Abstract
We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.
1. Introduction
We first consider the problem where , , , is the Caputo fractional derivative of order , , and . Namely, we are interested in providing sufficient conditions for which problem (1) admits no global solution. Next, we study the same question for the inhomogeneous problem where , , , , , , and .
Due to the importance of fractional calculus in applications (see e.g. [1–5]), in the past few decades, there has been a growing interest in the study of fractional differential equations. In particular, from the theoretical point of view, the existence of solutions for different classes of fractional differential equations was investigated in many contributions (see e.g. [6–12] and the references therein).
For the issue of nonexistence of solutions for fractional differential equations and inequalities, we refer to [13–22] and the references therein. In particular, in [17], Laskri and Tatar studied the problem where , , , and , is the Riemann-Liouville fractional derivative of order , and is the left-sided Riemann-Liouville fractional integral of order . It was shown that, if , then problem (3) does not admit nontrivial global solution. In [16], Kassim et al. studied the problem where , is an integer, , and . It was shown that, if then problem (4) does not admit nontrivial global solution. In [15], Furati and Kirane investigated the system of nonlinear fractional differential equations subject to the initial conditions where , , and . It was shown that, if then solutions to system (6) subject to (7) blow up in a finite time.
For the issue of nonexistence of global solutions for fractional in time evolution equations, we refer to [6, 23–25] and the references therein.
On the other hand, to the best of our knowledge, the nonexistence of global solutions for problems of types (1) and (2) was not yet investigated.
Before stating our main results, let us mention what we mean by global solutions to problems (1) and (2).
Definition 1. A function is said to be a global solution to problem (1), if satisfies for almost every where , and
Definition 2. A function is said to be a global solution to problem (2), if satisfies for almost every where , and
We first consider problem (1). We discuss separately the cases and .
Theorem 3. Let , and . If, then for all , problem (1) admits no global solution.
Theorem 4. Let , , , and . (i)If , then for all , the only global solution to problem (1) is (ii)If , then for allthe only global solution to problem (1) is .
Next, we consider problem (2).
Theorem 5. Let , , , , and . If , then for all , problem (2) admits no global solution.
Theorem 6. Let , , , , , , , and . If then problem (2) admits no global solution.
We discuss below some special cases of Theorem 6.
Corollary 7. Let , , , and . Let where and . (i)If , then for all , problem (2) admits no global solution(ii)If and , then for all , problem (2) admits no global solution(iii)If and , then for allproblem (2) admits no global solution.
Corollary 8. Let , , , and . Let where . (i)If , then for all , problem (2) admits no global solution(ii)Let (a)If , then for all , problem (2) admits no global solution(b)If and , then for all , problem (2) admits no global solution(c)If and , then for allproblem (2) admits no global solution.
The rest of the paper is organized as follows. In Section 2, we recall briefly some standard notions on fractional calculus and prove some properties. Section 3 is devoted to the Proofs of Theorems 3, 4, 5, and 6 and Corollaries 7 and 8.
2. Some Preliminaries
We denote by the space of absolutely continuous functions on . Given an integer , we denote by the space of functions which have continuous derivatives up to order on such that . Here, denotes the derivative of order of .
Let be fixed. Given and , the left-sided Riemann-Liouville fractional integral of order of is defined by for almost everywhere . Here, denotes the Gamma function. The right-sided Riemann-Liouville fractional integral of order of is defined by for almost everywhere . Notice that, if , then is defined for all . Moreover, one has . Similarly, if , then is defined for all . Moreover, one has .
Lemma 9 (see [5]). Let and , where . Then for almsot everywhere .
Lemma 10 (see [5]). Let , , and ( and in the case ). If and , then
Let and , where is an integer. The (left-sided) Caputo fractional derivative of order of is defined by for almost everywhere . Here, for , .
For ( is large enough), we define the function
Lemma 11. Let and . Then
Proof. We prove only (25). Namely, differentiating (25), (26) follows. Similarly, differentiating (26), (27) follows. Moreover, taking in (27) and using a similar calculation as in the proof of (25), (28) follows.
For , one has
Using the change of variable , one obtains
where is the beta function. Using the property
one obtains (25).
3. Proofs
The proofs of our results are based on the test function method (see e.g. [26])and some integral estimates.
Proof of Theorem 3. Let us suppose that is a global solution to (1). For , multiplying the differential inequality in (1) by , where is the function defined by (24), and integrating over , one obtains Without restriction of the generality, we may suppose that On the other hand, using Lemma 10, one obtains Using an integration by parts, the initial conditions and (25), it holds that On the other hand, by Lemma 9 and using the initial conditions, one obtains Therefore, by (35), one obtains Using an integration by parts, the initial conditions, (26) and Lemma 10, it holds that Similarly, one has Next, using (32), (38), and (39), one obtains On the other hand, using -Young inequality with , one obtains where is a positive real number that depends only on and . Similarly, one has Hence, it follows from (40), (41), and (42) that Since , one deduces from (43) that On the other hand, by (25), one has and which yield Since , one deduces that where . Next, using (28) with and , one obtains which yields where Similarly, one has where Therefore, it follows from (44), (48), (50), and (52) that which yields Notice that for all , one has Hence, using (56) and passing to the limit as in (55), one obtains , which contradicts the fact that . Therefore, one deduces that for all , problem (1) admits no global solution.
Proof of Theorem 4. Let . First, one observes that in this case is a global solution to (1). Suppose now that is a global solution to (1). Taking in (43), one obtains
for all , where and . Next, using (24) and the estimates (50) and (52), one obtains
Notice that since , one has
Moreover, if , then
Hence, passing to the infimum limit as in (58) and using Fatou’s lemma, one obtains
which yields
for almost everywhere . Then, using the initial conditions and Lemma 9, one deduces that
for almost everywhere . Since is continuous (), it holds that for all . This proves part (i) of Theorem 4.
Suppose now that . In this case, if , then (60) holds. Hence, proceeding as above, one obtains for all , which proves part (ii) of Theorem 4.
Proof of Theorem 5. It is sufficient to observe that any global solution to problem (2) is a global solution to problem (1). Hence, using Theorem 3, one deduces that problem (2) admits no global solution.
Proof of Theorem 6. Let us suppose that is a global solution to problem (2). Proceeding as in the Proof of Theorem 4 and using that , one obtains for all , where is a constant (independent on ). On the other hand, by (24), one has Moreover, by (50) and (52), one has Next, it follows from (64), (65), and (66) that which yields
Finally, passing to the supremum limit as in (68), using (14) and the fact , a contradiction follows.
Proof of Corollary 7. For all , one has
If , then
which yields
Hence, by Theorem 6, one deduces that for all , problem (2) admits no global solution, which proves part (i).
If , then
which yields
Therefore, if , one has
which yields
Hence, by Theorem 6, one deduces that for all , problem (2) admits no global solution, which proves part (ii). On the other hand, if , one has
which yields
Hence, by Theorem 6, one deduces that for all , problem (2) admits no global solution, which proves part (iii).
Proof of Corollary 8. For all , one has
which yields
Hence,
Notice that
Hence, by Theorem 6, one deduces that for all satisfying
problem (2) admits no global solution.
Consider the case . In this case, for all , one has
Then (82) is satisfied for all . Hence, one deduces that for all , problem (2) admits no global solution, which proves part (i).
Suppose now that . If , then for all , one has
Then, (82) is satisfied for all . Hence, one deduces that for all , problem (2) admits no global solution, which proves part (ii)(a). On the other hand, if and , then
Hence, (82) is satisfied for all . Therefore, for all , problem (2) admits no global solution, which proves part (ii)(b). Finally, if and , then for all , (82) is equivalent to
Hence, for all satisfying the above condition, problem (2) admits no global solution, which proves part (ii)(c).
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 contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No. RGP-1435-034.