Existence Results for Impulsive Fractional Integrodifferential Equations Involving Integral Boundary Conditions

Department of Mathematics & Centre for Research and Development, KPR Institute of Engineering and Technology, Coimbatore 641 407, Tamil Nadu, India Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, ailand Department of Mathematics, Sri Vasavi College, Erode, Tamil Nadu, India Department of Mathematics, K.S.Rangasamy College of Technology, Tiruchengode, Namakkal, Tamilnadu, India Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, ailand


Introduction
Fractional di erential equations play a key role to describe various problems in di erent areas of science. Fractional models are more useful than the classical models. Fractional di erential equations are used in economics, image processing, physics, and so on. For detailed information on fractional di erential equations and their applications, see [1][2][3][4][5][6][7][8][9].
A. Baleanu and D. Baleanu [10] introduces a nonsingular Caputo and Riemann-Liouville version of the fractional di erential operator using the Mittag-Le er function as its kernel. Bonyah et al. [11] developed a mathematical model for cancer and hepatitis coinfection diseases using the ABfractional derivative.
e AB-fractional derivative of the fractional-order tumor-immune-vitamin model was provided in [12] and discussed the existence, uniqueness, and Hyres-Ulam stability results also. rough the fractional tumor-immune dynamical model with AB-fractional derivative, researchers [13] created a chaotic and comparative work of tumour and e ector cells. e fractional AB derivative was used to explore the numerical solution of the fractional immunogenetic tumour model in [14]. Like this, several applications were mentioned in reference [15][16][17][18][19].
In [20], the authors deal with the transmission dynamics of the COVID-19 mathematical model under ABC-fractional-order derivative. With the AB-fractional derivative, Logeswari et al. [21] investigated the mathematical model for COVID-19 infection propagation over the world. ey also developed a framework for generating numerical outputs in order to forecast the e ect of the illness spreading across India. [22][23][24] are a few other key publications that sought to address the issue of various diseases modeled as FDEs involving AB-fractional derivative.
In [25], Tidke introduced the following linear evolution equation: where 0 < q < 1, the unknown x(.) takes values in the Banach space X; f ∈ C(J × X, X), and A(t) is a bounded linear operator on a Banach space X, and x 0 is a given element of X. e operator D q denotes the Caputo fractional derivative of order q and he has investigated about the existence, uniqueness, and other properties of solutions of fractional semilinear evolution equations in Banach spaces.
In [26], the authors Kucche et al. dealt with the following nonlinear implicit fractional differential equations and investigated the existence and interval of existence of solutions, uniqueness, and properties of continuous dependance.
In [27], Guo et al. discussed the impulsive fractional differential equations with boundary value problems of the form: where C D α t is the Caputo fractional derivative of order α ∈ (0, 1) with the lower limit zero, f: J × R ⟶ R is jointly continuous and t k satisfy , and x(t − k ) � lim ϵ⟶0 − x(t k + ϵ) represent the right and left limits of x(t) at t � t k . I k ∈ C(R, R), and a, b, c are real constants with a + b ≠ 0.
In [28], Yukunthorn et al. studied the impulsive Hadamard fractional differential equations with boundary value problems of the form: where C D p k t k is the Hadamard fractional derivative of order 0 < p k ≤ 1 on intervals J k � (t k , t k+1 ], k � 1, 2, . . . , m, with J 0 � [t 0 , t 1 ], 0 < t 1 < t 2 < t 3 < . . . < t k < . . . < t m < t m+1 � T which are the impulse points, J: � [t 0 , T], f: J × R ⟶ R is a continuous function, and φ k ∈ C(R, R), J q i t i is the Hadamard fractional integral of order q i > 0, i � 1, 2, . . . , m. e jump conditions are defined by . . , m. Motivated by the above works, we study multiderivative nonlinear impulsive FDEs including AB-fractional derivative (AB derivative) of the following form: * with integral boundary condition of the form: where J � [0, T], T > 0, 0 < α < 1, D α τ indicates the ABRfractional differential operator of order α and represent the right and left hand limits of τ(t) at τ � τ k . e fixed point theorem of Krasnoselskii is used to prove the existence of a solution.
e Gronwall-Bellman inequality and the properties of the fractional integral operator are used to prove that the solution is unique. e following is the outline of the paper: the required background for the development of the study is presented in Section 2. Section 3 discusses the existence and uniqueness of impulsive fractional integrodifferential equations. Section 4 describes the examples.

Preliminaries
e ABC-fractional derivative and the generalized Mittag-Leffler function are defined and discussed in this section.
Definition 4 (see [30,31]). A Mittag-Leffler generalized function [c] α,β (z) for the complex α, β with Re(α) > 0 can be defined as follows: where c k is the Pochhammer symbol given by We note that e following Laplace transformation results are required.
Lemma 6 (Krasnoselskii's Fixed Point eorem [6]). Let ω be a Banach space. Let S be a bounded, closed, convex subset of ω, and let F 1 , F 2 be maps of S into ω such that F 1 ω + F 2 υ ∈ S for every pair ω, υ ∈ S. If F 1 is contraction and F 2 is completely continuous, then the equation has a solution on S.
implies that with integral boundary condition of the form if and only if x is a solution of fractional integral equation
where β ∈ C, with Re(β) > 0. en, using definition of operator F and equation (64), we have is demonstrates that the operator equation F is invertible on C(J), has the unique solution is completes the proof. ω(τ)), τ ∈ J can be solvable on C(J) and has a solution given by where β ∈ C with Re(β) > 0 and τ 0 f(σ, ω(σ))d(σ), τ ∈ J.

Main Results
Theorem 4 (Existence theorem). Suppose f ∈ C(J × R × R, R) and the Lipschitz condition H ∈ C(J × R) is satisfied.
where p: J ⟶ R + , with L � supp(τ) and for the real constants L H , M H , C H > 0. If 0 < L < min 1, 1/2T { }, then the ABR-FDEs (5) and (7) has a solution in C(J) provided Proof. Assume that where M f � sup|f(τ, 0, 0)| and M * > 0 is a constant such that m i�1 |y i | ≤ M * . We have R > 0 because of the choice of L and condition (47).
Let us consider the set It is possible to prove that S is a closed, convex, and bounded subset of ω. We consider the operators F 1 : S ⟶ ω and F 2 : S ⟶ ω, both of which are defined by e operator equation for the equivalent fractional integral equation (55) to the ABR-FDEs (5) and (7) is as follows: We show that the operators F 1 and F 2 satisfy Lemma 6 requirements. e following steps have been used to demonstrate the same.
For every ω, υ ∈ PC(J) and τ ∈ J, we obtain using the Lipschitz condition on f, is gives Mathematical Problems in Engineering Step 2. Next, we show that F 2 is completely continuous.
One can readily demonstrate that the operator F 2 is completely continuous using the Ascoli-Arzela theorem and eorem 2.
Step 3. F 1 ω + F 2 υ ∈ S, for any ω, υ ∈ S. For any ω, υ ∈ S using eorem 2, we obtain i.e., by condition (47) with (48), we get From inequalities (54) and (55), we have is gives is shows that F 1 ω + F 2 υ ∈ S, for ω, υ ∈ S. en, the operator equation, has a fixed point in S, which is the solution of ABR-FDEs (5) and (7). is completes the proof. e uniqueness of solutions to ABR-FDEs (5) and (7) is demonstrated in two methods in the following theorem. We prove the result first using the properties of the fractional integral operator E 1 α,1,−α/1−α;0+ , and then using the Gronwall-Bellman inequality.

Conclusion
In this paper, we examined the impulsive fractional integrodifferential equations involving ABC derivative with integral boundary conditions. Recently ABC-derivative gained much attention due to the nonsingular property of the kernels. e existence of solution is investigated for the proposed equations by using Krasnoselskii fixed point theorem. e uniqueness of the result is derived with the help of Gronwall-Bellman inequality as well as the properties of fractional integral operator. In future, we extend this work with the delay properties involving Mittag-Leffler function.

Data Availability
ere is no data used for this manuscript.

Conflicts of Interest
e authors declare no conflicts of interest.