Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative

1.   Introduction

In this paper, we consider the multiplicity of positive solutions for the following singular tempered fractional equation with lower order tempered fractional derivative

where $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$, $h\in C([0, +\infty)\times[0, +\infty), [0, +\infty))$, $p \in L^1((0, 1), (0, +\infty))$, which implies that the weight function can have infinite many singular points in $[0, 1]$.

The equation (1.1) contains a tempered fractional derivative ${^R _0}{{\mathbb{D}_t}^{\alpha, \lambda}}$, which is actually obtained by multiplying an exponential factor in the Riemann-Liouville fractional derivative ${^R _0}\pmb{\mathscr{D}_t}^{\alpha}$, i.e., the following relationship exists between tempered fractional derivative and Riemann-Liouville fractional derivative

For the definition of the standard Riemann-Liouville fractional derivative and integral, we refer the reader to ^[1,2,3,4,5].

As the optimization of the Riemann-Liouville fractional derivative, the tempered fractional derivative has many advantages, which not only overcomes the defect of using the power law of the classical fractional derivative in the mathematical sense, such as the Riemann-Liouville fractional derivatives ^[6,7], the Caputo fractional derivatives ^[8], Hadamard fractional derivatives ^[9,10,11] etc, but also brings many practical applications. It especially describes the anomalous diffusion phenomena in Brownian motion with the semi-heavy tails or semi-long range dependence, such as the limits of random walk with an exponentially tempered jump distribution ^[12,13], transient super-diffusion ^[14], anomalous diffusions in heterogeneous systems ^[15], exponentially tempering Lévy flights with both the $\alpha$-stable and Gaussian trends ^[16], and pure jump Lévy process with the fractional derivatives in the risk management of financial derivatives traded over the counter ^[17].

In recent work ^[18], an upper and lower solutions technique has been employed to establish the existence of positive solutions for a singular tempered fractional turbulent flow model in a porous medium

with $\alpha\in(0, 1], \ \beta\in (1, 2]$ and the nonlinearity $f$ is decreasing in the second variable. In ^[19], in view of the monotone iterative method, the iterative properties of positive solutions for a tempered fractional equation

were established, where $\vartheta\in(1, 2], \ \delta\in (0, 1)$ with $\vartheta-\delta > 1$, $\mu$ is a positive constant, and $f:(0, 1) \times [0, +\infty)\times[0, +\infty)\to [0, +\infty)$ is a continuous and nondecreasing function with respect to the two space variables. Recently, by using spaces theories ^{[20,21,22,23,24]}, smooth theories ^[25,26,27], operator method ^[28,29], the method of moving sphere ^[30], critical point theories ^{[31,32,33,34]} and so on, some other types of fractional equations was also studied ^{[35,36,37,38,39,40,41,42,43,44,45,46]}.

However, when the nonlinearity contains lower order tempered fractional derivative, the results of multiplicity of positive solutions have not yet been obtained. In order to fill this gap, by applying the reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation are established in this paper. The interesting points are that the nonlinearity contains the lower order tempered fractional derivative and the weight function can have infinitely many singular points in $[0, 1]$.

2.   Preliminaries and lemmas

In this section, we first recall definitions and some useful properties of the Riemann-Liouville fractional derivative and integral.

Definition 2.1 ((2.1.1) on page 69 in ^[1]). The Riemann-Liouville fractional integral of order $\alpha > 0$ of a function $u:(0, +\infty)\rightarrow \mathbb{R}$ is given by

provided that the righthand side is pointwise and defined on $(0, +\infty)$.

Definition 2.2 ((2.1.5) on page 70 in ^[1]). The Riemann-Liouville fractional derivative of order $\alpha > 0$ of a function $u:(0, +\infty)\rightarrow \mathbb{R}$ is given by

where $n = [\alpha]+1$, $[\alpha]$ denotes the greatest integer part of the number $\alpha$, provided that the righthand side is pointwise and defined on $(0, +\infty)$.

The following properties of the Riemann-Liouville fractional derivative and integral can be found on pages 73–75 (Lemmas 2.3–2.5) in ^[1].

Lemma 2.1. Suppose $u(t) \in C[0, 1]\cap L^1[0, 1]$ and $\alpha > \beta > 0$. Let $n = [\alpha]+1$, then

$\textrm{(i)}$

where $c_i \in \mathbb{R}, i = 1, 2, 3, ..., n.$

$\textrm{(ii)}$

Lemma 2.2. Let $p \in L^1([0, 1], (0, +\infty))$, then the singular linear tempered fractional equation

has a unique positive solution $u(t)$ provided that $1 < \alpha-\beta\le2,$ which can be expressed by

where

is the Green function of (2.2).

Proof. In fact, it follows from $1 < \alpha-\beta\le2$, (1.2) and (2.1) that

Since $u(0) = 0$ and $u(1) = 0$, we have $b_2 = 0$ and

Thus

The following Lemma has been proven (see Lemma 2.3 of ^[6] or Lemma 3 of ^[4]).

Lemma 2.3. Suppose $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$, then $H(t, s)$ is a nonnegative continuous function in $[0, 1]\times[0, 1]$ satisfying, for any $(t, s)\in[0, 1]\times[0, 1],$

Suppose $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$. In order to use the reducing order technique, we introduce the following integral transformation

and then consider the following reducing order problem

Lemma 2.4. Suppose $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$. The reducing order problem (2.7) is equivalent to the singular tempered fractional equation (1.1). In particular, if $y$ is a positive solution of the problem (2.7), then $u(t) = e^{-\lambda t}I^{\beta}(e^{\lambda t}y(t))$ is a positive solution of the singular tempered fractional equation (1.1).

Proof. We first suppose that $u$ is a positive solution of the singular tempered fractional equation (1.1). Let

Noticing that $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$, let $n = [\alpha]+1$, i.e., $n$ is the smallest integer greater than or equal to $\alpha$, then it follows from (1.2) and Lemma 2.1 that

and

By (2.8) and (2.9), we have

thus, $y$ solves the equation (2.7).

Conversely, suppose that $y$ is any solution of the reducing order problem (2.7), then we have

Make integral transformation (2.6), similar to (2.8) and (2.9), we have

Substituting the above formulas into (2.7), we get

and

which implies that $u(t) = e^{-\lambda t}I^{\beta}(e^{\lambda t}y(t))$ solves the singular tempered fractional equation (1.1).

Suppose $E = C([0, 1];\mathbb{R})$ with the norm

Define a cone of $E$ and an operator $T$, respectively:

and

In order to obtain the existence of a positive solution of the Eq (1.1), Lemma 2.4 indicates that we only consider the fixed points of the operator $T$.

Now, we list the following hypotheses, which are used in the rest of this paper.

$\mathbf{(C1)} \; h\in C([0, +\infty)\times[0, +\infty), [0, +\infty))$ and $p \in L^1((0, 1), (0, +\infty))$.

$\mathbf{(C2)}$ There is a constant $n > 0$ such that for any $0\le u+v\le \left(1+\frac{e^\lambda}{\Gamma(\beta+1)}\right)n$,

where

$\mathbf{(C3)}$ There is a constant $\rho > 0$ such that for any

it implies

where

Lemma 2.5. Assume $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$ and $(\mathbf{C1})$ holds, then the operator $T : P \to P$ is completely continuous.

Proof. First by $(\mathbf{C1})$, $T$ is continuous on $[0, 1]$. For any $y \in P$, there exists a constant $M > 0$ such that $||y|| \leq M$, then

Let

then by (2.11), for any $y \in P$, we have

which implies that $T : P \to E$ is well-defined. In addition, by (2.5) and (2.12), we have

which implies that $T (P) \subset P.$

In the end, by using the standard arguments and combining the Ascoli-Arzela theorem, we know $T (P) \subset P$ is completely continuous.

Our proof of main results depends on the fixed point theorem of cone expansion and compression (see Theorem 2.3.3 on page 93 of ^[47]).

Lemma 2.6. ^[47] Suppose $P$ is a cone of real Banach space $E$, the bounded open subsets $\Omega_1, \Omega_2$ of $E$ satisfy $\theta\in \Omega_1, \overline{ \Omega}_1\subset\Omega_2$. Let $T: P\cap(\overline{\Omega}_2\setminus \Omega_1)\to P$ be a completely continuous operator such that either

(1) $\|Tz\|\leq \|z\|, z\in P\cap\partial\Omega_1$ and $\|Tz\|\geq \|z\|, z\in P\cap \partial\Omega_2,$ or

(2) $\|Tz\|\geq \|z\|, z\in P\cap\partial\Omega_1$ and $\|Tz\| \leq \|z \|, z\in P\cap \partial\Omega_2;$

then $T$ has a fixed point in $P\cap(\overline{\Omega}_2 \setminus \Omega_1).$

3.   Main results

For the convenience of the proof, we first introduce the following notations whenever the limits exist or not:

and then state our main results as follows.

Theorem 3.1. Assume that $\mathbf{(C1)}$ and $\mathbf{(C2)}$ hold, and

then the tempered fractional equation (1.1) has at least two positive solutions $u_1, u_2$; moreover, there exist four constants $A_1, B_1, A_2, B_2 > 0$ such that

Proof. It follows from $h_0 = +\infty$ that there exist $0 < m < n$ and a sufficiently large constant

such that for any $0 < u+v\le\left(1+\frac{e^\lambda}{\Gamma(\beta+1)}\right)m$,

Take $\Omega_m = \left \{ {y\in E:\left | \left | y \right | \right | < m } \right \}$, and $\partial \Omega _{m} = \left \{ y\in E: \left | \left | y \right | \right | = m \right \}$, then for any $y\in P\cap \partial\Omega_m$, one has

then

Thus,

So, for any $y\in P\cap \partial\Omega_m$, we have $||Ty||\ge ||y||$.

Next, let $\Omega_n = \left \{ {y\in E:\left | \left | y \right | \right | < n } \right \}$ and $\partial \Omega _{n} = \left \{ y\in E: \left | \left | y \right | \right | = n\right \}$, then it follows from (3.2) and $\mathbf{(C2)}$ that for any $y\in P\cap \partial\Omega_n$, one has

Therefore for any $y\in P\cap \partial\Omega_n$, we have $||Ty||\le ||y||$.

On the other hand, it follows from $h_\infty = +\infty$ that there exists $M > 0$ and

such that for any $u+v > M$, we have

Take $R = n+4^{\alpha-\beta} e^{\frac34\lambda}M$ and let $\Omega_R = \left \{ {y\in E:\left | \left | y \right | \right | < R } \right \}$ and $\partial \Omega _{R} = \left \{ y\in E: \left | \left | y \right | \right | = R\right \}$, then for any $y\in P\cap \partial\Omega_R$ and $[\frac14, \frac34]$, one has

Thus, for any $y\in P\cap \partial\Omega_R$, we have

Therefore, for any $y\in P\cap \partial\Omega_{R}$, we have $||Ty||\le ||y||$.

According to Lemma 2.6, $T$ has two fixed points $y_1\in P\cap (\overline{\Omega}_m\backslash \Omega_n)$ and $y_2\in P\cap (\overline{\Omega}_{R}\backslash \Omega_{n})$ with $m\le||y_1||\le n\le ||y_2||\le R$. Thus, it follows from Lemma 2.4 that the tempered fractional equation (1.1) has at least two positive solutions satisfying

and

Theorem 3.2. Assume that $\mathbf{(C1)}$ and $\mathbf{(C3)}$ hold and

then the tempered fractional equation (1.1) has at least two positive solutions $u_3, u_4$; moreover, there exist four constants $A_3, B_3, A_4, B_4 > 0$ such that

Proof. First, notice that $h_0 = 0$, for any $\epsilon > 0$. Let us select $0 < \kappa < \rho$ such that for any $0 < u+v < \left(1+\frac{e^\lambda}{\Gamma(\beta+1)}\right)\kappa$,

Choose $\epsilon$ such that

Now, let $\Omega_\kappa = \left \{ {y\in E:\left | \left | y \right | \right | < \kappa } \right \}$ and $\partial \Omega _{\kappa} = \left \{ y\in E: \left | \left | y \right | \right | = \kappa\right \}$, then for any $y\in P\cap \partial\Omega_\kappa$, the same as (3.2), we also have

which implies that for any $y\in P\cap \partial\Omega_\kappa$, the following estimation is valid:

Consequently, for any $y\in P\cap \partial\Omega_\kappa$, one gets

(3.7) implies that $||Ty||\le ||y||, \ \ y\in P\cap \partial\Omega_\kappa$.

Next, let $\Omega_\rho = \left \{ {y\in E:\left | \left | y \right | \right | < \rho } \right \}$, and $\partial \Omega _{\rho} = \left \{ y\in E: \left | \left | y \right | \right | = \rho\right \}$, then for any $y\in P\cap \partial\Omega_\rho$, we have

Thus, it follows from (3.8) that for any $y\in P\cap \partial\Omega_\rho$ and $[\frac14, \frac34]$, one has

So, for any $y\in P\cap \partial\Omega_\rho$, we have

which implies that $||Ty||\ge ||y||$ holds and $y\in P\cap \partial\Omega_\rho$.

On the other hand, since $h_\infty = 0$, for any $\epsilon > 0$, there exists $M^* > 0$ such that for any $u+v > M^*$

For the above $\epsilon > 0$, choose a sufficiently small one such that

and take

Next, let $\Omega_{R^*} = \left \{ {y\in E:\left | \left | y \right | \right | < {R^*} } \right \}$, and $\partial \Omega _{{R^*}} = \left \{ y\in E: \left | \left | y \right | \right | = {R^*}\right \}$, then for any $y\in P\cap \partial\Omega_{R^*}$, one has

Therefore, for any $y\in P\cap \partial\Omega_R$, we have $||Ty||\ge ||y||$.

According to Lemma 2.6, $T$ has two fixed points $y_3\in P\cap (\overline{\Omega}_\rho\backslash \Omega_\kappa)$ and $y_4\in P\cap (\overline{\Omega}_{R^*}\backslash \Omega_\rho)$ with $\kappa\le||y_3||\le \rho\le ||y_4||\le R^*$. Thus, it follows from Lemma 2.4 that the tempered fractional equation (1.1) has at least two positive solutions satisfying

and

4.   Examples

The tempered fractional diffusion equation has many important applications, including the tempered fractional Langevin and Vasicek differential equations ^[48] and the space-time tempered fractional diffusion-wave equation ^[49]. The new theorems established in the present paper are very useful in the area of tempered fractional calculus. We can find the multiple positive solutions for the singular tempered fractional equations with the lower order tempered fractional derivatives using the proposed theorems.

Let us apply the main results to solve two singular tempered fractional equations with lower order tempered fractional derivative.

Example 4.1. Let $\alpha = \frac53, \ \beta = \frac13, \ \lambda = 3,$ and

We consider the multiplicity of positive solutions for the following singular tempered fractional equation with the lower order tempered fractional derivative:

Conclusion. The singular tempered fractional equation (4.1) has at least two positive solutions $u_1, u_2$: moreover, there exist four constants $A_1, B_1, A_2, B_2 > 0$ such that

Proof. Here,

Clearly, $(\mathbf{C1})$ holds and

In the following, we verify the condition $(\mathbf{C2})$. In fact, take $n = 2$, then

and for any $0\le u+v\le \left(1+\frac{e^\lambda}{\Gamma(\beta+1)}\right)n = 2\left(1+\frac{e^3}{\Gamma(\frac43)}\right) = 46.9853$, we have

which implies that $(\mathbf{C2})$ holds.

Consequently, it follows from Theorem 3.1 that the tempered fractional equation (4.1) has at least two positive solutions $u_1, u_2$; moreover, there exist four constants $A_1, B_1, A_2, B_2 > 0$ such that

Example 4.2. Let $\alpha = \frac53, \ \beta = \frac13, \ \lambda = 3,$ and

We consider the multiplicity of positive solutions for the following singular tempered fractional equation with the lower order tempered fractional derivative

Conclusion. The tempered fractional equation (4.2) has at least two positive solutions $u_1, u_2$; moreover, there exist four constants $A_1, B_1, A_2, B_2 > 0$ such that

Proof. Here

Clearly, $(\mathbf{C1})$ holds and

Take $\rho = 2$, and we have

For any

one has

which implies that $(\mathbf{C3})$ holds.

Consequently, it follows from Theorem 3.2 that the tempered fractional equation (4.2) has at least two positive solutions $u_1, u_2$; moreover, there exist four constants $A_1, B_1, A_2, B_2 > 0$ such that

5.   Conclusions

This work studies the multiplicity of positive solutions for a class of singular tempered fractional equations with the lower order tempered fractional derivative. By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the equation are established. The interesting points are the nonlinearity contains the lower order tempered fractional derivative and the weight function can have infinitely many singular points in $[0, 1]$. However, in this study, the conditions $\alpha\in (1, 2], \beta\in (0, 1)$ with $\alpha-\beta > 1$ are required; if $0 < \alpha-\beta < 1$ or $h$ has singularity at space variables, these interesting problems are still worth future studying.

Use of AI tools declaration

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

Acknowledgments

This work is granted by the Natural Science Foundation of Shandong Province of China (ZR2022AM015) and an ARC Discovery Project Grant.

Conflict of interest

Xinguang Zhang is a guest editor of the special issue for ERA and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.