Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.


Introduction
In recent years, there are few studies on boundary value problems of conformable fractional differential equations under new definitions [1] [2] [3].And conformable fractional derivatives not only have good operational properties (Four Operational Rules of Derivatives, Chain Rule and Leibniz Rule), this definition can also construct fractional Newton equation and Euler-Lagrange equation from fractional variational method, this is of great significance to the study of uniform or uniformly accelerated motion of particles and to the solution of Newton's fractional-order mechanical problems [4] [5] (fractional-order harmonic oscillator, fractional-order damped oscillator and forced oscillator).And the method of upper and lower solution for monotone iteration can not only gives the existence theorem, but also determines the value range of the solution.Therefore, this method has gradually become an important method for studying nonlinear differential equations [6] [7] [8] [9].In addition, with the application of anti-periodic boundary value problems in various mathematical models and physical processes has been widely applied, the integral boundaries are also widely used in heat conduction, chemical engineering, groundwater flow, thermoelasticity, plasma physics and other fields.As a result, more and more studies have been made on this kind of problems [10] [11] [12] (anti-periodic boundary value problems, anti-periodic boundary value problems with integral boundaries).However, the indefinite sign of solutions of nonlinear differential equations determines that some problems (anti-periodic boundary value problems and their generalizations) cannot be studied directly by the method of upper and lower solutions for monotone iteration.But the development of nonlinear analysis theory provides a powerful tool for the study of these problems.In the generalized monotone iteration process, the method of coupled upper and lower solution becomes an important method to study this kind of problem by the flexible construction of the comparison theorem [13] [14] [15] [16].Motivated by the above work, in this paper, the existence of solutions for a class of boundary value problems of conformable fractional differential equations under a new definition is proved by using the method of coupled upper and lower solution, and the range of solutions is obtained.Throughout this paper, we consider the existence of solutions of boundary value problems for the following uniform fractional differential equations where ( ) ( ) is the conformable fractional derivatives of order δ for ( ) which is defined in [1], and

Preliminaries
In this section, we present some definitions and lemmas which will be used in the proof of our main results.
Definition 2.1.(See [1]) Given a function Then the con- formable fractional derivative of x of order δ is defined by  , then the function pair ( ) , y z is said to be coupled solutions of (1), if , ρ γ is said to be minimum and maximum coupled solutions of (1), if ( ) , ρ γ are coupled solutions of (1), and for any coupled solution ( ) , and assume 1 2 , x x to be δ-differentiable, then ax bx t ax t bx t x t δ exists, then for 0 t ≠ , we have ( ] Then ( ) p t is the solution of the initial value problem as follows Proof Assume that ( ) p t is given by (2), then p is differentiable for 0 t > , therefore we have

2, and ( )
p t subject to the condition ( ) , and the following inequalities hold true , then we have ( ) , then the following inequalities hold true ( ) ( ) ( ) for , y t z t as initial elements, the itera- tive sequences defined by [ ] . There is a unique solution to the boundary value problem as follows , , 0,1 , [ ] 2 and Lemma 2. 3.Where , , , , , where operators 1 2 , T T are given by ( ) ( [ ] Then the fixed point of operator T in D D × means the coupled solutions of (1).
Here we prove that 0 , , y y z z y z ≤ ≤ ≤ , and 1 1 , y z are coupled lower and upper solutions of (1).

y t f t y t M y t y t t y rz y s s z t f t z t M z t z t t z ry z s s
And 0 0 , y z are coupled lower and upper solutions of (1), then we have

y t y t M y t y t y y z t z t M z t z t z z
. And by Lemma 2.5, we have

. y t y t z t z t t ≤ ≤ ∈
So we can easily get that  3) and (5).i.e., 1 1 , y z are coupled lower and upper solutions of Journal of Applied Mathematics and Physics (1).
We also get that Let ( ) ( )

y t z t M y t z t y z y s z s s r y z
, then from formula (4), we have that , n n y z are coupled lower and upper solutions of (1) for any 2 n ≥ , which is similar to the proof above.And

≤ ≤ ≤
In summary, we have is equicontinuous, we can also get that In summary, by Ascoli-Arzela theorem [17], we can prove that { } { } ∈ .Next we take limits on both sides of (4), then from Lebesgue Dominated Convergence Theorem, we have

T y z T y z T y z y z
, y z are coupled solutions of (1).
2) Here we prove that ( ) * * , y z are coupled minimal and maximal solutions of (1) respectively in D.
Assume that ( ) , x x are a set of coupled solutions of (1), then the above problem is equivalent to prove that Consider that And from Definition 2.3, we have that x In that way, we have , y z are coupled minimal and maximal solutions of (1) respectively in D from Definition 2. 3.
3) Here we prove that if x is the solution of (1) in D, then * * y x z ≤ ≤ .In conclusion (2) above, let ( ) ( ) ( ) , because that x is the solution of (1) in D, therefore, ( ) , x x are a set of coupled solutions of (1).If ( ) On the basis of (1), we can also consider the existence of solutions of boundary value problems for the following uniform fractional differential equations:  + , so the original problem needs to be solved until the solution of the equation of order n before we construct the comparison theorem, which is the difficulty of (6).

X. Y. Jian
In addition, because that functions are convergent because of the monotonicity of Sequences, i.e., there are two fractional derivatives of order δ for continuous.Similarly, the existence of the solution can be proved by the method of coupled upper and lower solution, and the range of the solution can be obtained.Due to ( ] . . .