Necessary Conditions for the Existence of Global Solutions to Nonlinear Fractional Differential Inequalities and Systems
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 , 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 , in this paper, we first consider the fractional differential inequality where , , , , , , and . It is supposed that is a function and satisfies
(A1) for all
(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,
Example 3. Consider problem (3) with , , , and . Then
Example 4. Consider problem (3) with and where
Example 5. Consider problem (3) with and where
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).
We have the following result.
Example 8. Consider System (13) with
In this case, after elementary calculations, we obtain
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.
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 , Lemma 2.7). Let , , and (, , in the case ). If , then For ( is sufficiently large), let
The following results can be found in .
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 .
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,
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
Integrating over , we obtain
Integrating over , we obtain
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
Arguing similarly with the second inequality in (13), we obtain where
Next, by Hölder’s inequality, we have
Similarly, we have
For , let
On the other hand, using Young’s inequality, we obtain
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.
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
then (13) admits no global solution.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors made equal contributions and read and supported the last original copy.
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.
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