- Research
- Open access
- Published:
Singular fractional integro-differential inequalities and applications
Journal of Inequalities and Applications volume 2011, Article number: 110 (2011)
Abstract
In this article, fractional integro-differential inequalities with singular coefficients have been considered. The bounds obtained for investigating the behavior of the solution of a class of singular nonlinear fractional differential equations has been used, some applications are provided.
2010 Mathematics Subject Classification: 26A33; 34A08; 34A34; 45J05.
1. Introduction
Many physical and chemical phenomena can be modeled with fractional differential equations. However, finding solutions to such equations may not be possible in most cases, particularly the nonlinear ones. Instead, many researchers have been studying the qualitative attributes of the solutions without having them explicitly. In particular, the existence and uniqueness of solutions of a wide class of Cauchy-type problems have been intensively investigated; see for example [1] and the references therein. Also classes of boundary value problems have been considered. For example in [2, 3], the authors established the existence and uniqueness of the solution for a class of linear and superlinear fractional differential equations.
Inequalities play an important role in the study of existence, uniqueness, stability, continuous dependence, and perturbation. In [4–7], bounds for solutions of fractional differential inequalities of order 0 < α < 1 are obtained. Those bounds are generalizations and extensions of analogous bounds from the integer order case [8, 9]. In [5], a number of Bihari-type inequalities for the integer order derivatives are extended to non-integer orders. However, the coefficients of these inequalities are assumed to be continuous at the left end of the interval of definition.
In this article, we extend these inequalities to ones with singular integrable coefficients of the form
and
where 0 < α < 1, 0 ≤ β0 < β1 < ... < β k < α, 0 ≤ γ0 < γ1 < ... < γ m < α, n ≥ 1 is an integer, and a, b ∈ C(0, T] ∩ L1(0,T). Also we give some applications.
The rest of the article is organized as follows. In Section 2, we introduce some definitions and results that we use in our proofs. Section 3 contains the main results. The last section is devoted to some applications.
2. Preliminaries
In this section, we introduce some notations, definitions, and lemmas which will be needed later. For more details, we refer the reader to [1, 8, 10, 11].
We denote by L p , 1 ≤ p ≤ ∞, the Lebesgue spaces, and by AC[a, b] the space of all absolutely continuous functions on [a, b], -∞ < a < b < ∞.
Definition 1. Let f ∈ L1(a, b), the integral
is called the Riemann-Liouville fractional integral of order α of the function f. Here, Γ(α) is the gamma function.
Definition 2. The expression
is called the Riemann-Liouville fractional derivative of order α of the function f.
Note that . We use the notation f α to denote . We set
Definition 3. Let 0 < α < 1. A function f ∈ L1(a, b) is said to have a summable fractional derivative on (a, b) if .
Definition 4. We define the space , α > 0, 1 ≤ p < ∞, to be the space of all functions f such that for some φ ∈ L p (a, b).
Theorem 5. A function f is in , if and only if f1-α∈ AC[a, b], and f1-α(a) = 0 (see [[11], Theorem 2.3, p. 43]).
Lemma 6. If α > 0 and β > 0, then
is satisfied at almost every point t ∈ [a, b] for f ∈ L1 (a, b), 1 ≤ p ≤ ∞ (see [[1], p. 73]).
Lemma 7. If f ∈ AC [a, b], then I1-αf ∈ AC [a, b], 0 < α < 1 (see [[11], Lemma 2.1, p. 33]).
Corollary 8. If f ∈ L1 (a, b) has a summable fractional derivative , 0 < α < 1, on (a, b), then for 0 ≤ β < α < 1 we have
Proof. Since , then we can write
Also from Lemmas 6 and 7 we have
Thus, f has a summable fractional derivative given by
Lemma 9. Let v, f, g and k be non-negative continuous functions on [a, b]. Let ω be a continuous, non-negative and non-decreasing function on [0, ∞), with ω(0) = 0 and ω(u) > 0 for u > 0, and let F(t) = max0≤s≤tf(s) and G(t) = max0≤s≤tg(s). Assume that
Then
where , 0 < v0 ≤ v, H-1 is the inverse of H and T > a is such that , for all t ∈ [a, T) (see [[8], Corollary 5.5]).
Let I ⊂ R, and g1, g2: I → R\{0} We write g1 ∝ g2 if g2/g1 is non-decreasing in I.
Lemma 10. Let f(t) be a positive continuous function on [a, b], and k j (t, s), 1 ≤ j ≤ n, be non-negative continuous functions for a ≤ s ≤ t < b which are monotonic non-decreasing in t for any fixed s. Let g j (u), j = 1, 2, ..., n, be non-decreasing continuous functions on [0, ∞), with g j (0) = 0, g j (u) > 0 for u > 0, and g1 ∝ g2 ∝... ∝ g n in (0, ∞). If u(t) is a non-negative continuous functions on [a, b] and satisfy the inequality
then
where ,
and T is chosen so that the function c j (t), j = 1, 2, ..., n, are defined for a ≤ t < T (see [[8], Theorem 10.3]).
Lemma 11. For non-negative a i , i = 1, 2, ..., k,
Definition 12. We denote by CL1(a, b) the space of all functions f such that .
Lemma 13. If , then .
Proof. Clearly if α ≥ 1, then . For 0 < α < 1, it follows from Fubini's theorem that . So, it remains to show that is continuous at every t0 ∈ (a, b]. We have the following two cases.
Case 1. t0 ∈ (a, b), and t ∈ (t0, b]. Then
Clearly the right-hand side →0 as t → t0. This implies that and thus the continuity.
Case 2. t0 ∈ (a, b], and t ∈ (a, t0), the proof is similar to that of case 1.
Remark 1.
-
1.
If f ∈ C (a, b) and then f ∈ CL 1 (a, b).
-
2.
If f ∈ C (a, b), and then (t-a) σ f ∈ C[a, b] for all σ > 0.
Lemma 14. Let 0 < σ < α < 1, and (t-a) σ f(t)∈ C[a, b]. Then is continuous on [a, b].
(This lemma is proven in [12].)
Next we extend the inequalities in [8] (Lemmas 1.1 and 4.1) to functions in C(0, T].
Lemma 15. Let f(t) and g(t) be continuous functions in (0, T), T > 0. Let v(t) be a differentiable function for t > 0 such that . If
then,
Proof. We write (1) as
and obtain
By integrating both sides over (ε, t), ε > 0, we obtain
The result follows by taking the limit as ε → 0.
Remark 2.
-
1.
If v 0 < ∞, and f, g ∈ CL 1(0, T), then the right-hand side is bounded.
-
2.
If v 0 = 0, and g ∈ CL 1(0, T), then the first term of the right hand said equal to zero.
Lemma 16. Let v(t) be a positive differentiable function on (0, T) such that , and
where the functions h and k are continuous functions on (0, T), and p ≥ 0, p ≠ 1, is a constant. Then,
where q = 1-p and T is chosen so that the expression between the brackets is positive in the interval (0, T).
Proof. Let , then and
By Lemma 15, we obtain
or
where ≤ (respectively, ≥) hold for q > 0 (respectively, q < 0). In both cases, this estimate implies the result.
Below, we use the terms non-increasing and non-decreasing to refer to monotonic functions only.
3. Main results
In this section, we present and prove our main results. Without loss of generality, we take the left end of the intervals to be 0 and drop the subscript a+.
Theorem 17. Let a, b ∈ CL1(0, T), T > 0, be non-negative functions, and tσ b(t) ∈ C[0, T], where 0 ≤ β0 < β1 < ... < β k < α < 1. Let c ∈ C[0, T] be a non-negative function. Let u ∈ L1(0, T) be such that u1-α∈ AC[0, T] and satisfy the inequality
where n > 1 an integer.
Then,
provided that g ∈ L1(0, T), and
where
and
Proof. Let
Then, clearly ϕ(0) = 0,
and
By Corollary 8 and Equation 7 we have
Substituting (8) into (9), and using Lemma 11, we obtain
Since ϕ(t) is non-decreasing, we can write (10) as
where g(t) and h(t) are as defined by (4) and (5).
By integrating both sides of (11) over (0, t) we obtain
where . Since g(t) is non-negative and integrable, l(t) is non-decreasing and continuous on [0, T]. Thus . Also from the assumptions and Lemma 14, h(t) ∈ C[0. T].
By applying Lemma 9 with ω(v) = vn we obtain
where and . That is
as long as
Our result follows from (8) and the bound in (13).
Corollary 18. If in addition to the hypotheses of Theorem 17, u ∈ Iα (L1(0,T)) then g(t) reduces to
Proof. This follows from Theorem 5.
Remark 3. If , for all 0 ≤ j ≤ k, and , then g ∈ L1(0, T).
For n = 1 we have the following inequality
Theorem 19. Let a, b ∈ CL1(0, T) be non-negative functions. Let c ∈ C(0, T] be a non-negative function. Let u ∈ L1(0, T) be such that u1-α∈ AC[0, T], 0 < α < 1, and satisfy the inequality
with 0 ≤ β0 < β1 < ... < β k < α. Then
where
and
Proof. This follows by applying Lemma 15 to (11).
Corollary 20. If k = 0 and β0 = β in Theorem 19, then g(t) and h(t) reduce to
and
Corollary 21. If in addition to the hypotheses of Theorem 19, u ∈ I α (L1(0, T)), then g(t) reduces to
Proof. This follows from Theorem 5.
For the next theorem we use the following expressions. Let
Theorem 22. Let a ∈ C(0, T) be such that is non-zero and finite. Let c ∈ C(0, T] be a non-negative function. Let u ∈ L1(0, T) be such that u1-α∈ AC[0, T], 0 < α < 1, and satisfy the inequality
where 0 ≤ β0 < β1 < ... < β k < α.
-
(a)
If a(t) is positive and non-decreasing then
where T1 is the largest value of t for which .
-
(b)
If a(t) is non-negative and non-increasing then
where T2 is the largest value of t for which
Proof.
-
(a)
When a(t) is positive and non-decreasing we can write the inequality (19) as
(20)
Let ψ(t) denote the right-hand side of (20). Then ψ(0) = 1,
and
Since ψ(t) is non-decreasing then by Corollary 8 we can write (22) in the form
where L1(t) and L2(t) are as defined in (18).
Using Lemma 16 (with p = 2) we obtain
as long as .
-
(b)
When a(t) is non-negative and non-increasing we can write (19) in the form
(23)
Denoting the right-hand side of (23) by φ(t), we have
and φ(t) = a0. By differentiation of φ we obtain
Then, we proceed as in the first part of the proof.
Corollary 23. Let a ∈ C(0, T) be such that is non-zero and finite. Let c ∈ C(0, T] be a non-negative function. Let u ∈ Iα (L1(0, T)), 0 < α < 1, satisfy the inequality (19). Let
Then,
-
(a)
If a(t) is positive and non-decreasing; and D 1 ∈ L 1(0, T), then
as long as .
-
(b)
If a(t) is non-negative and non-increasing; and D 2 ∈ L 1(0, T), then
as long as .
Proof. The result follows from Theorem 5 and Corollary 8.
For the next theorem, we introduce the following expressions.
Theorem 24. Let a ∈ C(0, T)with non-zero and finite. Let c ∈ C[0, T] be non-negative. Let u ∈ L1(0, T) be such that u1-α∈ AC[0, T], 0 < α < 1, and satisfy the inequality
where 0 ≤ β0 < β1 < ... < β k < α, 0 ≤ γ0 < γ1 < ... < γ m < α.
-
(a)
If a(t) is positive and non-decreasing; K 1 ∈ L 1(0, T), and K 2, K 3 ∈ C[0, T), then
where T3 is the largest value of t for which the bracket is positive.
-
(b)
If a(t) is non-negative and non-increasing; a(t) K 1(t) ∈ L 1(0, T), a(t) K 2(t), a(t) K 3(t) ∈ C[0, T), then
t ∈ (0, T4), where T4 is the largest value of t for which the bracket is positive.
Proof.
-
(a)
Suppose a(t) is positive and non-decreasing. Then, we can write the inequality (25) as
(26)
Let ψ(t) denote the right-hand side of (26). Then, ψ(0) = 1,
and
Since ψ is non-decreasing, by Corollary 8 we have
and
By substituting (29) and (30) into (28) and since ψ is non-decreasing, we obtain
where K1(t), K2(t) and K3(t) are as defined in (24). By integrating (31) we obtain
Applying Lemma 10 with
We obtain our result.
-
(b)
Suppose a(t) is non-negative and non-increasing. Then, we can write (25) in the form
(33)
Denoting the right-hand side of (33) by φ(t), we have |Dα u(t)| ≤ φ(t), φ(0) = a0, and
The reset of the proof is similar to that of the first part.
Corollary 25. Let a ∈ C(0, T), with non-zero and finite. Let c ∈ C[0, T] be non-negative. Let u ∈ Iα (L1(0, T)), 0 < α < 1, satisfy the inequality (25). Let
-
(a)
If a(t) is positive and non-decreasing; and if K 1 ∈ L 1(0, T), then
as long as .
-
(b)
If a(t) is non-negative and non-increasing and K 2 ∈ L 1(0, T), then
as long as .
Proof. The result follows from Theorem 5 and Corollary 8.
4. Applications
In this section, we illustrate our previous results by some applications. In particular, we show how to use these results to prove existence and determine the asymptotic behavior for some classes of fractional differential equations.
We consider the following Cauchy-type problem
where 0 ≤ β0 < β1 < ... < β k < α < 1, and f is a continuous function in all its variables.
Proposition 26. If u(t) has a assumable fractional derivative Dβu in (0, T), 0 ≤ β ≤ 1, then for α ≥ β,
(See [[11], p. 48].) In particular, we have
Theorem 27. If u ∈ L1(0, T) such that I1-αu ∈ AC[0, T]and satisfy the problem (34), then
Theorem 28. Suppose
with a, b ∈ CL1(0, T) and c ∈ C[0, T] are non-negative, and tσ b(t) ∈ C[0, T], . Further suppose the following hold.
-
(a)
for all 0 ≤ j ≤ k,
-
(b)
,
-
(c)
, where g(t) and h(t) are as defined by (4) and (5).
If u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then this solution exists for t ∈ (0, T0), where T0 is the largest value in (0, T) such that .
Proof. Following the proof of Theorem 17, we have
for all 0 < t < T0. By theorem 27, we have
Therefore, u(t) is bounded by a continuous function.
Example 1. Consider the problem
with f satisfying the inequality
Then, we can take T0 = 0.17826. Thus, if u ∈ L1(0, π) is a local solution with a summable fractional derivative D0.8u(t), this solution exists for 0 < t < 0.17826.
Theorem 29. Suppose that
where 0 ≤ β0 < β1 < ... < β k < α < 1, a ∈ C(0, T) with non-zero and finite, and c ∈ C(0, T) is non-negative. Let L k (t), k = 1, 2, 3, and T j , j = 1, 2 be as in Theorem 22.
Then, we consider the following cases.
-
(a)
a(t) is positive and non-decreasing, L 1, L 2 ∈ CL 1(0, T), and
In this case, if u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then this solution exists for all t ∈ (0, T1).
-
(b)
a(t) is non-negative and non-increasing, L 1, L 3 ∈ CL 1(0, T), and
In this case, if u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then this solution exists for all t ∈ (0, T2)
Proof. The result follows from Theorems 22 and 27.
Example 2. Consider the problem
with f satisfying the inequality
We have L1(t) = 0, L3(t) = 1.14 t0.17 ∈ CL1(0, 1), and 1-0.97 t0.17 > 0, t ∈ (0, 1.02638).
Thus, if u ∈ L1(0, 2) is a local solution with a summable fractional derivative D0.7u(t), then this solution exists for all t ∈ (0, 1.02638).
Theorem 30. Suppose that
where 0 ≤ β0 < β1 < ... < β k < α < 1, 0 ≤ γ0 < γ1 < ... < γ m < α < 1, a ∈ C(0, T) with non-zero and finite, and c ∈ C[0, T] is non-negative. Let K i (t), i = 1, 2, 3, , , T3 and T4 be as in Theorem 24. Then, we consider the following cases.
-
(a)
a(t) is positive and non-decreasing, for all i, j, K 2, K 3 ∈ C[0, T), and
In this case, if u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then this solution exists for all t ∈ (0, T3).
-
(b)
a(t) is non-negative and non-increasing, a(t) K 2(t), a(t) K 3(t) ∈ C[0, T), for all i, j, and
In this case, if u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then this solution exists for all t ∈ (0, T4).
Proof. The result follows from Theorems 24 and 27.
Example 3. Consider the problem
with f satisfying the inequality
By calculation we have for t ∈ (0, 0.404986). Thus, if u ∈ L1(0, 1) is a local solution with a summable fractional derivative D0.6u(t), then this solution exists for all t ∈ (0, 0.404986).
Example 4. Consider the problem
with f satisfy the inequality
It follows that if u ∈ L1(0, 1) is a local solution with a summable fractional derivative D0.7u(t), then this solution exists for all t ∈ (0, 1).
Finally, we show how the results in Section 3 can provide information about the behavior of the solutions for large values of t. For this purpose, we utilize the following lemma which is proved in [13].
Lemma. 31. Let α, λ, ω > 0, then
Theorem 32. Suppose f in (34) satisfy the inequality
with 0 ≤ β0 < β1 < ... < β k < α < 1, a, b ∈ CL1(0, T) and c ∈ C(0, T] are non-negative. Further, suppose the following.
-
1.
where M is a positive constant, and g(t), h(t) are as defined by (16) and (17).
-
2.
-
3.
If u ∈ L1(0, T) is a local solution of (34) that has a summable fractional derivative Dα u(t), then
where C is positive constant.
Proof. From Theorems 27 and 19, we have
where
Since then
and
The conditions 2-3 and Lemma 31 yield the result.
References
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. In Mathematics Studies. Volume 204. Elsevier; 2006.
Baleanu D, Mustafa O, Agarwal R: Asymptotically linear solutions for some linear fractional differential equations. Abstr Appl Anal 2010., 8: (Article ID 865139)
Baleanu D, Mustafa O, Agarwal R: An existence result for a superlinear fractional differential equation. Appl Math Lett 2010,23(9):1129–1132. 10.1016/j.aml.2010.04.049
Anastassiou GA: Fractional Differentiation Inequalities. Springer; 2009.
Furati KM, Tatar N-E: Some fractional differential inequalities and their applications. Math Inequal Appl 2006,9(4):577–598.
Furati KM, Tatar N-E: Inequalities for fractional differential equations. Math Inequal Appl 2009,12(2):279–293.
Furati KM: A fractional differential inequality with application. Nonlinear Anal Theory Methods Appl 2011,74(13):4330–4337. 10.1016/j.na.2011.03.027
Bainov D, Simeonov P: Integral Inequalities and Applications. Volume 57. Kluwer Academic Publishers; 1992.
Pachpatte BG: Inequalities for differential and integral equations. In Mathematics in Science and Engineering. Volume 197. Edited by: Ames WF. Academic Press; 1998.
Kolomogorov AN, Fomin SV: Fundamentals of the Theory of Functions and Functional Analysis (Russian). Moscow 1986.
Samko SG, Kilbas AA, Marichev OI: Fractional Integral Derivatives, Theory and Applications. Gorden and Breach, Amsterdam; 1993.
Bai C, Fang J: The existence of a positive solution for singular coupled system of nonlinear fractional differential equations. Appl Math Comput 2004, 150: 611–621. 10.1016/S0096-3003(03)00294-7
Kirane M, Tatar N-e: Global existence and stability of some semilinear problems. Arch Math 2000,36(1):33–44.
Acknowledgements
The authors are very grateful for the financial support provided by the King Fahd University of Petroleum and Minerals and Princess Nora Bint Abdurrahman University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors worked jointly on drafting and approving the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Al-Jaser, A., Furati, K.M. Singular fractional integro-differential inequalities and applications. J Inequal Appl 2011, 110 (2011). https://doi.org/10.1186/1029-242X-2011-110
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2011-110