Monotone Iterative Technique for a New Class of Nonlinear Sequential Fractional Differential Equations with Nonlinear Boundary Conditions under the ψ -Caputo Operator

: The main crux of this work is to study the existence of extremal solutions for a new class of nonlinear sequential fractional differential equations (NSFDEs) with nonlinear boundary conditions (NBCs) under the ψ -Caputo operator. The obtained outcomes of the proposed problem are derived by means of the monotone iterative technique (MIT) associated with the method of upper and lower solutions. Lastly, the desired ﬁndings are well illustrated by an example.


Introduction
Currently, the study of initial or boundary value problems (BVPs) for fractional differential equations (FDEs) has received great recognition due to their important applications in various areas, such as mathematical, physical, and engineering models; see [1][2][3]. For more recent developments on this topic, one can see the monographs [4][5][6][7] and the references therein. As a consequence of the advancement made in the field of fractional calculus (FC), several new fractional operators have appeared ranging from Riemann-Liouville, Caputo, Hadamard and Hilfer to ψ-Caputo and ψ-Hilfer operators. For more clarifications and basic properties of these new fractional operators, the reader is referred to the following references [8][9][10][11][12][13]. The latest operators have the ability to recover the aforementioned operators. In this respect, real-world events are often nonlinear, and thus, they can be modeled by nonlinear FDEs. Recently, plenty of scholars studied the aforesaid field looking for some qualitative properties of their solutions. Generally speaking, getting the exact solution of FDEs involving nonlinearities is a tough task. Namely, in order to bypass the absence of exact solutions of nonlinear FDEs, many researchers have devoted themselves to developing various techniques to compute the approximate solutions to such problems of the considered FDEs. Among them, the monotone iterative technique (MIT) [14,15] linked with the method of upper and lower solutions is employed as a fundamental mechanism to prove the existence as well as the approximation of solutions to many applied problems of nonlinear differential equations and integral equations. In other words, the suggested approach has many interesting advantages. The main advantage of this tool is that it not only proves the existence of solutions but it can also provide calculable monotone sequences that converge to the extremal solutions. Recent results by means of the MIT are obtained in [16][17][18][19][20][21][22] and the references therein. To the best of our knowledge, NSFDEs involving the ψ-fractional operator were not given enough consideration and were only studied by a few researchers [23], and it is the motivation of this paper. So, in this manuscript, we will explore the existence of extremal solutions for the following NSFDEs in the ψ-Caputo sense involving NBCs: where c D ;ψ a + is the ψ-Caputo fractional derivative of order ∈ (0, 1] (which will be specified in Definition 2), M : ∆ × R −→ R, W : R × R −→ R are both continuous functions, ω is a positive real number, and θ 1 ∈ R.
This manuscript has the following structure: Section 2 offers some basic definitions and useful tools that are required in this paper. Section 3 is devoted to the principal findings concerning the existence of extremal solutions for the proposed model (1). An example is proposed in Section 4 to highlight the usefulness of our theoretical outcomes. At last, the manuscript ends with a brief conclusion and some suggestions for future work are also pointed out.

Preliminaries
Below, we provide some definitions and fundamental lemmas which will be employed and used as helping tools in our proofs later.
Lemma 2 ( [5,21]). Let u ∈ (0, 1), v > u be arbitrary and ∈ R. The functions E u , E u,u and E u,v are nonnegative and have the properties listed below: This holds as long as the integral on the right-hand side exists.

Definition 6 ([11]
). Let u and v be two functions which are piecewise continuous at each interval [a, b] and of ψ(r)-exponential order. We define the generalized convolution of u and v by

Lemma 3 ([11]
). Let u and v be two functions which are piecewise continuous at each interval [a, b] and of ψ-exponential order. Then, In the following Lemma, we present the generalized Laplace transforms of some elementary functions Lemma 4 ([11]). The following properties are satisfied: In following theorems, we state the generalized Laplace transforms of the generalized fractional integrals and derivatives.
In particular, if 0 < ≤ 1, then has a unique solution given explicitly by Proof. Performing the generalized Laplace transform to both sides of Equation (4) and then using Lemma 4, one obtains Taking the inverse generalized Laplace transform on both sides of the last expression, we get [1];ψ (a) ≥ 0, then γ(r) ≥ 0 for all r ∈ ∆.

Main Results
An upper solution q of the problem (1) can be defined in a similar way by reversing the above inequality.
According to Lemma 8, we arrive at p 1 (r) ≤p 1 (r), r ∈ ∆. Secondly, we need to show that p 1 andp 1 are the lower and upper solutions of problem (1), respectively. Taking into account that M is an increasing function with respect to the second variable, we get This means that p 1 is a lower solution of problem (1). Analogously, we can verify that p 1 is an upper solution of problem (1).
Thirdly, we show that the sequences {p n } and {p n } converge uniformly to their limit functions p * andp * , respectively. We show that the sequences {p n } and {p n } converge uniformly to their limit functions p * andp * , respectively.
It remains to be shown that the sequences {p n } and {p n } are equicontinuous on ∆. To do this, choosing r 1 , r 2 ∈ ∆, with r 1 ≤ r 2 . By (11) and Lemma 2, we have By the continuity of the function |θ 1 |(ψ(r) − ψ(a))E 1,2 −ω(ψ(r 2 ) − ψ(a)) on ∆, the right-hand side of the previous inequality approaches to zero when r 2 → r 1 independently of {p n }. Hence, the family {p n } is equicontinuous on ∆. Likewise, we can demonstrate that {p n } is equicontinuous. Therefore, by Ascoli-Arzela's Theorem, there exist subsequences {p n k } and {p n k } which converge uniformly to p * andp * , respectively, on ∆. This together with the monotonicity of sequences {p n } and {p n } implies lim n→∞ p n (r) = p * (r) and lim n→∞p n (r) =p * (r), uniformly on r ∈ ∆ and the limit functions p * ,p * satisfy problem (1).
Lastly, we prove the minimal and maximal property of p * andp * on [p 0 ,p 0 ]. To do this, let p ∈ [p 0 ,p 0 ] be any solution of (1). Suppose for some n ∈ N * that p n (r) ≤ p(r) ≤p n (r), r ∈ ∆.

An Example
To illustrate our abstract results (Theorem1), let us consider problem (1) with specific data. More precisely, taking Taking p 0 (r) = 1 and q 0 (r) = 1 + r, it is easy to see that p 0 and q 0 are lower and upper solutions of problem (14), respectively, and p 0 ≤ q 0 . So, condition (H 1 ) holds. In addition, it is obvious that the function M : ∆ × R −→ R is a continuous and nondecreasing function with respect to the second variable. Hence, condition (H 2 ) is satisfied. Moreover, for p 0 (a) ≤ u 1 ≤ u 2 ≤ q 0 (a), Therefore, the hypothesis (H 2 ) of Theorem 1 is fulfilled with c = 1 and d = 0. Thus, all assumptions of Theorem 1 are valid. As a result, the suggested problem problem (14) has extremal solutions on [p 0 , q 0 ].

Conclusions
The existence of extremal solutions for a new class of nonlinear sequential fractional differential equations (NSFDEs) with nonlinear boundary conditions (NBCs) containing the ψ-Caputo operator is the topic of our study. To arrive at the principal findings of this study, we used the interlinking between the monotone iterative technique (MIT) and the method of upper and lower solutions. We also tested the applicability and efficiency of the mentioned method by an example. For future research, we plan to look at the same outcomes for our present model (1) using other modern fractional operators. It would be also intriguing to construct numerical approaches to approximate the solutions suggested by our Theorem 1.