In recent years, there are few studies on boundary value problems of conformable fractional differential equations under new definitions    . 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   (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     . 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    (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     . 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 for which is defined in  , and , , , , is continuous.
In this section, we present some definitions and lemmas which will be used in the proof of our main results.
Definition 2.1. (See  ) Given a function . Then the conformable fractional derivative of x of order is defined by
for all , . If the conformable fractional derivative of x of order exists, then we simply say that x is δ-differentiable. If x is δ-differentiable in some , , and exists, then we define
Definition 2.2. Let , then are said to be coupled lower and upper solutions of (1), respectively, if
Definition 2.3. Let , then the function pair is said to be coupled solutions of (1), if
Let , then is said to be minimum and maximum coupled solutions of (1), if are coupled solutions of (1), and for any coupled solution .
Lemma 2.1. (See  ) Let , and assume to be δ-differentiable, then
for , .
Lemma 2.2. (See  ) If x is differentiable, , then .
Lemma. 2.3 (See  ) If exists, then for , we have .
Lemma 2.4. Assume that , and , , , Define function as follows:
Then is the solution of the initial value problem as follows
Proof Assume that is given by (2), then p is differentiable for , therefore we have
from Lemma 2.2, and subject to the condition
Lemma 2.5. (Comparison Theorem) Let , and the following inequalities hold true
then , for .
Proof Let , then we have for , and we can draw a conclusion from (2.1) and Lemma 2.3.
Theorem 3.1. Assume that are coupled lower and upper solutions of (1.1) with for , let . And if , then the following inequalities hold true
for and . If we take as initial elements, the iterative sequences defined by
are and , then
1) and uniformly and ;
2) are coupled minimal and maximal solutions of (1.1) respectively in D;
3) If is the solution of (1.1) in D, then we have ; i.e., we have
Proof 1). There is a unique solution to the boundary value problem as follows
which is given by
for and from Lemma 2.2 and Lemma 2.3. Where , . Define operator
where operators are given by
respectively. Then the fixed point of operator T in means the coupled solutions of (1).
Here we prove that , and are coupled lower and upper solutions of (1).
And are coupled lower and upper solutions of (1), then we have
for . And by Lemma 2.5, we have
So we can easily get that
from formula (3) and (5). i.e., are coupled lower and upper solutions of (1).
We also get that
from formula (5) and . Similarly, we have . by Lemma 2.5.
Let , then from formula (4), we have that are coupled lower and upper solutions of (1) for any , which is similar to the proof above. And
In summary, we have
for . Therefore, function sequences are uniformly bounded, i.e.,
for and . Because f is continuous, we have
for and . In addition, because that functions and are continuous, we have
if and . Hence, is equicontinuous, we can also get that is equicontinuous similarly.
In summary, by Ascoli-Arzela theorem  , we can prove that are convergent because of the monotonicity of Sequences, i.e., there are two functions, such that
and. Next we take limits on both sides of (4), then from Lebesgue Dominated Convergence Theorem, we have
if. i.e., are coupled solutions of (1).
2) Here we prove that are coupled minimal and maximal solutions of (1) respectively in D.
Assume that are a set of coupled solutions of (1), then the above problem is equivalent to prove that
Whereas, therefore. Assume that for, here we prove that.
And from Definition 2.3, we have that
Then from (3), we get that
In that way, we have
according to Lemma 2.5. By Mathematical Induction, we can get
for. In addition, because of the convergence of iterative sequences, we have
for. Therefore, 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. In conclusion (2) above, let, because that x is the solution of (1) in D, therefore, are a set of coupled solutions of (1). Obviously, x subject to
In summary, Theorem 3.1 is proved.
Theorem 3.2. Assume that is increasing in x on , and, then there exists a unique solution of (1) in.
Proof By Theorem 3.1, we get that and. And we have for. Then we have that for. Here we prove that.
If is increasing in x on, assume that, then we have
considering the convergence of iterative sequences. Let, then we have by Lemma 2.2, i.e., the function is monotonically decreasing. Hence, , therefore, we draw a contradiction from the conclusion that, which can be obtained from the condition and the boundary value conditions above. Therefore, we have, i.e., is the solution of (1).
On the basis of (1), we can also consider the existence of solutions of boundary value problems for the following uniform fractional differential equations:
where is the conformable fractional derivatives of order for which is defined in  , and, , , is 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, 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).
To illustrate our main results, we present the following example.
Example 4.1. Consider the boundary value problem of conformable fractional differential equations under the following new definitions
It is obvious that are coupled lower and upper solutions of (7), and from the condition, we can get that there exists a constant for, such that the formula (4) of Theorem 3.1 holds. Hence, problem (4) has at least one solution for by Theorem 3.2.
Example 4.2. Consider the boundary value problem of conformable fractional differential equations under the following new definitions
where, it is easy to get that
which yield to
Therefore, are coupled lower and upper solutions of (8), it is obvious that the formula (4) of Theorem 3.1 holds. Hence, problem (8) has at least one solution for by Theorem 3.2.
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