Initial value problems for fractional p-Laplacian equations with singularity

1.   Introduction and preliminaries

Interest in the subject of fractional differential equations has increased greatly over the past decades. Fractional differential equations appear in many fields such as physics, aerodynamics, electro-dynamics, and control theory (see ^{[1,14,17,21,24,31,32,33]}).

We studied the following singular nonlinear initial value problem involving p-Laplacian

$f:[0, 1]\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\phi_p(t) = |t|^{p-2}t, \, p > 1$, and $D^\alpha_Cx(t)$ denotes the Caputo fractional derivative which is defined by $D^\alpha_Cx(t): = I^{1-\alpha}Dx(t)$ for $0 < \alpha\leq 1$, where $I^\alpha$ stands for the Riemann-Liouville fractional integral of order $\alpha > 0$ defined by

where $L^p: = L^p[0, 1] = \{x(t): \int_0^1|x(t)|^p < \infty, \, 1\leq p < \infty\}$ and $\Gamma(\alpha): = \int_0^\infty s^{\alpha-1}exp(-s)\, ds, \, \alpha > 0$ (see ^{[14,31,32,33]}). A literature review will be given at the end of this section. Since the problem studied in this paper contains a singular term, we believe that the obtained results are new.

We denote the domain of an operator $T$ by $Dom(T)$. The domain of the Caputo derivative is defined by $Dom(D^\alpha_C) = \{x(t): x\in AC[0, 1]\}$, where $AC[0, 1]$ is the set of absolutely continuous functions on the interval $[0, 1]$.

In general, for any $m\in \mathbb{N}$, and $0 < \alpha\leq 1,$ $D^{m+\alpha}_Cx(t) = I^{1-\alpha}D^{m+1}x(t)$ with $Dom(D^{m+\alpha}_C) = \{x(t): D^m x\in AC[0, 1]\}.$

Besides the Caputo derivative, we will also use the Riemann-Liouville and generalized Caputo derivatives.

For $0 < \alpha\leq 1$, the Riemann-Liouville derivative is defined by $D^\alpha x(t): = DI^{1-\alpha}x(t), \, \hbox{with} \, Dom(D^\alpha) = \{x(t): I^{1-\alpha}x\in AC[0, 1]\}$. In general, for any $m\in \mathbb{N}$, and $0 < \alpha\leq 1, \,$ $D^{m+\alpha}x(t) = D^{m+1}I^{1-\alpha}x(t), \, \hbox{with} \, Dom(D^{m+\alpha}) = \{x(t): D^m(I^{1-\alpha}x)\in AC[0, 1]\}$.

For $0 < \alpha\leq 1$, the generalized Caputo derivative is defined by $D^\alpha_*x(t): = D^\alpha(x(t)-x_0), \, \hbox{with} \, Dom(D^\alpha_*) = \{x(t): I^{1-\alpha}(x-x_0)\in AC[0, 1]\}$, where $x_0 = x(0).$

We studied initial value problems on the interval $[0, 1]$, but all results in this paper are also valid for any interval $[0, T]$.

Let us give some properties of $D^\alpha, \, D^\alpha_C, \, D^\alpha_*$, and $I^\alpha$ that we will use frequently in this paper.

Proposition 1.1. $a) \, I^\alpha:L^p[0, 1]\rightarrow L^p[0, 1], \, 1\leq p\leq \infty$ is a bounded operator for all $\alpha > 0$ (^[36], Proposition 3.2).

$b) \, I^\alpha:C[0, 1]\rightarrow C[0, 1]$ is a bounded operator for all $\alpha > 0$ (^[36], Proposition 3.2), where $C[0, 1]$ denotes the normed space of all continuous functions defined on the interval $[0, 1]$, with the norm $||x|| = max_{t\in [0, 1]}|x(t)|.$

$c)$ If $\alpha, \beta > 0$ and $x\in L^1 [0, 1]$, then $I^\alpha I^\beta x(t) = I^{\alpha+\beta}x(t)$ for a.e. $t\in [0, 1]$. Moreover, if $\alpha+\beta \geq 1$ and $x\in L^1 [0, 1]$, then $I^\alpha I^\beta x(t) = I^{\alpha+\beta}x(t)$ for all $t\in [0, 1]$ (^[36], Lemma 3.4).

$d)$ For $0 < \alpha\leq 1$ and $m\in \mathbb{N}$, the operator $D^{m+\alpha}$ is the left inverse of the operator $I^{m+\alpha}$ in $L^1[0, 1]$, i.e., $D^{m+\alpha}I^{m+\alpha}x(t) = x(t)$, for all $x\in L^1[0, 1]$ (^[36], Lemma 4.2).

$e)$ Let $0 < \alpha < 1$. The operator $D^{\alpha}_*$ is the left inverse of the operator $I^{\alpha}$ in the space $C[0, 1]$, i.e., $D^{\alpha}_* I^\alpha x(t) = x(t)$, for all $x\in C[0, 1]$ (^[36], Lemma 4.5).

$f)$ If $0 < \alpha < 1$ and $x(t)\in AC[0, 1]$, then $D^\alpha_*x(t) = D^\alpha (x(t)-x(0)) = D^\alpha_C x(t)$ (^[36], Proposition 4.4). Hence, the operator $D^{\alpha}_C$ is the left inverse of the operator $I^{\alpha}$ in the space $AC[0, 1]$, i.e., $D^{\alpha}_C I^\alpha x(t) = x(t)$, for all $x\in AC[0, 1]$.

But, in general, $D^\alpha_C$ is not a right inverse for $I^\alpha$. However, the following formula holds.

In general, fractional differential equations include the Caputo fractional derivative, generalized Caputo fractional derivative, and Riemann-Liouville fractional derivative. Initial and boundary value problems containing Riemann-Liouville fractional derivatives were studied by many authors (see ^{[3,6,7,12,20,22,38]} and references therein). The application areas of Riemann-Liouville fractional differential equations have been gradually expanding in recent years (see ^[26,27]).

In the past decades, different aspects of Caputo fractional differential equations were studied by a number of researchers. For example, initial value problems were studied in ^{[13,15,23,28,36,37]}. The articles ^{[2,4,5,8,16,25,30,34,35]} deal with boundary value problems for Caputo fractional differential equations. However, none of these articles contain the fractional p-Laplacian.

Chen et al. ^[9] studied the existence of antiperiodic solutions for the Lienard-type p-Laplacian equation. Chen et al. ^[10] studied the solvability of periodic boundary value problem for the fractional p-Laplacian equation in the following form:

The existence of solutions for the anti-periodic boundary value problem of a fractional p-Laplacian equation was studied by Chen et al.^[11]. They studied the following problem:

Some anti-periodic problems were also considered in ^[19,30].

Note that, none of these articles contain singularity. Since we are dealing with singular initial value problems, we would like to give a brief overview of some related results.

In ^[3], Agarwal et al. studied the existence of positive solutions for the following singular fractional boundary value problem with the Riemann-Liouville fractional derivative:

In ^[4], Agarwal et al. proved the existence of positive solutions to the singular fractional boundary value problem with the Caputo fractional derivative:

Webb ^[36] studied initial value problems for Caputo fractional differential equations with singular nonlinearities in the forms:

and

The proof of the existence of a solution to these problems is based on the Leray-Schauder fixed point theorem and Gronwall type inequalities. Gronwall inequalities are widely used to obtain a priori bounds (see ^{[18,29,37,38,39]}).

We note that, for the main concepts used in this article, we generally followed ^[11,36,37].

This paper consists of three sections. The first section includes the introduction and preliminary information. The second section includes a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. The existence of solutions to initial value problems is discussed in Section 3.

2.   Volterra integral equations

In this section, we establish a relationship between Caputo fractional differential equations and Volterra integral equations.

Theorem 2.1. Let $f\in C([0, 1]\times \mathbb{R})$ and $0\leq \gamma < \alpha, \, \beta \leq 1.$ If a function $x(t)$ satisfies the initial value problem

then $x(t)$ satisfies the Volterra integral equation

where $x_1 = \phi_p(x_\alpha)$, $p, q > 1, \, and\, 1/p+1/q = 1$.

Proof. The condition $x\in AC[0, 1]$ implies $x\in Dom(D^\alpha_C)$ and $D^\alpha_C x\in AC[0, 1]$ implies that $\phi_p(D^\alpha_Cx)\in AC[0, 1]$. Consequently, $\phi_p(D^\alpha_Cx)\in Dom(D^\beta_C)$. This means that under these conditions the problem (2.1) is well-defined. Here, for the inclusion $\phi_p(D^\alpha_Cx)\in AC[0, 1]$, we have used the composition rule: If the functions $F:[c, d]\rightarrow \mathbb{ R}$ and $G:[a, b]\rightarrow [c, d]$ are absolutely continuous, then $F(G(t))$ is also absolutely continuous.

Let $x(t)$ satisfy (2.1). Since $D^\beta_C\phi_p(D^\alpha_Cx(t))\in L^1$, we can apply $I^\beta$ to (2.1). Then

or the same

$D\bigl(\phi_p(D^\alpha_Cx(t))\in L^1$, then by Proposition 1.1 c) we have

Hence, by using $\phi_p(D^\alpha_Cx)\in AC[0, 1]$ we get

and

where $x_1 = \phi_p(D^\alpha_C x(0)) = \phi_p(x_\alpha).$

Applying the operator $I^\alpha$ to both sides of (2.3) and using Proposition 1.1 f) we get the Volterra integral equation (2.2):

□

Thus, (2.1) implies (2.2). But in general (2.2) does not imply (2.1). More precisely, if $x\in C[0, 1]$ satisfies the Volterra integral equation (2.2) then, in general, it does not follow that $x\in AC[0, 1]$ and $D^\alpha_C x\in AC[0, 1]$. However, if we use the generalized Caputo derivative $D^\alpha_*$ instead of the Caputo derivative $D^\alpha_C$ then we can show that they are equivalent. The equivalence result is given in Theorem 2.2.

Additionally, we note that $I^\alpha:AC[0, 1]\rightarrow AC[0, 1]$ and consequently

On the other hand, if $0 < \alpha < 1$ and $x\in Dom(D^\alpha_C)$, then $D^\alpha_*x(t) = D^\alpha_C x(t)\, \, \,$ (Proposition 1.1 f)). This means that the operator $D^\alpha_*$ is an extension of the operator $D^\alpha_C$.

Now we give the equivalence result.

Theorem 2.2. Let $f\in C([0, 1]\times \mathbb{R})$ and $0\leq \gamma < \alpha, \, \beta \leq 1.$ A function $x\in C[0, 1]$ satisfies the Volterra integral equation

if and only if $x\in C[0, 1]$ is a solution of the following initial value problem

where $x_\alpha^* = \phi_q(x_1)$.

Proof. Let $x\in C[0, 1]$ satisfy the Volterra integral equation (2.4). By using the substitution $s = \sigma t$ we can write

This equality together with the conditions $f\in C([0, 1]\times \mathbb{R})$ and $0\leq \gamma < \beta \leq 1$ yields

It follows from (2.6) that $I^\alpha\phi_q\bigl[I^\beta(t^{-\gamma}f(t, x(t)))+x_1\bigr]\Bigl |_{t = 0} = 0$ and consequently $x(0) = x_0.$ Applying $I^{1-\alpha}$ to (2.4) and using Proposition 1.1. c) we get

This means that $x\in Dom(D^\alpha_*)$. By Proposition 1.1 d)

So, $D^\alpha_*x\in C[0, 1]$ and $D^\alpha_*x(0) = \phi_q(x_1) = x_\alpha^*.$ Applying $\phi_p$ to both sides of (2.7) and using the fact that $t^{-\gamma}f(t, x(t))\in L^1$ we obtain

It follows from the second equation in (2.8) that

Conversely, assume that the conditions $x\in Dom(D^\alpha_*)$, $\phi_p(D^\alpha_*x)\in Dom(D^\beta_*)$, and $D^\alpha_*x\in C[0, 1]$ are satisfied and Eq (2.5) holds. By using these conditions, the definition of $D^\beta_*$ and Proposition 1.1, we obtain from Eq (2.5) that

Now, using the definition of $D^\alpha_*$ and the condition $x\in Dom(D^\alpha_*)$ we obtain that

Taking the derivative from the last equation we get

□

Finally, we compare the problems (2.1) and (2.5).

Theorem 2.3. Let $0\leq \gamma < \alpha, \, \beta < 1$. If a function $x(t)$ is a solution of the problem with Caputo derivative:

then it is a solution of the problem with generalized Caputo derivative:

Conversely, if a function $x(t)$ is a solution of the problem with the generalized Caputo derivative and additionally $x\in AC[0, 1], \, D^\alpha_C x\in AC[0, 1]$, then it is a solution of the problem with the Caputo derivative.

Proof. This fact immediately follows from the definition of the domain of the operators $D^\alpha_C, \, D^\alpha_*$, and Proposition 1.1. f), i.e., if $0 < \alpha < 1$ and $x(t)\in AC[0, 1]$ then $D^\alpha_*x(t) = D^\alpha (x(t)-x(0)) = D^\alpha_C x(t).$ □

3.   Solvability of initial and boundary value problems

Our starting initial value problem is:

We consider the homogeneous initial value problem because by using the substitution $x(t) = y(t)+\frac{x_\alpha}{\alpha \Gamma (\alpha)}t^\alpha+x_0$ one can transform the non-homogeneous initial conditions $x(0) = x_0, \, D^\alpha_Cx(0) = x_\alpha$ into the homogeneous conditions.

Theorem 2.3 gives us the basis for defining a generalized solution concept as follows.

Definition 3.1. A function $x(t)$ is called a generalized solution to the problem (3.1) if it is a solution to the following problem.

By Theorem 2.2, the problem (3.2) is equivalent to the Volterra integral equation:

For this reason, we will study the problem (3.3) instead of (3.2).

The main theorem regarding the existence of a solution to the Volterra integral equation is as follows.

Theorem 3.1. Let $f$ be continuous on $[0, 1]\times \mathbb{R}$ and there exist nonnegative functions $a, b\in C[0, 1]$ such that $|f(t, u)|\leq a(t)+b(t)|u|^{p-1},$ for all $t\in [0, 1]$ and $u\in \mathbb{R}$. If

then the Volterra integral equation (3.3) has a solution in $C[0, 1]$. So, the problem (3.1) has a generalized solution in $C[0, 1]$.

Proof. We use Schaefer's fixed point theorem to show the existence of a solution to problem (3.3). A version of Schaefer's fixed point theorem is as follows:

Let $X$ be a Banach space. If

ⅰ) $T:X\rightarrow X$ is a continuous compact operator,

ⅱ) the set $\cup_{0\leq \lambda\leq 1}\{x\in X: x(t) = \lambda Tx(t)\}$ is bounded,

then $T$ has a fixed point in $X$.

We have

Denoting $Tx(t) = I^\alpha\phi_q\bigl[I^\beta(t^{-\gamma}f(t, x(t)))\bigr]$ we can write this equation in the following form

Now, we need to show that $T$ has a fixed point in $C[0, 1].$

ⅰ) We have to prove that $T:C[0, 1]\rightarrow C[0, 1]$ is a continuous compact operator. First, let us show that $T:C[0, 1]\rightarrow C[0, 1]$ is continuous, i.e.,

A convergent sequence in a normed space is bounded. Hence,

Since the function $f$ is continuous on $[0, 1]\times [-M, M]$, it is uniformly continuous on this compact set. It means that for $\varepsilon > 0$ there exists $n_\varepsilon\in \mathbb{N}$ such that

By the definition of $T$,

Let

The function $f$ is bounded on $[0, 1]\times [-M, M]$. Hence $|f(t, s)|\leq L, \, \hbox{for all}\, (t, s)\in [0, 1]\times [-M, M].$

Then

and

where $B(\nu, \mu) = \int_0^1 (1-s)^{\nu-1}s^{\mu-1}\, ds = \frac{\Gamma(\nu)\Gamma(\mu)}{\Gamma (\nu+\mu)}\,$ for $\nu > 0, \, \mu > 0.$

Finally, it follows from the continuity of the function $\phi_p$ and continuity of the operator $I^\alpha: C[0, 1]\rightarrow C[0, 1]$ that the operator $T$ is continuous.

Now we prove that the operator $Tx(t) = I^\alpha\phi_q\bigl[I^\beta(t^{-\gamma}f(t, x(t)))\bigr]$ is compact. For this, we need to show that for any bounded set $\Omega \subset C[0, 1]$ the set $\overline{T(\Omega)}$ is compact. By the Arzela-Ascoli theorem, $\overline{T(\Omega)}$ is compact if and only if

a) $T(\Omega)$ is bounded in $C[0, 1]$ (it is the same that $T(\Omega)$ is uniformly bounded).

b) $T(\Omega)$ is equicontinuous.

We first show that $T(\Omega)$ is bounded in $C[0, 1]$. Boundedness of $T(\Omega)$ means that

where $C$ does not depend on $x$. This is a trivial fact. However, we will give a short proof. Since $\Omega$ is bounded, we have $\|x\|\leq C$ for all $x\in \Omega.$ From the continuity of $f$ it follows that it is uniformly continuous on $[0, 1]\times [-C, C]$. Then $|f(t, x(t)|\leq M$ for all $x\in \Omega$ and all $t\in [0, 1]$. Hence,

On the other hand, $I^\alpha: C[0, 1]\rightarrow C[0, 1]$ is a bounded operator and

Then, by using (3.5), (3.6), and the monotonicity of $s^{q-1}$, we obtain that

This means that $T(\Omega)$ is uniformly bounded.

b) Now we show that $T(\Omega)$ is equicontinuous. To prove this, we use the same technique as in ^[11] (see theorem 3.1 in ^[11]).

Let $0\leq t_1 < t_2\leq 1$ and $x\in \Omega$. By (3.5) we have

Then, using (3.7) we get

This inequality together with uniform continuity of $t^\alpha$ on $[0, 1]$ yields that $T(\Omega)$ is equicontinuous.

ⅱ) Let us prove that the set $\cup_{0\leq \lambda\leq 1}\{x\in C[0, 1]: x(t) = \lambda Tx(t)\}$ is bounded. Let $x(t) = \lambda Tx(t), \, \lambda\in (0, 1].$ By the condition of the theorem $|f(t, u)|\leq a(t)+b(t)|u|^{p-1},$ for all $t\in [0, 1]$ and all $u\in \mathbb{R}.$ Then

Setting $\tau = \sigma s$, we obtain that

where we have used $B(\beta, 1-\gamma) = \frac{\Gamma(\beta)\Gamma(1-\gamma)}{\Gamma(1-\gamma+\beta)}.$ Since $(p-1)(q-1) = 1$, we obtain from the last inequality that

and consequently,

If the set $\cup_{0\leq \lambda \leq 1}\{x\in C[0, 1]: x(t) = \lambda Tx(t)\}$ were unbounded then there would be a sequence in this set that converges to infinity. Then taking limit as $\|x\|\rightarrow \infty$ in (3.8) and using the condition of the theorem $\frac{\Gamma(1-\gamma)\|b\|}{\Gamma(\alpha+1)^{p-1}\Gamma(1-\gamma+\beta)} < 1$, we get the following contradiction

Hence, the set $\cup_{0\leq \lambda \leq 1}\{x\in C[0, 1]: x(t) = \lambda Tx(t)\}$ is bounded and, by Schaefer's theorem, the operator $T$ has a fixed point. □

Using the methods applied in Theorem 3.1, we obtain the following result for the smallest eigenvalue of the fractional p-laplacian operator.

Corollary 3.1. Let $\mu_1$ be the smallest eigenvalue of the following eigenvalue problem:

Then

Proof. Let $0 < \mu_1 < \Gamma(\alpha+1)^{p-1}\Gamma(\beta+1)$ be an eigenvalue of the given problem. Then,

By Theorem 2.2, this problem is equivalent to the Volterra integral equation:

According to Theorem 3.1, we have $a = 0, b = \mu_1$ and by the assumption $0 < \mu_1 < \Gamma(\alpha+1)^{p-1}\Gamma(\beta+1)$ the condition

is satisfied. Then under these conditions we have proved in Theorem 3.1 that the set $\cup_{0\leq \lambda\leq 1}\{x\in C[0, 1]: x(t) = \lambda Tx(t)\}$ is bounded, where $Tx(t) = I^\alpha\phi_q\bigl[I^\beta(\mu_1 |x|^{p-2}x)\bigr]$. But, since $\mu_1$ is an eigenvalue, the set $x(t) = \lambda Tx(t)\}$ is unbounded for $\lambda = 1$. This is a contradiction. Consequently, $\mu_1\geq \Gamma(\alpha+1)^{p-1}\Gamma(\beta+1).$ □

Finally, we give an example as an application of Theorem 3.1.

Example 3.1. If $0 < b < \frac{1}{2}$ then the following problem has a solution in $C[0, 1]$.

Let us check the conditions of Theorem 3.1. In this case

$f(t, x(t)) = 1+b x^2(t), \, p = 3, \, \gamma = \frac{1}{3}, \, \, \alpha = \beta = \frac{1}{2}$ and $0 < \gamma < \alpha, \, \beta < 1$. Then

$\Gamma(\frac{2}{3}) = 1, 35411.., \, \, \Gamma\Bigl(\frac{3}{2}\Bigr) = 0, 88622.., \, \, \Gamma\Bigl(\frac{7}{6}\Bigr) = 0, 92771..$ and

Consequently, the condition

is satisfied and the problem has a solution.

4.   Conclusions

The main subject of this paper was the initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. The most important difference of this paper from other studies on this subject is that the equation

contains the singular term $t^{-\gamma}$.

In general, Volterra integral equations are used to solve problems given for fractional differential equations. However, their equivalence is not always shown in the literature. In this article, the equivalence of such problems was discussed in full detail. We would like to emphasize that we benefited from the techniques of Webb ^[36] in this subject.

For the case $f = \mu|x|^{p-2}x, \, \gamma = 0$, a result related to the lower bound of the eigenvalues is given as Corollary 3.1.

By the methods, applied in the article, similar results can be obtained for higher order equations and different initial and boundary value problems.

Use of AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares that he has no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.