right-hand side of inequality (51) tends to zero that is |N(w(t 2 ))− N(w (t 1 ))| ⟶ 0 as t 1 ⟶ t 2 . Hence, by the Ascoli − Arzela ' theorem N: B ⟶ B is completely continuous.
Step 4: to complete the proof, it remains to show that the set E � w ∈ : w � ζNw, for 0 < ζ < 1 { } is bounded. Let w ∈ E, then for any t ∈ S, we have Using 0 < ζ < 1 and (47) and (49), from (52), we get the following result: is shows that the set E is bounded. erefore, by Schaefer's fixed point theorem, problem (2) has at least one solution. e following and our second result is based on the Banach fixed point theorem.

Theorem 2. If the assumptions (H 1 ) − (H 3 ) and the inequality
are satisfied, then (2) has an unique solution.
Proof. To show that the operator N as defined above has a unique fixed point, we consider w 1 , w 2 ∈ B.
For t ∈ t q , t q+1 ], we have Mathematical Problems in Engineering 9 where which implies us, from (55), we have By further simplification, we obtain the following inequality: erefore, by Banach contraction theorem, problem (2) has a unique fixed point.

Stability Analysis
In this section, we study Hyers-Ulam stability of problem (2).