The Extremal Solution To Conformable Fractional Differential Equations Involving Integral Boundary Condition

In this article, by using the monotone iterative technique coupled with the method of upper and lower solution, we obtain the existence of extremal iteration solutions to conformable fractional differential equations involving Riemann-Stieltjes integral boundary conditions. At the same time, the comparison principle of solving such problems is investigated. Finally, an example is given to illustrate our main results. It should be noted that the conformal fractional derivative is essentially a modified version of the first-order derivative. Our results show that such known results can be translated and stated in the setting of the so-called conformal fractional derivative.


Introduction
Fractional calculus is an excellent tool for the description of the process of mathematical analysis in various areas of finance, physical systems, control systems and mechanics, and so forth [1][2][3][4][5].Many methods are used to study various fractional differential equations, such as fixed point index theory [6], iterative method [7][8][9], theory of linear operator [10,11] sequential techniques, and regularization [12], fixed point theorems [13][14][15][16][17], the Mawhin continuation theorem for resonance [18][19][20][21][22], the variational method [23].The definition of the fractional order derivative used in the aforementioned results is either the Caputo or the Riemann-Liouville fractional order derivative.Recently, Khalil et al. [24] gave a new simple fractional derivative called "the conformable fractional derivative" depending on the familiar limit definition of the derivative of a function and that break with other definitions.This new fractional derivative is called "the conformable fractional derivative" and this new theory is improved by Abdeljawad [25].However, a conformal fractional derivative is not a fractional derivative, it is simply a first derivative multiplied by an additional simple factor.Therefore, this new definition seems to be a natural extension of the classical derivative.However, it has the advantage of being different from other fractional differentials.Firstly, it can integrate the standard properties of fractional derivatives.It is suitable for many extensions to the classical theorem of calculus, such as the derivative of the product and compound of two functions, the Rolle's and the mean value theorem, conformable integration by parts, fractional power series expansion and many more.Secondly, the conformal fractional derivative of the real function is zero, and the Riemann-Liouville fractional derivative does not satisfy this property.For the two iterative conformal differentials, the semigroup property is not satisfied, and the Caputo differential satisfies this.In particular, for functions that are not differentiable, in conformal sense; however, the function is differentiable.Some functions are not infinitely differentiable at some points; where there is no Taylor power series expansion, in conformal calculus theory, they do exist.This led many people wanting to explore it.Please see [26][27][28][29][30][31][32][33][34] for recent developments on conformable differentiation.For example, a mean value theorem of the conformable fractional calculus on arbitrary time scales is proved in [33], and whose results reconciled with familiar classical results when the operator T α is of order α = 1 and the time scale coincides with the set of real numbers.In [34], Asawasamrit and Ntoutas introduced a new definition of exponential notations and by employing the method of lower and upper solution combined with the monotone iterative technique, some new conditions for the existence of solutions are presented.
Motivated by the above works, we consider the existence of solutions for the following nonlinear conformable fractional differential equation involving integral boundary condition, using the method of upper and lower solutions and its associated monotone iterative technique where x(t)dµ(t) denotes the Riemann-Stieltjes integral with positive Stieltjes measure of µ, and D α f (t) stands for the conformable fractional derivative.Based on a comparison result, two monotone iterative sequences are obtained using the upper and lower solutions, and these two sequences approximate the extremal solutions of the given problem.For applications of the method of upper and lower solutions and monotone iterative technique to differential equations and differential systems such as ordinary differential equations [35][36][37], ordinary differential systems [38], fractional differential equations [39][40][41][42], fractional differential systems [43].
Then there exists a constant ξ ∈ (a, b), such that If inequalities (2), (3) are reversed, then u is an upper solution of problem (1).
Next, we present the following existence and uniqueness results for linear equations.
Proof.Multiplying both sides of the first equation of the problem ( 4) by e t 0 s α−1 M(s)ds and using Lemma 2, we can get In other words, by means of the product rule (item (iv) of Lemma 2), the above equality turns to ( Applying the conformable fractional integral of order α to both side of (5), we have Then From the boundary condition of (4), we have On account of condition α = 0, then , thus problem (4) has a unique solution.The proof is finished.
In the next Lemma, we discuss comparison results for the linear problem which play a key role in the proof of the main result.
The proof is complete.

Main Results
In this section, we prove the existence of extremal solutions for conformable fractional differential equations involving integral boundary condition.For convenience, we list some assumptions.
According to (H 3 ) and Definition 3, we deduce that v 1 is a lower solution of (1).Similarly, w 1 is a upper solutions of (1).By the above arguments and mathematical induction, it is clear that Thirdly, we show that lim n→∞ v n = v, lim n→∞ w n = w.Hence, we need to conclude that v n , w n are uniformly bounded and equicontinuous on [0, 1].Obviously, the uniform boundedness of sequences v n , w n follows from (8).Thus, there exists L > 0 such that Using Theorem 1, we get where Let n → ∞ in (9) .We have This shows that v is a solution of the nonlinear problem (1).Similarly, we obtain w is a solution of the nonlinear problem (1) too.And Finally, we are going to prove that v, w are minimal and maximal solutions of (1) in the sector [v 0 , w 0 ].In the following, we show this using induction arguments.Suppose that g(t) is any solution of (1) in the [v 0 , w 0 ] that is ). Let ), ), ).
Then v 0 , w 0 are lower and upper solutions of (10).When M(t) = 1, it is easy to verify that assumption (H 3 ) holds.In addition, By Theorem 2, problem (10) has an extremal iterative solution in [v 0 , w 0 ].

Conclusions
In this article, on the integral boundary value problem for conformable fractional differential equations, we use the monotone iterative technique to investigate the existence results for extremal solutions for Equation (1).At the same time, two sequences are obtained using the upper and lower solutions, and these two sequences approximate the extremal solutions of nonlinear differential equations.It is clear that the method of using the upper and lower solutions is a very effective method for studying the solvability of conformable fractional differential equations.However, almost all the results derived in the paper are more-or-less straightforward extensions of well-known results from the theory of the first-order differential equations, since the conformal fractional derivative is essentially a modified version of the first-order derivative.
By taking n → ∞ in the above inequalities, we get that v ≤ g ≤ w.That is v, w are extremal solutions of problem (1) in [v 0 , w 0 ].Thus, the proof is finished.Example 1.Consider the following nonlinear problem: