On the positive solutions for IBVP of conformable differential equations

1.   Introduction

Since the inception of the fractional derivatives by the scientist Leibniz in the year 1695, this type of derivative has known development in all branches of mathematics and even included applications in engineering and science (see ^{[3,10,16,18,19,20,21]}).

Recently, a new definition has drawn much interest from many researchers, namely conformable fractional derivative introduced in ^[14] by Khalil et al. Since that time, several equations and applications have been studied and several articles have been published regarding this type of derivative (see ^{[1,4,5,6,7,9,12,17,23,24,25]}). Many researchers study existence and positivity problems using the upper and lower solutions technique due to its effectiveness and good results (see ^{[2,8,11,13,15,22]}). The upper and lower solutions method is associated with the use of fixed point theory to prove the existence and uniqueness of the solution.

Xu and Sun ^[22] proved the existence of positive solutions for the IBVP of the fractional differential equations

where $D^{s}$ is the standard Riemann-Liouville derivative and $s\in \left(1, 2\right]$.

Zhong and Wang ^[25] used the fixed point theorem in a cone to show the existence of positive solutions of the BVP

where $s\in \left(1, 2\right]$, $T_{s}$ denotes the conformable derivative of order $s$ and $\lambda$ is positive number.

The purpose of this paper is to examine an integral boundary value problem of conformable differential equations defined as follows

where $s\in \left(1, 2\right]$, the functions $p, q:\left[0, 1\right] \times \left[0, \infty \right) \rightarrow \left[0, \infty \right)$ are continuous such that $q\left(r, y\right)$ is non-decreasing on $y$.

In this context, the main contributions of this paper is to apply the conformable derivative introduced in ^[14] to an integral boundary value problem which is the generalization of the problem (1.2). So, our study is organized as follow. After recalling some definitions and results of the conformable derivative in Section 2, and we give, in Section 3, the proof of our results concerning the existence and uniqueness of positive solutions. In Section 4, we write the conclusion in which we explain the contribution of this research.

2.   Preliminaries

The purpose of this section is to provide the most important materials and preliminaries results for understanding conformable derivatives, (See ^[1,14,25]).

Definition 2.1. ^[1,14] Let $s\in \left(1, 2\right]$. The conformable fractional derivative of a function $p:\left[0, \infty \right) \rightarrow \mathbb{R}$ of order $s$ is defined by

If $T_{s}p\left(r\right)$ exists on $\left(0, \infty \right)$, then $T_{s}p\left(0\right) = \lim_{r\rightarrow 0^{+}}T_{s}p\left(r\right)$.

Definition 2.2. ^[1,14] Let $s\in \left(m, m+1\right]$, $m\in \mathbb{N} _{0}$ and function $p:\left[0, \infty \right) \rightarrow \mathbb{R}.$

(a) The conformable fractional derivative of a function $p$ of order $s$ is defined by

(b) The fractional integral of a function $p$ of order $s$ is defined by

Lemma 2.1. ^[1,25] Let $s$ be in $\left(m, m+1\right]$.

(a) If $p$ is a continuous function on $\left[0, \infty \right)$, then, for all $r > 0$, $T_{s}I_{s}p\left(r\right) = p\left(r\right).$

(b) $T_{s}r^{k} = 0$ for $r$ in $\left[0, 1\right]$ and $k = 0, 1, 2, ..., m$.

(c) If $T_{s}p\left(r\right)$ is continuous on $\left[0, \infty \right)$, then

for some real numbers $c_{k}, k = 0, 1, 2, ..., m$.

Lemma 2.2. Let $y\in C\left(\left[0, 1\right], \mathbb{R} \right)$. Then $y$ is a solution of (1.3) if and only if

where

Proof. Note that

So

Then, by Lemma 2.1 we have

the boundary conditions $y\left(0\right) = 0$, implies $c_{0} = 0$ and

Hence

So,

Lemma 2.3 (^[25]). For any $\left(r, \zeta \right)$ in $\left(0, 1\right] \times \left(0, 1 \right]$,

where $\omega \left(r\right) = r\left(1-r\right)$.

3.   Main results

Let the norm

of the Banach space $X = C\left(\left[0, 1\right] \right)$. Denote $\Omega : = \left\{ y\in X:y\left(r\right) \geq 0, r\in \left[0, 1\right] \right\}$.

Let $a, b\in \mathbb{R} ^{+}$ with $a < b$. For any $y\in \left[a, b\right]$, we define the upper control function with

and the lower control function with

Clearly, $U\left(r, y\right)$ and $L\left(r, y\right)$ are monotonous non-decreasing on $y$ and

We need the following hypothesis:

${\rm{(A)}}$ Let $\underline{y}\left(r\right), \overline{y}\left(r\right) \in \Omega$ with $a\leq \underline{y}\left(r\right) \leq \overline{y} \left(r\right) \leq b$ and

for all $r\in \left[0, 1\right]$.

The function $\underline{y}\left(r\right)$ called the lower solution and the function $\overline{y}\left(r\right)$ is the upper solution of (1.3).

We need the following Lemma in the proof of the Theorem below

Lemma 3.1. For each $r_{1}, r_{2}\in \left[0, 1\right]$, $r_{1} < r_{2}$, the function $\mathcal{Q}$ defined by (2.1) satisfies

Proof. Let $\zeta\in \left[0, 1\right]$, for each $r_{1}, r_{2}\in \left[0, 1\right]$, such that $r_{1} < r_{2}$ we have three cases:

Case 1. For $0\leq \zeta \leq r_{1} < r_{2}\leq 1$, we have

Case 2. For $0\leq r_{1} < r_{2}\leq \zeta \leq 1$, we have

Case 3. For $0\leq r_{1} < \zeta \leq r_{2}\leq 1$

Theorem 3.1. The problem (1.3) has at least one positive solution $y\in \Omega$ if (A) holds. Furthermore,

Proof. Let

It is easy to see that $\left\Vert y\right\Vert \leq b$, so $\Sigma \subset X$ is closed, convex and bounded. If $y\in \Sigma$, $\exists R_{p}, R_{q} > 0$ two constants such that

and

From Lemma 2.2 we define the operator $\digamma$ as

The continuity of $p$ and $q$ give the continuity of the operator $\digamma$ on $\Sigma$. Then, for $y\in \Sigma$ we have

Hence $\digamma \left(\Sigma \right)$ is uniformly bounded.

Now, for each $y\in \Sigma$, $r_{1}, r_{2}\in \left[0, 1\right]$, $r_{1} < r_{2}$, we get

Therefore, $\digamma \left(\Sigma \right)$ is equicontinuous. By Ascoli-Arzele Theorem, $\digamma :\Sigma \rightarrow X$ is compact.

Next we will show that $\digamma \left(\Sigma \right) \subset \Sigma$. Let $y\in \Sigma$, then from the hypothesis (A) we get

Similarly, $\left(\digamma y\right) \left(r\right) \geq \underline{y} \left(r\right)$.

As a conclusion, by the Schauder fixed point theorem, $\digamma$ has at least one fixed point, $y\in \Sigma$. So, the Eq (1.3) has at least one positive solution for all $y\in X$ and $\underline{y}\left(r\right) \leq y\left(r\right) \leq \overline{y}\left(r\right)$, $\ $for all $r\in \left[0, 1\right]$.

Corollary 3.1. Assume that there exist continuous functions $h_{1}$, $h_{2}$, $h_{3}$ and $h_{4}$ such that

and at least one of $h_{1}\left(r\right)$ and $h_{3}\left(r\right)$ is not identically equal to $0$. Then (1.3) has at least one positive solution $y\in X$ and

Proof. Consider the problem

which is equivalent to

By the definitions of control function, we have

where $a, b$ are minimal and maximal of $x\left(r\right)$ on $\left[0, 1 \right]$. Therefore we have

Thus (3.5) is an upper solution of (1.3). On the other hand, we can prove

is a lower solution of (1.3). According to Theorem 3.1, (1.3) has at least one positive solution $y\in X$ and we obtain (3.3).

Corollary 3.2. Suppose that

(i) (3.2) and $0\leq h_{1}\left(r\right) \leq p\left(r, z\right), r\in \left[0, 1\right]$ hold,

(ii) $p\left(r, z\right)$ uniformly converges to $h\left(r\right)$ on $\left[0, 1\right]$ as $z\rightarrow \infty$,

(iii) at least one of $h_{1}\left(r\right)$ and $h_{3}\left(r\right)$ is not identically equal to $0$.

Then (1.3) has at least one positive solution $y\in X$.

Proof. From (ⅱ), there exist $\eta$, $K > 0$ such that

hence

Let $v = \max_{\left(r, z\right) \in \left[0, 1\right] \times \left[0, K\right] }p\left(r, z\right)$, hence

From Corollary 3.1, the IBVP (1.3) has at least one solution, $y\in X$, satisfies

Theorem 3.2. Let (A) holds and assume that for any $r\in \left[0, 1\right]$, $z, z^{\ast }\in \Sigma$,

where $L_{p}, L_{q} > 0$ are constants satisfie

Then the IBVP (1.3) has a unique positive solution on $\Sigma$.

Proof. We show in Theorem 3.1 that $\digamma :\Sigma \rightarrow \Sigma$. So, for any $r\in \left[0, 1\right]$, $z, z^{\ast }\in \Sigma$, we have

Since (3.6) is hold, then $\digamma$ is a contraction mapping that has unique fixed point $y\in \Sigma$. Therefore, the IBVP (1.3) has a unique positive solution on $\Sigma$.

In order to illustrate our results, we provide an example.

Example 3.1. Consider the IBVP

where $q\left(r, y\right) = \frac{\pi }{2}+r+\tan ^{-1}y\left(r\right)$, $p\left(r, y\right) = r^{3}+\frac{ry\left(r\right) }{4+y\left(r\right) }$. We can find $q$ is non-decreasing on $y$, and

for $\left(r, y\right) \in \left[0, 1\right] \times \left[0, \infty \right)$.

So, the IBVP (3.7) has at least one solution according to the above Corollaries. In addition, we have

this implies $L_{q}+\frac{L_{p}}{s\left(s+1\right) } = \frac{1}{2}+\frac{ \frac{1}{4}}{\frac{7}{4}\left(\frac{7}{4}+1\right) } < 1$, so the IBVP(3.7) has a unique positive solution due to Theorem 3.2.

4.   Conclusions

In this paper, we study an integral boundary problem with a conformable fractional derivative, such that our problem is more general than the problem studied in ^[25], so if $q\left(r, y\left(r\right) \right) \equiv 0$ or constant then we obtain results for the problem (1.2). Especially, the uniqueness has not been studied in the work ^[25].

The method of upper and lower solutions is more applicable and easily used for more general problems. Also, the fixed point theorems play an important role to show the existence and the uniquenes.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no competing interests