Nonexistence for fractional differential inequalities and systems in the sense of Erdélyi-Kober

1.   Introduction

Fractional differential equations arise in the mathematical modeling of various problems such as biology, fluid mechanics, electrochemistry, finance, and many other areas of applications; see, e.g., ^[1,2,3]. This fact motivated the study of fractional differential equations in various directions: existence of solutions ^[4,5], comparison principles ^[6,7], inverse problems ^[8,9], inequalities ^[10], etc.

One of the most important topics of the theory of differential and partial differential equations is the issue of nonexistence of solutions, which was initiated by the famous Liouville theorem for harmonic functions (see, e.g., ^[11]). Nonexistence theorems have several applications, in particular, in the study of blow-up of solutions (see, e.g., ^[12]). The study of nonexistence of solutions to fractional differential equations and inequalities was initiated by Kirane and his collaborators. Next, this topic was developed by many authors. For instance, Kirane and Malik ^[13] investigated the profile of the blowing-up solutions to the nonlinear system of fractional differential equations

where $0 < \alpha, \beta < 1$, ${}^C\!D_0^\alpha$ (resp. ${}^C\!D_0^\beta$) is the Caputo fractional derivative of order $\alpha$ (resp. $\beta$), $p, q > 1$, and $u_0, v_0\in \mathbb{R}$. Laskri and Tatar ^[14] considered nonlinear fractional differential inequalities of the form

where $0 < \alpha < 1$, $D_0^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha$, $b\in \mathbb{R}$, $m > 1$, and $\gamma\in \mathbb{R}$. Using the test function method (see ^[12]), the authors obtained sufficient conditions under which (1.1) admits no global solution. Kassim, Furati, and Tatar ^[15] studied fractional differential inequalities of the form

where $0 < \beta\leq \alpha$, $n = -[-\alpha]$, $\gamma\in \mathbb{R}$, $m > 1$, and $b_i\in \mathbb{R}$, $i = 0, 1, \cdots, n-1$. More recent works can be found in ^{[16,17,18,19]} (see also the references therein).

In all the above cited works, the fractional derivatives were considered in the sense of Riemann-Liouville or Caputo (see, e.g., ^[20] for the definitions of these operators). Very recently, in ^[21], the authors studied fractional differential inequalities of the form

where $0 < \alpha < 1$, $a > 0$, $\sigma > 0$, $\eta\in \mathbb{R}$, $p > 1$, $u_a > 0$, and $V$ is a measurable positive function. Here, $D_{a; \sigma, \eta}^\alpha$ denotes the Erdélyi-Kober fractional derivative of order $\alpha$ and parameters $\sigma$ and $\eta$, and $I_{a; \sigma, \eta+\alpha}^{1-\alpha}$ denotes the left-sided Erdélyi-Kober fractional integral of order $1-\alpha$ and parameters $\sigma$ and $\eta+\alpha$. It was shown that, if

then (1.2) admits no weak solution. In particular, when

where $C_V > 0$ is a constant, it was proved that, if one of the following conditions:

holds, then (1.2) admits no weak solution.

The aim of the first part of this work is to obtain sufficient conditions for the nonexistence of weak solutions to the inhomogeneous version of (1.2) (with $V\equiv 1$), namely,

subject to the initial condition

where $a > 0$, $\sigma > 0$, $\eta\in \mathbb{R}$, $0 < \alpha < 1$, $p > 1$, $u_a\in \mathbb{R}$, and $f\in L^1_{\rm loc}([a, \infty))$. Our motivation for considering problems of type (1.4) is to study the influence of the inhomogeneous term $f$ on the large-time behavior of solutions to (1.2) with $V\equiv 1$. We show that, if $f(t)\geq C_f t^{-\sigma \eta} (t^\sigma-a^\sigma)^\gamma$, $t > a$, where $C_f > 0$ is a constant, and $\gamma > \max\{\eta, -1\}$, then for all $p > 1$, (1.4) and (1.5) admit no weak solution.

We next extend our study to systems of fractional differential inequalities of the form

subject to the initial conditions

where $0 < \alpha, \beta < 1$, $p, q > 1$, $g, h$ are positive measurable functions, and $u_a, v_a\in \mathbb{R}$. Our motivation for considering systems of the form (1.6) is to extend the obtained results in ^[13] from the Caputo sense to the Erdélyi-Kober sense.

We finally mention that some existence and nonexistence results for a class of nonlinear Erdélyi-Kober type fractional differential equations on unbounded domains were established in ^[22], making use of some tools from fixed point theory. The approach that we use in this paper is based on nonlinear capacity estimates specifically adapted to Erdélyi-Kober fractional derivatives.

The organization of the rest of the paper is as follows. In Section 2, some notions and properties related to Erdélyi-Kober fractional operators are recalled. The definitions of weak solutions to the considered problems as well as the obtained results are presented in Section 3. Some important lemmas are established in Section 4. Finally, the proofs of our obtained results are given in Section 5.

2.   Preliminaries and notation

In this section, we recall briefly some basic notions and properties related to Erdélyi-Kober fractional operators, and fix some notation. For more details, we refer to ^[20].

Let $a, T\in \mathbb{R}$ be fixed such that $0 < a < T$. We first recall the Riemann-Liouville fractional integral operators.

The left-sided and right-sided Riemann-Liouville fractional integrals of order $\kappa > 0$ of a function $f\in L^1([a, T])$ are defined respectively by

and

for almost everywhere $t\in [a, T]$, where $\Gamma$ denotes the gamma function.

The left-sided and right-sided Erdélyi-Kober fractional integrals of order $\alpha > 0$ and parameters $\sigma > 0$ and $\eta\in \mathbb{R}$ of a function $f\in L^1([a, T])$, are defined respectively by

and

for almost everywhere $t\in [a, T]$.

Some relations between the Riemann-Liouville and Erdélyi-Kober fractional integrals can be easily obtained. Using the change of variable $z = s^\sigma$, for $a < t < T$, we obtain

where

Using the same change of variable, we get

where

We have the following integration by parts rule (see ^[21]).

Lemma 2.1. Let $\mu, \sigma > 0$ and $\eta\in \mathbb{R}$. Let $k, m\geq 1$ and $\frac{1}{k}+\frac{1}{m}\leq 1+\mu$ ($k\neq 1$ and $m\neq 1$ if $\frac{1}{k}+\frac{1}{m} = 1+\mu$). If $f\in L^k([a, T])$ and $g\in L^m([a, T])$, then

The proof of the following result can be found in ^[21].

Lemma 2.2. Let $0 < \mu < 1$, $\sigma > 0$, and $\eta\in \mathbb{R}$. For $\lambda\gg 1$ ($\lambda$ is sufficiently large), let

For all $a < t < T$, we have

Let $0 < \alpha < 1$, $\sigma > 0$, $\eta\in \mathbb{R}$ and $f$ be a function such that

where $AC([a, T])$ denotes the space of absolutely continuous functions on $[a, T]$. The (left-sided) Erdélyi-Kober fractional derivative of order $\alpha$ and parameters $\sigma$ and $\eta$ of $f$ is defined by (see, e.g., ^[20])

for almost everywhere $t\in [a, T]$.

Throughout this paper, we shall use the following notations. By $C$, we mean a positive constant independent of $T$ and the solutions $u$ and $v$. Its value is not necessarily the same from one line to another. By $\lambda\gg 1$, where $\lambda > 0$, we mean that $\lambda$ is sufficiently large.

3.   The results

In this section, we state our obtained results for problems (1.4)–(1.7).

Let us define weak solutions to (1.4) and (1.5). For all $T > a$, we introduce the set of functions

Definition 3.1. We say that $u$ is a weak solution to (1.4) and (1.5), if $u\in L^p_{\rm loc}([a, \infty))$ and

for all $T > a$ and $\psi\in \Psi_T$.

Notice that, if $u$ satisfies (1.4) and (1.5), then for all $T > a$, multiplying (1.4) by $\psi\in \Psi_T$, integrating by parts over $(a, T)$, using Lemma 2.1 and (1.5), we obtain (3.1).

Our main result for problems (1.4) and (1.5) is stated in the following theorem.

Theorem 3.2. Let $a > 0$, $\sigma > 0$, $\alpha\in (0, 1)$, and $p > 1$. Let $f\in L^1_{\rm loc}([a, \infty))$ be such that

for almost everywhere $t > a$, where $C_f > 0$ is a constant. If

then (1.4) and (1.5) admit no weak solution.

Remark 3.3. From Theorem 3.2, we show that the value of the parameter $\alpha$ has no effect on the nonexistence result.

Remark 3.4. In the homogeneous case ($f\equiv 0$), problem (1.4) under the initial condition (1.5), reduces to problem (1.2) with $V\equiv 1$. In this case, from the obtained result in ^[21], if one of the conditions:

holds, then we have no weak solution. From Theorem 3.2, we show that under conditions (3.2) and (3.3), the effect of the inhomogeneous term on the large-time behavior of solutions is considerable. Namely, in this case, for every $p > 1$, the inhomogeneous problems (1.4) and (1.5) admit no weak solution.

We now define weak solutions to (1.6) and (1.7).

Definition 3.5. We say that the pair of functions $(u, v)$ is a weak solution to (1.6) and (1.7), if $u\in L^q_{\rm loc}([a, \infty), h(t)\, dt)\cap L^1_{\rm loc}([a, \infty))$, $v\in L^p_{\rm loc}([a, \infty), g(t)\, dt)\cap L^1_{\rm loc}([a, \infty))$ and

for all $T > a$ and $\psi\in \Psi_T$.

Notice that, if $(u, v)$ satisfies (1.6) and (1.7), then for all $T > a$, multiplying the first inequality in (1.6) by $\psi\in \Psi_T$, integrating by parts over $(a, T)$, using Lemma 2.1 and (1.7), we obtain (3.4). Similarly, multiplying the second inequality in (1.6) by $\psi$ and integrating by parts over $(a, T)$, we get (3.5).

Our main result for (1.6) and (1.7) is stated in the following theorem.

Theorem 3.6. Let $a > 0$, $\sigma > 0$, $\eta\in \mathbb{R}$, $0 < \alpha, \beta < 1$, and $p, q > 1$. Assume that $g^{\frac{-1}{p-1}}, h^{\frac{-1}{q-1}}\in L^1_{\rm loc}([a, \infty))$ and $u_a, v_a\geq 0$. If one of the following conditions:

(i) $v_a > 0$ and

(ii) $u_a > 0$ and

is satisfied, then (1.6) and (1.7) admit no weak solution.

Remark 3.7. If $\alpha = \beta$, $p = q$, $g = h = V$, $u_a = v_a$, and $u = v$, then system (1.6) under the initial conditions (1.7) reduces to (1.2). In this case, (3.6) and (3.7) reduce to (1.3). Then, we recover the nonexistence result obtained in ^[21] for (1.2).

We now consider singular weight functions of the forms

where $\gamma, \rho\geq 0$ are constants. It can be easily seen that $g^{\frac{-1}{p-1}}, h^{\frac{-1}{q-1}}\in L^1_{\rm loc}([a, \infty))$. From Theorem 3.6, we deduce the following result.

Corollary 3.8. Let $a > 0$, $\sigma > 0$, $\eta < -1$, $0 < \alpha, \beta < 1$, $p, q > 1$, and $u_a, v_a\geq 0$. Let $g$ and $h$ be the functions defined by (3.8), where $\gamma, \rho\geq 0$. If one of the conditions:

(ⅰ) $v_a > 0$ and

(ⅱ) $u_a > 0$ and

holds, then (1.6) and (1.7) admit no weak solution.

We provide below an example to illustrate the above result.

Example 3.9. Consider the system of fractional differential inequalities

where $a > 0$, $\sigma > 0$, $\gamma, \rho\geq 0$, subject to the initial conditions (1.7) with $u_a, v_a > 0$. System (3.10) is a special case of (1.6), where $g, h$ are defined by (3.8), $\alpha = \frac{1}{2}$, $\beta = \frac{1}{4}$, $\eta = -2$, $p = 2$, and $q = 3$. From Corollary 3.8, if

then system (3.10) under the initial conditions (1.7) admits no weak solution. In this case, we have

and

which shows that (3.9) is equivalent to (3.11).

4.   Auxiliary results

In this section, we establish some important lemmas that will be used later in the proofs of our main results.

For all $T > a$ and $\psi\in \Psi_T$, let

and

We have the following a priori estimate for problems (1.4) and (1.5).

Lemma 4.1. If $u$ is a weak solution to (1.4), (1.5), and $u_a\geq 0$, then

for all $T > a$ and $\psi\in \Psi_T$, provided $J(\psi) < \infty$.

Proof. Let $u$ be a weak solution to (1.4), (1.5), and $u_a\geq 0$. Let $T > a$ and $\psi\in \Psi_T$, where $J(\psi) < \infty$. By (3.1), we have

Since $u_a\geq 0$, the above inequality yields

On the other hand, we have by Young's inequality that

which implies by (4.5) that

Since $1-\frac{1}{p} > 0$, the above inequality yields (4.4). □

We also have the following a priori estimate for problems (1.6) and (1.7).

Lemma 4.2. If $(u, v)$ is a weak solution to (1.6), (1.7), and $u_a, v_a\geq 0$, then

and

for all $T > a$ and $\psi\in \Psi_T$, provided $K_i(\psi) < \infty$, $i = 1, 2$.

Proof. Let $(u, v)$ be a weak solution to (1.6) and (1.7). Let $T > a$ and $\psi\in \Psi_T$ be such that $K_i(\psi) < \infty$, $i = 1, 2$. By (3.4), we have

On the other hand, by Hölder's inequality, we get

which implies by (4.8) that

Similarly, by (3.5), we have

Making use of Hölder's inequality, we obtain

which implies by (4.10) that

Since $u_a\geq 0$, it follows from (4.9) that

The above estimate together with (4.11) implies that

We now use Young's inequality to get

that is,

which yields (4.6). Similarly, since $v_a\geq 0$, it follows from (4.11) that

The above estimate together with (4.9) gives us that

which implies by Young's inequality that

and (4.7) follows. □

For $T > a$ with $T\gg 1$ and $\lambda\gg 1$, let us consider test functions of the form

where $\varphi$ is the function defined by (2.1).

Lemma 4.3. The function $\psi$ defined by (4.12) belongs to $\psi_T$.

Proof. The result follows immediately from (2.1) and (4.12). □

Let us now estimate the integral terms $J(\psi)$ and $K_i(\psi)$, $i = 1, 2$.

Lemma 4.4. We have

Proof. By (4.12), for all $a < t < T$, we have

which implies by Lemma 2.2 with $\mu = 1-\alpha$ that

Then, by (2.1), it holds that

Using (4.1) and integrating over $(a, T)$, we obtain

which proves (4.13). □

Lemma 4.5. Assume that $h^{\frac{-1}{q-1}}\in L^1_{\rm loc}([a, \infty))$. We have

Proof. By (4.14), it holds that

Using (4.2) and integrating over $(a, T)$, we obtain

which proves (4.15). □

Similarly, by (2.1), (4.3), and (4.12), we obtain the following estimate of $K_2(\psi)$.

Lemma 4.6. Assume that $g^{\frac{-1}{p-1}}\in L^1_{\rm loc}([a, \infty))$. We have

The following estimates follow immediately from Lemmas 4.5 and 4.6.

Lemma 4.7. Assume that $h^{\frac{-1}{q-1}}\in L^1_{\rm loc}([a, \infty))$ and $g^{\frac{-1}{p-1}}\in L^1_{\rm loc}([a, \infty))$. We have

and

5.   Proofs of the obtained results

This section is devoted to the proofs of Theorems 3.2 and 3.6, and Corollary 3.8.

Proof of Theorem 3.2. We use a contradiction argument. Namely, suppose that $u$ is a weak solution to (1.4) and (1.5). By Lemmas 4.1 and 4.5, we have

where for $T, \lambda\gg 1$, the function $\psi$ is defined by (4.12). On the other hand, by (2.1), (3.2), and (4.12), we have

Furthermore, we have

Making the change of variable $s = \frac{t^\sigma-a^\sigma}{T^\sigma-a^\sigma}$ and using that $\gamma > -1$ (by (3.3)), we obtain

where $B$ is the beta function. Hence, by (5.2), we have

We now use Lemma 4.4, (5.1), and (5.3) to get

that is,

where

and

Note that due to (3.3), we have $\tau_i < 0$, $i = 1, 2$. Hence, passing to the limit as $T\to \infty$ in (5.4), we reach a contradiction. This completes the proof of Theorem 3.2. □

Proof of Theorem 3.6. We also use a contradiction argument. Namely, suppose that $(u, v)$ is a weak solution to (1.6) and (1.7).

We first consider the case (ⅰ). By Lemma 4.2, for all $T > a$ and $\psi\in \Psi_T$, we have

provided $K_i(\psi) < \infty$, $i = 1, 2$. In particular, since $g^{\frac{-1}{p-1}}, h^{\frac{-1}{q-1}}\in L^1_{\rm loc}([a, \infty))$, then by Lemmas 4.3, 4.5, and 4.6, (5.5) holds for the function $\psi$ defined by (4.12). Since $\psi(a) = a^{\sigma(\eta+1)-1} > 0$, then (5.5) reduces to

which implies by the first estimate in Lemma 4.7 that

Passing to the infimum limit as $T\to \infty$ in the above inequality and using (3.6), we obtain (recall that $v_a\geq 0$)

which contradicts the fact that $v_a > 0$.

Consider now the case (ⅱ). Similarly to the previous case, we obtain by Lemmas 4.2, 4.3, 4.5, and 4.6 that

where $\psi$ is defined by (4.12). Then, from the second estimate in Lemma 4.7, we deduce that

Passing to the infimum limit as $T\to \infty$ in the above inequality and using (3.7), we obtain (recall that $u_a\geq 0$)

which contradicts the fact that $u_a > 0$.

Hence, in both cases (ⅰ) and (ⅱ), we reach a contradiction. This completes the proof of Theorem 3.6. □

Proof of Corollary 3.8. We only give the proof of the case (ⅰ). The proof of the case (ⅱ) follows using a similar argument. By the definition of $h$, for $T\gg 1$, we have

which implies that

Similarly, by the definition of $g$, we have

Then, it follows from (5.6) and (5.7) that

where

Observe that from (3.9), we have $\eta_i < 0$, $i = 1, 2, 3, 4$. Hence, from (5.8), we deduce that

which shows that (3.6) is satisfied. Then, Theorem 3.6 applies. □

6.   Conclusions

Using nonlinear capacity estimates, sufficient conditions for the nonexistence of weak solutions were obtained for the inhomogeneous Erdélyi-Kober fractional differential inequality (1.4) subject to the initial condition (1.5) (see Theorem 3.2) and the system of Erdélyi-Kober fractional differential inequalities (1.6) under the initial conditions (1.7) (see Theorem 3.6). By comparing Theorem 3.2 with the recent result obtained in ^[21] for the homogeneous problem (1.2) with $V\equiv 1$, we observe that, if the inhomogeneous term $f$ satisfies (3.2) and (3.3), then the nonexistence holds for every $p > 1$. However, in the homogeneous case, the nonexistence holds for a certain range of $p$. Furthermore, Theorem 3.6 recovers the nonexistence result established in ^[21] (See Remark 3.7).

In this paper, we only studied the nonexistence of solutions to the considered problems. It would be interesting to extend this study in order to get sufficient conditions for the existence of solutions. We hope that in a future work, this question will be solved.

Author contributions

Mohamed Jleli: Conceptualization, methodology, investigation, formal analysis; Bessem Samet: Conceptualization, methodology, validation, investigation, writing review and editing. All authors have read and approved the final draft of the paper for publication.

Use of AI tools declaration

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

Acknowledgments

The first author is supported by Researchers Supporting Project number (RSP2024R57), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no competing interests.