#### Abstract

In this paper, fractional differential inequalities and systems of fractional differential inequalities involving fractional derivatives in the sense of Caputo are investigated. Namely, necessary conditions for the existence of global solutions are obtained. Our approach is based on the test function method and some integral inequalities.

#### 1. Introduction and Main Results

In several studies, the usefulness of fractional derivatives in the mathematical modeling of various phenomena from physics and engineering has been demonstrated (see, e.g., [1–7], and the references therein). Due to this fact, the study of fractional differential equations has received a great deal of attention from many researchers. The existence of solutions is one of the most important topics of fractional differential equations. The study of sufficient conditions for the existence of solutions has been investigated by many authors using different approaches from functional analysis (see, e.g., [8–18], and the references therein). The study of necessary conditions for the existence of global solutions in the context of fractional differential equations has been initiated by Kirane and his collaborators (see, e.g., [19–24], and the references therein).

In [19], Furati and Kirane investigated the system of fractional differential equations: where , , and . Here, , , denotes the Caputo fractional derivative of order . Namely, it was shown that if then (1) admits no global solution.

Motivated by Furati and Kirane [19], in this paper, we first consider the fractional differential inequality where , , , , , , and . It is supposed that is a function and satisfies

(A1) for all

(A2)

(A3) for all

Our aim is to study the influence of on the large time behavior of solutions. By a global solution to (3), we mean a function (an absolutely continuous function) satisfying the fractional differential inequality in (3) for almost everywhere , and the initial condition . Our aim is to derive sufficient conditions for which (3) admits no global solution. Namely, the following result is obtained.

Theorem 1. *Suppose that
**Then (3) admits no global solution.*

We provide below some examples where (4) is satisfied.

*Example 2. *Consider problem (3) with , , , and . Then,

Hence, by Theorem 1, we deduce that for all , (3) admits no global solution.

*Example 3. *Consider problem (3) with , , , and . Then

Hence, by Theorem 1, we deduce that for all , (3) admits no global solution.

*Example 4. *Consider problem (3) with and
where

In this case, after elementary calculations, we obtain where is a constant independent of . Hence, by Theorem 1, we deduce that for all , (3) admits no global solution.

*Example 5. *Consider problem (3) with and
where

In this case, after elementary calculations, we obtain where is a constant independent of . Hence, by Theorem 1, we deduce that for all , (3) admits no global solution.

In the second part of this paper, we extend the previous study to the system of fractional differential inequalities: where for , , , , , , , and . Moreover, it is supposed that is a function and satisfies (A1)–(A3). Notice that in the special case , , and , (13) reduces to (1).

By a global solution to (13), we mean a pair of functions satisfying the fractional differential inequalities in (13) for almost everywhere , and the initial condition .

We have the following result.

Theorem 6. (i)*Let and . If**then (13) admits no global solution
*(ii)*Let and . If**then (13) admits no global solution*

We provide below some examples for which (14) or (15) are satisfied.

*Example 7. *Consider (1), that is, System (13) with , , , and , for all . Then

Hence, by Theorem 6, we deduce that for all , (1) admits no global solution. This improves [19] (Theorem 1), where the nonexistence of a global solution was obtained only when .

*Example 8. *Consider System (13) with

In this case, after elementary calculations, we obtain

Hence, by Theorem 6, we deduce that for all and , (13) admits no global solution.

The rest of this paper is organized as follows. In Section 2, we recall briefly some notions related to fractional calculus and provide some lemmas that will be used in the proofs of our main results. In Section 3, we prove Theorems 1 and 6.

#### 2. Preliminaries

We first recall some basic notions and properties related to fractional calculus (see, e.g., [25, 26]).

Let and . The left-sided Riemann-Liouville fractional integral of order of a function is defined by for almost everywhere , where is the Gamma function. The right-sided Riemann-Liouville fractional integral of order of a function is defined by for almost everywhere .

The Caputo fractional derivative of order of a function is defined by for almost everywhere .

Lemma 9 (see [25], Lemma 2.7). *Let , , and (, , in the case ). If , then
**For ( is sufficiently large), let
*

The following results can be found in [19].

Lemma 10. *Let . Then
**for all .*

#### 3. Proofs of the Main Results

The proofs of our main results are based on the test function method developed by Mitidieri and Pohozaev [27].

*Proof of Theorem 1. *We use the contradiction argument. Namely, suppose that is a global solution to (3). Multiplying the fractional differential inequality in (3) by the function defined by (24) with , and integrating over , , we obtain
By the initial condition , we have

Notice that by the definition of , we have and . Therefore, it holds that

Next, using Lemma 9, we obtain

Integrating by parts and using the initial condition , we obtain

On the other hand, by Lemma 10, we have and . Therefore,

Combining (29) with (31), we deduce that

Hence, it follows from (28) and (29) that

Consider now the terms from the left side of (26). By a change of variable, using the properties (A1)–(A3), and the decay property of , we obtain

Next, by (24), and using that (we have also ), we obtain

Combining (34) with (35), we deduce that

Hence, it follows from (26), (33), and (36) that where

Next, using Young’s inequality, we obtain where denotes a positive constant (independent of ) whose value may change from line to line. Then, it follows from (37), (39), and (40) that where

Let us estimate the terms , . For all , by (24) and (38), we have

Integrating over , we obtain

Next, by (38) and Lemma 10, for all , we have

Integrating over , we obtain

Therefore, it follows from (41), (44), and (46) that which yields

Hence, we deduce that which contradicts (4). The proof is completed.

*Proof of Theorem 6. *Suppose that is a global solution to (13). Multiplying the first fractional differential inequality in (13) by the function defined by (24) with , and integrating over , , we obtain

Following the same steps of the Proof of Theorem 1, we obtain

Hence, it follows from (50), (51), and (52) that where

Arguing similarly with the second inequality in (13), we obtain where

Next, by Hölder’s inequality, we have

Similarly, we have

For , let

It follows from (53), (57), and (58) that

Similarly, by (55), (59), and (60), we obtain

Now, we consider the case and . In this case, using (62) and (63), we obtain

On the other hand, using Young’s inequality, we obtain

Therefore, combining (64) with (65), it holds that which yields

Next, using the estimates (44) and (46), we obtain (after elementary calculations)

Combining (67) with (68), we obtain which contradicts (14).

In the second case, when and , following the same steps as used in the previous case, we obtain which contradicts (15). The proof is completed.

#### 4. Conclusion

In this paper, problems (3) and (13) are investigated. Namely, using the test function method and some integral inequalities, sufficient conditions for the nonexistence of global solutions (or equivalently, necessary conditions for the existence of global solutions) to the considered problems are obtained. For problem (3), we proved that (see Theorem 1) under assumptions (A1), (A2), and (A3), if then (3) admits no global solution. For the system of fractional differential inequalities (13), always under assumptions (A1), (A2), and (A3), we proved that (see Theorem 6), if and

or and

then (13) admits no global solution.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors made equal contributions and read and supported the last original copy.

#### Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Al Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-01.