Kyungpook Mathematical Journal 2021; 61(1): 139-153
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Existence of Positive Solutions for a Class of Conformable Fractional Differential Equations with Parameterized Integral Boundary Conditions
Department of Physics, University of Sciences and Technology of Oran-MB, El Mnaouar, BP 1505, 31000 Oran, Algeria Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, 31000 Oran, Algeria
e-mail : firstname.lastname@example.org
Received: June 10, 2019; Revised: June 2, 2020; Accepted: June 2, 2020
In this paper, we study the existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions. By using the properties of Green's function with the fixed point theorem in a cone, we prove the existence of a positive solution. We also provide some examples to illustrate our results.
Keywords: conformable fractional derivatives, integral boundary value problems, positive solutions, fixed point theorems, cone.
Fractional calculus and fractional differential equations are recently experiencing rapid development. There are several notions of fractional derivatives, some classical, such as the Riemann-Liouville or Caputo definitions, and some novel, such as conformable fractional derivatives , β-derivatives , or others [12, 20]. Recently, the new definition of a conformable fractional derivative, given by [1, 2, 18], has drawn much interest from many researchers [6, 7, 17, 22, 23, 24, 26]. Recent results on conformable fractional differential equations can also be found in [3, 8, 11].
In 2017, X. Dong et al. studied the existence and multiplicity of positive solutions for the following conformable fractional differential equation with
In , the authors considered the following three-point boundary value problem for a conformable fractional differential equation
In , D. R. Anderson et al., considered the following conformable fractional-order boundary value problem with Sturm-Liouville boundary conditions
In a recent paper , using the well-known topological transversality theorem, L. He et al., obtained the existence of solutions for the fractional differential equation
with one of the following boundary value conditions
In the same year, Q. Song et al.  investigated the following fractional Dirichlet boundary value problem
Very recently, in 2018, W. Zhong and L. Wang  discussed the existence of positive solutions of the conformable fractional differential equation
subject to the boundary conditions
where the order α belongs to
Inspired and motivated by the above recent works, we intend in the present paper to study the existence of positive solutions for the boundary value problem of conformable fractional differential equation
For the case of
In Section 2, we present the necessary definitions and we give some lemmas in order to prove our main results. In particular, we state some properties of Green's function associated with BVP (1.1) and (1.2). In Section 3, some sufficient conditions are established for the existence of positive solution to our BVP when
provided the limits of the right side exists.
() Let α be in
Moreover, in checking the second boundary condition, we get
Substituting the value of
The proof is therefore complete.
for all where and
On the other hand, we have
(iii) It follows immediately from (ii).
On the other hand, from (2.10) and Lemma 2.6 for any
From (2.11), we obtain
(C2) There exists
such that for and .
3. Existence Results
Throughout this section, we assume that
, and the parameter
Given a positive number
and also, define the operator
If the hypothesis (H) holds, then
In order to discuss the complete continuity of the operator
where the operators
By Lemma 2.7, it follows that
If the hypothesis (H) holds, then the operator
For convenience, we introduce the following notations
Now, we will state and prove our main results.
Then, by Lemma 2.6 and Lemma 2.7, for each
We next show that the operator
So, by virtue of (3.4), we get
Hence, the condition (C1) in Lemma 2.8 is satisfied. By Lemma 2.8 and Lemma 3.3, the operator
By (3.6) and Lemma 2.6, for
Thus the operator
Now, we show that the operator
Hence, by (3.7), we have
We now choose the function
If the above fact is not true, then there exist a function
Then, by Lemma 2.6 and Lemma 2.7, for each
From Theorem 3.1 and Theorem 3.2, we can obtain the following corollary.
Consider the following boundary value problem
As a second example, we consider the fractional boundary value problem
By simple calculations, we find that
Hence, we get
In addition, we have
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