THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution §Corresponding author. This is an Open Access article in the “Special Issue on Fractal and Fractional with Applications to Nature” published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited. 2040025-1 Fr ac ta ls 2 02 0. 28 . D ow nl oa de d fr om w or ld sc ie nt if ic .c om by 3 4. 21 9. 60 .4 2 on 1 2/ 01 /2 1. R eus e an d di st ri bu tio n is s tr ic tly n ot p er m itt ed , e xc ep t f or O pe n A cc es s ar tic le s. December 14, 2020 18:2 0218-348X 2040025 J. Jiang, J. L. G. Guirao & T. Saeed are obtained for the variable order fractional linear differential equations according to Arzela– Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.


INTRODUCTION
It is well known that constant order fractional calculus has been viewed as a basic tool to describe the natural phenomenon of the practical problems in the recent years. [1][2][3][4][5] However, there exist many complex behavior in engineering practice which cannot be described by the constant order fractional order models. Thus, the variable-order (VO) fractional operator is proposed to model the complex phenomena.
The VO fractional operator originated at the end of the 20th century. In 1998, Lorenzo and Hartley proposed the concept of VO fractional operator. 6 In 2002, the concept is discussed more deeply, which connects with the physical process. 7 Then, the VO fractional operator was successfully applied to model the practical problems in the engineering fields. It attracted extensive attention between researchers.
In the recent years, the theory of VO fractional derivatives and integrals has obtained a wide range of study, but it is still in the infancy at present. At the turn of the century, it is used to simulate the temporal or spatial correlation phenomena. 8,9 According to the concept which is put forward by Lorenzo and Hartley, 6 the VO fractional operator is treated as a generalization of the constant order fractional operator. 10,11 So far, there exist several definitions of VO fractional derivative which includes Riemann-Liouvile, Caputo, Marchaud, 12 and Coimbra type fractional operator definition, 13 each of which have a special meaning to meet the desired objectives. 14,15 With the deepening of research, a series of research hotspots are emerging such as the existence, uniqueness of solutions of differential operators with variable fractional order.
In the past several decades, there exist several results on the existence of the solution for constant-order fractional differential equation. [16][17][18][19][20][21][22][23][24][25] Moreover, the boundary value problem has become an important research topic in the area of constant order fractional differential equations. For example, Jiang et al. 26 studied the two-point boundary value problems for fractional differential equation with causal operator by lower and upper solution method and the monotone iterative technique. Shuqin Zhang 27 studied the nonlinear boundary value problem of a VO fractional differential equation, and obtained the existence result. Besides, Zhang 28 obtained the existence results of solutions for the boundary value problem based on the Schauder fixed point theorem. However, there exist a few results on the boundary value problem of a VO differential equation with causal operator by monotone iterative technique. Motivated by this, we apply the monotone iterative technique to investigate boundary value problem of a VO fractional differential equation.
According to the definitions of the VO fractional operator, it is well known that the VO fractional operator is more complex than the constant order fractional operator. Because its kernel is related to a variable exponent. Moreover, the VO fractional operator is irreversible, and thus the VO fractional differential equations can not be simply transformed into the equivalent Volterra integral equations. Thus, this requires that the research on the existence of solutions for VO fractional differential equations needs other methods.
The rest of this paper is presented as follows: In Sec. 2, some definitions and lemmas are listed. In Sec. 3, the necessary inequality and the existence results of the solution are obtained for the VO fractional differential equation. Section 4 gives the theorem of the existence of extremal solution for the VO fractional differential equation. Section 5 illustrates an example to verify the theoretical results.
The Existence of the Extremal Solution for the Boundary Value Problems In this paper, we consider the following system: where 0 < q 1 ≤ q(t, s) ≤ q 2 < 1.

PRELIMINARIES
There exist many kinds of definitions for VO fractional derivative and integral. 7,14,29 In this paper, the Caputo-type definition is adopted due to its extensive application in engineering fields.
. The Caputo-type definition of VO fractional integration is defined as follows: where Γ(·) is the Gamma function, t ∈ [0, T ].
and I q(t,s) t if the integration is point-wise defined.
. The Caputo-type definition of VO fractional derivatives is defined if the integration is point-wise defined as follows: Due to the complexity of the VO fractional operators, generally, the VO fractional calculus D q(t,s) , I q(t,s) does not satisfy the following property: Thus, it requires more technique to deal with the boundary problem of VO fractional system.
Remark 1. If the VO parameter q(t, s) is a constant, then, the above definition of the VO fractional operator is reduced to the definition of the usual constant order Caputo fractional order calculus.
For the fractional order differential equations, Arzela-Ascoli theorem is an important tool in discussing the existence of solutions. Thus, in this study, it is applied to prove the existence of the extremal solution for the VO fractional differential equation.
is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence.

THE EXISTENCE RESULTS OF THE SOLUTION FOR VO FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS
In this section, some necessary theorems that are used to prove the existence of the extremal solutions are provided in the following.
for 0 < q 1 ≤ q(t, s) ≤ q 2 < 1, when q(t, s) = 1, the above inequality is also satisfied. Simultaneously, the following inequalities are true: Proof. Obviously, the following inequality is true: In order to obtain the main results, the process is divided into the following situations.
Through a series of mathematical transformations, it follows that Then, applying the fractional operator I q 2 t 0 + to both sides of the differential inequality (12), it is obtained from the condition (10) that Based on the boundary condition of the inequality (9), then, it results that by integrating the inequality (12) from both sides which implies that the contradiction Based on the inequality (12), for t ∈ [0, T ], it is obtained that Integrating the above inequality from both sides, it implies that Integrating the inequality (12) from 0 to T , then, we can get which implies Thus, integrating the inequality (12) can product the following inequalities: Together with the assumption Based on the inequality (12), it is obtained that and then, we get which is a contradiction. Thus, the proof is completed.
Based on Theorem 3.1, the following general inequality can be obtained.
When q(t, s) = 1, the above inequality (13) is also satisfied. Combined with the following condition: Then Proof. According to inequality (13), we have . Put m(t) = eT t 0 a(s)ds x(t), then, m(t) satisfies the following equations: with Q(t) = q(t)m(t).
Then, by simple calculating, the system (13) is simplified to the form: Based on Theorem 3.1, the proof is completed.
Theorem 3.2 provides a basic tool for the following lemma according to Arzela-Ascoli theorem.
Proof. We claim that there's only one solution at most for the linear problems (16).
In fact, a contradiction is made that it has two different solutions denoted by x 1 , x 2 ∈ C 1 ([0, T ], R).
Set X(t) = x 1 (t) − x 2 (t), then, we have the following problem: According to Theorem 3.2, it is obtained that As the same way, put X(t) = x 2 (t)−x 1 (T ), then, we can get X(t) ≤ 0. Thus, x 1 (t) = x 2 (t), which implies that the linear problem has at most one solution.
In the next step, we will prove that there exists at least one solution for the linear problem (16).
Set v(t) = e t 0 a(s)ds u(t), by the simple calculating, then, (e t 0 a(s)ds θ)(t).
Then the system (16) is transformed into the following form: where δ 1 = δe − T 0 a(s)ds . Based on the property of the VO fractional operator, the following is to prove the existence result by the iteration method as follows: where Now, we will prove that the sequence {v n (t)} is uniformly bounded and equi-continuity.
Assume {v n−1 (t)} is uniformly bounded on [0, T ] and let |v n−1 (t)| ≤ G n−1 , Then, v n (t) satisfies the following inequalities: which implies that the sequence {v n (t)} is uniformly bounded. For 0 ≤ t 1 ≤ t 2 ≤ T , we have the following inequalities: Fractals 2020. 28 s)) ds q(t 1 , s)) ds Assume lim t 1 →t 2 |v n−1 (t 2 ) − v n−1 (t 1 )| = 0. Since the Gamma function Γ(t) is continuous on (0, 1], then, Besides, the continuity of the exponential function results in the following conclusion: And, According to the condition of Theorem 3.3, the operators B and θ * are bounded, then, lim t 1 →t 2 |Q( Thus, we have lim t 1 →t 2 |v n (t 2 ) − v n (t 1 )| = 0, which implies that the set {v n } is equi-continuous. By Arzela-Ascoli theorem, there exists a converge subsequence which converges uniformly to a continuous function v * for t ∈ [0, T ]. Taking the limit from the both sides Eq. (19) when n → ∞, then, we have that v * is the solution of the system (18). The proof is completed. if it satisfies:

THE EXISTENCE OF THE EXTREMAL SOLUTIONS FOR THE BOUNDARY PROBLEM
H(x(0), x(T )) ≤ 0.
Reversely, it is a lower solution of the boundary problem (1).
for each solution x of the boundary value problem (1). Reversely, it is a minimal solution.
We claim that each solution x(t) of the linear boundary problem (20) belongs to the set Due to the condition (H 3 ), we can get that Thus, and According to the condition (H 4 ), it is obtained that Then, Set P (t) = u 0 (t) − x(t), we have the following boundary problem: c D q(t,s) P (t) ≤ −a(t)P (t) − (LP )(t), Thus, in light of Theorem 3.2, we have the inequality P (t) ≤ 0, and then u 0 (t) ≤ x(t), t ∈ [0, T ]. As the same method, it can be showed that In the following step, we construct two kinds of sequences {u n }, {v n } which satisfy the boundary value problem and From the above results of the boundary problem (20), it is obtained that each of the boundary value problem (22) and (23) have a solution in the sector We claim that In fact, the process is divided into the following cases based on the induction method: (i) We show that u 0 ≤ u 1 . Since u 0 is the lower solutions of problem (1), then, we have      c D q(t,s) u 0 (t) ≤ (Qu 0 )(t), c D q(t,s) u 1 (t) = (Qu 0 )(t) − (L(u 1 − u 0 ))(t) −a(t)[u 1 (t) − u 0 (t)].
The boundary condition is listed as It is showed that u 0 ≤ u 1 , for t ∈ [0, T ] according to Theorem 3.2.
Set P n = u n −v n . Similarly, the following inequality is true: