On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

On the other hand, in the case where a physical procedure is described by IVPs for FDEs, at that point it is desirable that any mistakes made in the estimation of initial data do not impact the solution so much. Mathematically, this is known as continuous dependence of solution of an IVP on the data introduced in the proposed problem. Actually, nonlocal conditions come up when estimations of the function on the limit are associated with values in the domain. It is seen as more reasonable than the classical initial conditions for the forming of some physical phenomena in specific problems of wave spread and thermodynamics. In crossing, we saw that the nonlocal condition m k=1 β k κ(τ k ) = κ 0 that may be applied in physical models yields preferred impact over the initial conditions κ(0) = κ 0 .
In this regard, many interested authors have presented excellent results on the existence and continuous dependence of solution of FDEs with the nonlocal conditions and classical fractional operators. For the recent review of these studies, we refer to [36][37][38][39][40][41][42][43][44].
Recently, ABC-fractional IVP is one of the studied problems by Thabet et al. [14] which is of type Through the above discussions, and motivated by [14,40], in this work, we will prove some new results based on a novel version of fractional operators. More precisely, we consider the following ABC-type nonlocal fractional problem: . . , m), and κ ∈ C[0, χ] such that the operator ABC D 0,θ exists and The main aim of this work is to study the existence, uniqueness of solutions and their continuous dependence on the nonlinear nonlocal problem (1.1)-(1.2) in the frame of ABC fractional derivative by means of Schauder's and Banach's fixed point theorems. To the best of our knowledge in the subject, no one considered the existence and data dependence of the ABC-type fractional problem with nonlocal conditions. Therefore, the acquired results are recent studies and an extension of the development of FDEs involving an ABC fractional derivative. Furthermore, the analysis of the results is restricted to a minimum of hypotheses.
The rest of the paper is arranged as follows. In Sect. 2, we recall some useful preliminaries related to the main outcomes. Section 3 is dedicated to obtaining the solution representation to a given problem. Then the existence and uniqueness results are proved via functional integral equation with the aid of some fixed point approaches. Moreover, we discuss the continuous dependence of solutions for the problem at hand. Illustrative examples are given in Sect. 4. Finally, concluding remarks are mentioned in Sect. 5.

Main results
This section is devoted to obtaining formula of the solution to ABC-type nonlocal problem (1.1)-(1.2). Then we prove the existence and uniqueness of solution for problem (1.1)-(1.2) by means of Schauder's fixed point theorem (Theorem 2.8)and Banach's fixed point theorem (Theorem 2.7). Moreover, we also discuss the continuous dependence of solutions to such equations on arbitrary data.

Solution representation
Then the solution of ABC-type nonlocal problem (1.1)-(1.2) can be indicated by the fractional integral equation where F κ is the solution of the functional integral equation and Applying AB I 0,θ on both sides of (1.1) and using Lemma 2.2, we have Multiplying β k and taking the sum to both sides of (3.4), we can write By nonlocal condition (1.2), we obtain which implies The proof is completed. Now, we consider the following hypotheses: (H 1 ) There exists a constant L 1 > 0 such that for all θ ∈ [0, χ] and x, x * , y, y * ∈ R.
(H 2 ) There exists a constant κ > 0 such that

Existence results
In this subsection, we prove the existence and uniqueness of solution to ABC-type non- The following result is based on Theorem 2.8.
The following result is based on Theorem 2.7.
Proof We shall use Theorem 2.7 to prove that T defined by (3.6) has a fixed point. Let κ, κ * ∈ C[0, χ] and θ ∈ [0, χ]. Then On the other hand, we have, for each θ ∈ [0, χ], By replacing (3.12) in (3.11), we get Consequently, by (3.10), T is a contraction. As a consequence of Theorem 2.7, we conclude that T has a fixed point which is a solution of problem (1.1)-(1.2).

Continuous dependence
This portion is devoted to discussing the continuous dependence of the solution for ABCtype nonlocal problem (1.1)-(1.2).  (3.14) and the solution of ABC-type nonlocal problem (1.1)-(3.13) is where F κ and F κ are the solutions of and Hence, However, we have from (H 1 ) that Thus By replacing (3.17) in (3.16), we get Since ϒ < 1, we get Proof In view of Lemma 3.1, the solution of ABC-type nonlocal problem (1.1)-(1.2) is and the solution of ABC-type nonlocal problem (1.1)-(3.18) is where F κ and F κ are the solutions of Hence, Substituting from (3.21) in (3.20), we get Now, we have from (3.19) that By (H 1 ), we obtain and (1 -1 ) .

Concluding remarks
We can conclude that the main outcomes of this manuscript have been effectively accomplished. The existence and uniqueness of solutions for the nonlocal Cauchy problem for a nonlinear implicit FDE involving the ABC fractional derivative have been proved through some fixed point techniques (Theorems 2.8, 2.7) and some outcomes related to AB operators. Then, as an application, the continuous dependence of solution to such equations on arbitrary data involved therein was discussed. This paper adds and contributes to growth FDEs, particularly in the case of nonlocal implicit FDEs involving a novel fractional derivative presented recently by Atangana and Baleanu [11]. There are some works that carried out reported studies on the existence and continuous dependence of solutions of classical FDEs, and one of the destinations of this paper is to contribute with the goal that it can have a more prominent degree of studies identified with FDEs involving generalized fractional operators. As a future direction, the studied problem would be interesting if it were studied on generalized fractional operators of variable order recently introduced by Yang and Machado [8] and its generalization by Sousa and Oliveira [47].