Monotone Iterative Method for ψ -Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

: The main contribution of this paper is to prove the existence of extremal solutions for a novel class of ψ -Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to conﬁrm the validity of our main results.


Introduction
In the mathematical modeling of real life phenomena, the study of fractional differential equations has gained notable importance among interested researchers. It is realized that the use of fractional calculus methods is quite prominent in modeling various processes. The main reason for the widespread use (applications) of fractional operators is the fact that, unlike "integer" operators, these operators possess non-local behavior which enables us to trace the past effects of the involved phenomena [1][2][3][4][5]. Based on some classical approaches, such as Riemann-Liouville, Caputo, and Hadamard fractional operators, there are many new definitions which have attempted to provide a general platform that includes these classical operators [6][7][8]. For the sake of consolidating these different definitions under one single fractional operator, the ψ-fractional operator has been introduced [9,10]. The main feature of the the ψ-fractional operator is that the function in its integral kernel an be adapted to accommodate other definitions when replacing it by specific functions. Other significant features include non-local behavior and the semigroup property which are clearly preserved. It has been recognized that these types of operators have been successfully used to describe and model many real life phenomena; therefore, several related research works have been produced [11][12][13]. Along with the recent developments in fractional differential equations, researchers have contributed many research studies that discus the solutions' behavior in terms of different types of fractional differential equations [14][15][16][17][18][19][20][21][22][23][24][25][26][27].
There are different types of fractional differential equations involving different fractional operators, which are associated with the various types of initial and boundary conditions that have been investigated by many researchers in various research works [28][29][30]. By exploring the literature, one can figure out that the existence of solutions has been the main target of investigations. In order to prove this, researchers often utilize some fixed point hypothesis along with certain mathematical inequalities. To the best of our observations, the monotone iterative technique combined with the method of upper and lower solutions has not been used to study the existence of solutions for ψ-Caputo fractional differential equation with nonlinear boundary conditions. For more expository details on monotone iterative method, the readers can consult some interesting research works [31][32][33][34][35].
Oriented by the above discussion, we study the following ψ-Caputo fractional differential equation (CpFDE) with nonlinear boundary conditions: for ϑ ∈ Ω := [a, b], where c D τ;ψ a + and c D λ;ψ a + denote the ψ-Caputo fractional derivatives of order τ and λ, respectively, such that τ, λ ∈ (0, 1], σ > 0, F ∈ C(Ω × R 2 , R), G, H ∈ C(R 2 , R). It is worth mentioning here that, unlike the above mentioned relevant works, the CpFDE (1) is subject to nonlinear boundary conditions. The above equation is the deterministic fractional differential equation, therefore, a fractional differential equation with its deterministic solution is only considered in this work without the involvement of any random processes.
The remaining part of the paper is organized as follows: In Section 2, we present some definitions and lemmas that will be used to prove our results. In Section 3, we prove our main results, which conveys the existence of extremal solutions for CpFDE (1). For this purpose, we use the monotone iterative method together with the technique of upper and lower solutions. In Section 4, we apply our results by providing an example and illustrate the solutions of behavior graphically.

Relevant Preliminaries
In the current section, we state some basic concepts of fractional calculus that are related to our work. Let Ω = [a, b], 0 ≤ a < b < ∞ be a finite interval and ψ : Ω → R be an increasing differentiable function such that ψ (ϑ) = 0, for all ϑ ∈ Ω. Definition 1 ( [9]). The Riemann-Lebesgue (RL) fractional integral of order τ > 0 for an integrable function z : Ω → R with respect to ψ is described by the following: where Definition 2 ( [9]). Let ψ, z ∈ C n (Ω, R). The RL fractional derivative of a function z of order n − 1 < τ < n with respect to ψ is given by the following: where n = [τ] + 1, n ∈ N and D ϑ = d dϑ .

Definition 3 ( [9]
). Let ψ, z ∈ C n (Ω, R). The Caputo fractional derivative of z of order n − 1 < τ < n with respect to ψ is defined by the following . From the definition, it is clear that the following is the case.
Some basic properties of the ψ-fractional operators are listed in the following Lemma.
We denote the set X by the following.
Equipped with the norm, we have the following: where x ∞ = max ξ∈Ω |x(ξ)| and one can conclude that (X, · X ) is a Banach space.
Lemma 2. For a given ∈ C(Ω, R), λ, τ ∈ (0, 1] and σ > 0, the linear fractional initial value problem is as follows: for ϑ ∈ Ω is equivalent to the following Volterra integral equation. Moreover, the explicit solution of the Volterra integral Equation (7) can be represented by the following.
Proof. Applying the ψ-Riemann-Liouville fractional integral of order τ to both sides of (6) and by using Lemma 1, we obtain the following.
Hence, we have the following.
The converse can be proven by direct computation. Now, we apply the method of successive approximations in order to prove that the integral Equation (7) can be expressed by the following.
For this, we set the following.
It follows from Equation (11) and Lemma 1 that the following is the case.
In continuing this process, we derive the following relation.
Taking the limit as m → ∞, we obtain the following explicit solution z(ϑ) of the integral Equation (7).
Then, the proof is completed. 1], and σ > 0. If ∆ ∈ C(Ω, R) fulfills the following inequalities: x ∈ R, we allow the following: Then, it follows by Equations (8) and (9) that the conclusion of Lemma 3 holds.
Let (X , T ) be a topological Hausdorff space and g 1 , g 2 : X → R be a lower semicontinuous function and an upper semi-continuous function, respectively. This means that for every r ∈ R, the subsets of the following: are open in X . Suppose that g 1 (x) ≤ g 2 (x) for all x ∈ X and we allow the interval [g 1 , g 2 ] consist of those upper or lower semi-continuous functions h : X → R such that In addition, suppose that the sequence {Ω(h n )} n∈N ⊂ [g 1 , g 2 ] consist of lower semi-continuous functions that increase pointwise to Ω(h) whenever the sequence {h n } n∈N ⊂ [g 1 , g 2 ] consist of lower semi-continuous functions that increase pointwise to h. A similar assumption is made when the sequence {h n } n∈N ⊂ [g 1 , g 2 ] consist of upper semi-continuous functions, which decreases pointwise to h ∈ [g 1 , g 2 ]. In particular, assume that Ω(h) is lower semi-continuous whenever h is lower semi-continuous and that Ω(h) is upper semi-continuous whenever h is so. Then for every n ∈ N, we have the following.
Moreover, the function h prefix is upper semi-continuous and the function h postfix is lower semi-continuous. Consequently, if Ω has at most one fixed point, then and, therefore, this unique fixed point is a continuous function.

Main Results
In this section, we prove the existence of extremal solutions for problem (1). Before proceeding, we provide the definitions of lower and upper solutions of the problem (1).

Definition 5.
A function z 0 ∈ X is called a lower solution of (1), if it satisfies the following: for ϑ ∈ Ω.

Definition 6.
A functionz 0 ∈ X is called an upper solution of (1), if it satisfies the following: for each ϑ ∈ Ω.
Then, there exist monotone iterative sequences {z n } and {z n }, which converge uniformly on Ω to the extremal solutions of (1) in the sector [z 0 ,z 0 ], where Proof. For any z 0 ,z 0 ∈ X, we define the following: and the following as well.
By Lemma 2, we know that (13) and (14) have unique solutions in X that are the following.
Again, this is because the following is the case.
Notice the following inequalities: and the following is the case.
Therefore, z 1 (ϑ) is a lower solution of (1). Analogously, it can be obtained thatz 1 (ϑ) is an upper solution of (1). By the above arguments and mathematical induction, we can show that the sequences z n (ϑ),z n (ϑ), (n ≥ 1) are lower and upper solutions of (1), respectively, and the relations (15) and (16) are true. On the contrary, by employing the earlier arguments, together with Ascoli-Arzela's Theorem, we can show that the following: when n → ∞. Finally, it remains to show that z * andz * are extremal solutions of (1) in [z 0 ,z 0 ]. To conduct this, let z ∈ [z 0 ,z 0 ] be any solution of (1). Suppose for some n ∈ N * that the following is the case: for ϑ ∈ Ω. Setting ∆(ϑ) = z(ϑ) − z n+1 (ϑ). It follows that the following obtains.
Notice the inequalities in the following: and the following.

Illustration
The theoretical outcomes are verified by a particular example with specific values. All the experiments are carried out in MATLAB Ver. 8.5.0.197613 (R2015a) on a computer equipped with a CPU AMD Athlon(tm) II X2 245 at 2.90 GHz running under the operating system Windows 7.
In order to illustrate Theorem 1, we take the following: for ϑ ∈ [0, 1] and obtain.

Conclusions
This paper is devoted to the study of a new type of ψ-Caputo fractional differential equation. The addressed problem is considered in the framework of nonlinear boundary value conditions. Our work is different from the existing results in the literature, which are basically based on fixed point approaches. However, we have proved our main results with the help of monotone iterative techniques along with the method of upper and lower solutions. It is observed that these methods are closely related to the Tarski-Knaster theorem and that, essentially speaking, also results in fixed point results. The main results have been verified and demonstrated by an example with explicit numerical values.