Monotone Iterative and Upper–Lower Solution Techniques for Solving the Nonlinear ψ − Caputo Fractional Boundary Value Problem

: The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the ψ − Caputo derivative C D σ ; ψ a + (cid:36) ( t ) = V ( t, (cid:36) ( t )) under integral boundary conditions (cid:36) ( a ) = λ I ν ; ψ (cid:36) ( η ) + δ . Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ψ ∗ ( t ) as t, Caputo, 2 t , √ t, and Katugampola (for ρ = 0.5 ) derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.


Introduction
The notion of fractional calculus refers to the last three centuries and it can be described as the generalization of classical calculus to orders of integration and differentiation that are not necessarily integers. Many researchers have used fractional calculus in different scientific areas [1][2][3][4].
In the literature, various definitions of the fractional-order derivative have been suggested. The oldest and the most famous ones advocate for the use of the Riemann-Liouville and Caputo settings. One of the most recent definitions of a fractional derivative was delivered by Kilbas et al., where the fractional differentiation of a function with respect to another function in the sense of Riemann-Liouville was introduced [5]. They further defined appropriate weighted spaces and studied some of their properties by using the corresponding fractional integral. In [6], Almaida defined the following new fractional derivative and integrals of a function with respect to some other function: where n = [σ] + 1 and respectively. He called the fractional derivative the ψ−Caputo fractional operator. In the above definitions, we get the Riemann-Liouville and Hadamard fractional operators whenever we consider ψ * (x) = x or ψ * (x) = ln x, respectively. Many researchers used this ψ−Caputo fractional derivative (see [7][8][9][10][11][12][13] and the references therein). Abdo et al., in [14], investigated the BVP for a fractional differential equation (FDE) involving ψ * operator and was given as D n−σ,ψ * t + 1 (t) = V (t, (t)), t ∈ ı =: [t 1 , t 2 ], [k] ψ * (t 1 ) = k t 1 , (k = 0, 1, 2, · · · , n − 2), ψ * (t 2 ) = t 2 . For more details on the development of the theory of fractional differential equations, one can refer to [15][16][17][18][19][20]. In order to establish existence theory, researchers have used diverse techniques of nonlinear analysis consisting of fixed-point theory, hybrid fixed-point theory, topological degree theory, and measure of noncompactness [21][22][23][24]. However, the use of the monotone iterative technique (MIT) along with the method of upper and lower solutions (u-l solutions) for solving a BVP involving the operator remains rare.
In the present paper, we are interested in the MIT blended with the method of upper and lower solutions to prove the existence of extremal solutions for the following BVP of an FDE involving the operator where C D σ;ψ * is the operator (1) of order 0 < σ, ν ≤ 1, I σ;ψ * is the operator (2), the function V : [t 1 , t 2 ] × R → R is continuous, λ and δ are real constants, and η ∈ (t 1 , t 2 ). It is worth mentioning that the MIT is efficiently used in the literature to investigate the existence of extremal solutions to many applied problems of nonlinear equations [25][26][27][28][29][30][31][32][33][34][35][36][37][38]. The rest of this paper is organized as follows. In Section 2, we recall some preliminary concepts, definitions, and lemmas that will act as prerequisites to proving the main results. The main results are stated and proved in Section 3. Finally, we give numerical examples to illustrate the correctness of the outcome.

Main Results
First, we start the following key fixed-point theorem.
Theorem 1 ([16,17]). Consider ı ⊂ O of an ordered Banach space B and a nondecreasing mapping u : ı → ı. If each sequence [u n ] ⊂ u(ı) converges whenever [ n ] is a monotone sequence in ı, then the sequence of the u-iteration of t 1 converges to the least fixed point * of u, and the sequence of the u-iteration of t 2 converges to the greatest fixed point * of u. Moreover, * = min{t ∈ ı : t ≥ ut}, and * = max{t ∈ ı : t ≤ ut}.
In fact, a function ∈ C(ı, R) is said to be a solution of Equation (3) if satisfies the we prove the the next key lemma of a solution for problem (3).
Proof. Lemma 4 implies that the problem (9) has the following unique solution: Moreover, Lemma 5 follows from (13).

Definition 1.
A function 0 ∈ C(ı, R) and 0 ∈ C(J, R) is said to be a lower solution (l-solution) and upper solution (u-solution) of problem ∀t ∈ ı, respectively.
This completes the proof.

Some Relevant Examples
Example 1. Consider the following problem: where and V : ı × R → R is given by for t ∈ ı, ∈ R. We take 0 (t) = 0 as the lower solution and 0 (t) = 1 + 2t as the upper solution of problem (21), and we take 0 ≤ 0 for t ∈ ı. So, (H1) of Theorem 2 holds. Now, we consider three cases for ψ * : Note that ψ * 1 (t) = t and ψ * 3 (t) = √ t give the Caputo and Katugampola (for ρ = 0.5) derivatives in this example.

Conclusions
In this study, we investigated the existence of solutions for a nonlinear FDE in the frame of the ψ−Caputo derivative with integral boundary conditions. To prove the main theorems, the monotone iterative and the upper-lower solution techniques in the sense of the ψ−Caputo fractional operator were used. Based on certain conditions, we constructed mis that uniformly converged to the extremal solutions of BVP. The results were tested by constructing two equations corresponding to BVP (3). Different values for ψ, such as the ψ * (t) = t, Caputo, ψ * (t) = 2 t , ψ * (t) = √ t, and Katugampola (for ρ = 0.5) derivatives and the upper and lower solutions, were examined and illustrated for the purpose of verification. We conclude that the results reported in this paper are of great significance for the relevant audience and can be applied to different types of fractional differential problems.   (ψ, v), v, e) * (eval(subs(ψ, v, t)) − subs(ψ, v, e)) ( n − σ − 1) * eval(subs(di f f ( , v, n), v, e)), t 1 , t); 9: E = 1/γ(n − σ) * F; 10: end if 11: mathbbD = E; 12: return mathbbD