On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders 0 < θ ≤ 1 and 1 < θ ≤ 2. We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.


Introduction
Fractional calculus [1][2][3] has continued to attract the attention of many authors in the past three decades. Recently, new fractional derivatives (FDs) which interpolate the Riemann-Liouville, Caputo, Hilfer, Hadamard, and generalized FDs have appeared, see [4][5][6][7][8][9]. Some investigators have recognized that innovation for novel FDs with various nonsingular or singular kernels is necessary to address the need to model more realistic problems in various areas of engineering and science. Caputo and Fabrizio [10] introduced a new kind of FDs where the kernel is based on the exponential function. Losada and Nieto [11] studied some properties of this new operator. In [12,13], the authors presented new interesting FDs where the kernel relies on Mittag-Leffler function, the so-called Atangana-Baleanu-Caputo (AB-Caputo) which is basically a generalization of the Caputo FD. Then in [14,15], the authors deliberated the discrete versions of those new operators. For modeling in the framework of nonsingular kernels and fractal-fractional derivatives, we refer to [16][17][18]. There are many works pertinent to ABC problem in medical science and engineering. Hence we highlight medical, as well as engineering, applications by referring to [19][20][21].
The BVP of AB-Caputo FD, presented by Abdeljawad in [40], is also one of the recent problems through which the higher fractional orders are addressed: Motivated by the above arguments, the intent of this work is to investigate two AB-Caputo-type implicit FDEs with nonlinear integral conditions described by Some fixed point theorems (FPTs) are applied to establish the existence and uniqueness theorems for the problems (1.1) and (1.2). The proposed problems are more general, and the results generalize those obtained in recent studies; we also provide an extension of the development of FDEs involving this new operator. Moreover, the analysis of the results was limited to the minimum assumptions.
The rest of the paper is structured as follows. In Sect. 2, we give some useful preliminaries related to main consequences. Section 3 is devoted to obtaining formulas of solution to the proposed problems. Moreover, the existence and uniqueness theorems for the problems at hand are proved by means of various techniques for FPTs. Ultimately, illustrative examples are offered in Sect. 4.

Background materials and preliminaries
Here we recollect some requisite definitions and preliminary concepts related to our work.
Let Z = [a, T] ⊂ R, C(Z, R) be the space of continuous functions ς : Z → R with the norm Clearly, C(Z, R) is a Banach space with the norm ς . Definition 2.1 ([12, 13]) Let ϑ ∈ (0, 1] and p ∈ H 1 (Z). Then the AB-Caputo and AB-Riemann-Liouville FDs of order ϑ for a function p are described by and respectively, where E ϑ is called the Mittag-Leffler function and described by The associated AB fractional integral is specified by where N(ϑ) > 0 is a normalization function satisfying N(0) = N(1) = 1.

Definition 2.2 ([13])
In particular, if a = 0, the Laplace transform of AB-Caputo FD of p(r) is specified by

Definition 2.3 ([40])
Let ϑ ∈ (n, n + 1] and p be such that p n ∈ H 1 (Z). Set v = ϑn where v ∈ (0, 1]. Then the AB-Caputo and AB-Riemann-Liouville FDs of order ϑ for a function p are described by respectively. The associated AB fractional integral is specified by
Proof Let p(r) = (ra) k . By Definition 2.3, we have Then the solution of the following problem is given by is given by  [56]) Let J be a Banach space, and K be a nonempty closed subset of J . If B : K − → K is a contraction, then there exists a unique fixed point of B.

Main results
This section is devoted to obtaining formulas of solutions to linear problems corresponding to (1.1) and (1.2). Moreover, we prove the existence and uniqueness theorems to suggested problems by applying Theorems 2.1 and 2.2.

1) if and only if ς is a solution of the ABC-problem
Proof Assume ς satisfies the first equation of (3.2). From Lemma 2.6, we have Also, Taking r → a on both sides of (3.4), we have Using the integral condition, we obtain From (3.3) and (3.5), and from fact that (a) = 0, we get Thus (3.1) is satisfied. Conversely, suppose that ς satisfies equation (3.1). Applying ABC D ϑ a + on both sides of (3.1), then using Remark 2.2 and Lemma 2.3, we find that Thus, we can simply infer that if and only if ς is a solution of the ABC-problem Proof Assume ς satisfies the first equation of (3.7). From Lemma 2.7, we have Using the integral condition ς(T) = T a g(s) ds, we get (3.10) Substituting the values of c 1 and c 2 into (3.8), we obtain Thus (3.6) is satisfied. Conversely, assume that ς satisfies (3.6). Applying ABC D ϑ a + on both sides of (3.6), then using Lemmas 2.2, 2.3, and 2.5, we find that Clearly, ς(a) = 0. Thus, we can simply infer that Before proceeding with the main findings, we are obligated to provide the following assumptions: (

Existence and uniqueness theorems for (1.1)
As a result of Theorem 3.1, we have the following theorem:  Proof Set By Theorem 3.3, we define the operator T : D → D by This T is well defined, that is, T(D) ⊆ D. Indeed, for any ς ∈ C(Z, R), f (·, ς(·), ABC D ϑ a + ς(·)) is continuous. Besides, by Lemma 2.4, Tς ∈ C(Z, R). Also, by Lemma 2.1 and Remark 2.1, we end up with Since f (r, ·, ·) is continuous on [a, T], one has ABC D ϑ a + (Tς)(r) ∈ C(Z, R). Now, we need to prove that T is a contraction. Let ς, ς ∈ D and r ∈ Z. Then

Using (A 1 ) and the fact that
By (A 2 ) and (3.11), for r ∈ Z, Condition (A 3 ) shows that T is a contraction. Hence, by Theorem 2.1, T has a unique fixed point.
Also, by (3.13), From (3.17), then for υ ∈ B ξ , Inequalities (3.19) and (3.20) give Using (A 3 ) and (3.15), for r ∈ Z and ς, υ ∈ B ξ , Step 2. We prove that T 1 is a contraction. From (A 1 ), we have Step 3. T 2 is compact and continuous. The map T 2 : B ξ → B ξ is continuous due to the continuity of f . Next, T 2 is uniformly bounded on B ξ by (3.20), because for any ς ∈ B ξ and r ∈ Z, we have This leads to a conclusion that T 2 is uniformly bounded on B ξ . Now, we show that T 2 (B ξ ) is equicontinuous. In order to establish that, let ς ∈ B ξ and a ≤ r 1 < r 2 ≤ T. Then Using (3.17), for ς ∈ B ξ , Observe that |(T 2 ς)(r 2 ) -(T 2 ς)(r 1 )| → 0 as t 2 → t 1 . In light of the former steps, together with Arzela-Ascoli theorem, we derive that (T 2 B ξ ) is relatively compact, and hence T 2 is completely continuous. So, Theorem 2.2 shows that (1.1) has at least one solution.