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Wenzhe Xie, Jing Xiao, Zhiguo Luo, "Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem", Abstract and Applied Analysis, vol. 2014, Article ID 540351, 9 pages, 2014. https://doi.org/10.1155/2014/540351
Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem
By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.
Fractional calculus, which is a powerful tool for the description of memory and hereditary properties of materials and processes, has attracted the attention of many researchers and has been successfully applied in various fields, such as economics, engineering, and physical sciences. For the development of the theory of fractional calculus, we refer the readers to the monographs [1, 2] and references therein.
Recently, some papers have dealt with the existence of the solutions for fractional boundary value problems mainly by means of fixed point theorems [3–5], Leray-Schauder continuation principle , critical point theory [7, 8], and the method of upper and lower solutions [9–12]. Therein, various kinds of boundary value problems for nonlinear fractional differential equations have been studied, and some excellent results have been established. We are particularly interested in the case where the nonlinear term depends explicitly on the fractional derivative of lower order. Specifically, Su and Zhang in  deal with a boundary value problem of a fractional differential equation with the nonlinear term dependent on a fractional derivative of lower order on the semi-infinite interval: where , , and and are the standard Riemann-Liouville fractional derivatives. The existence results for solutions are obtained by using Schauder's fixed point theorem on an unbounded domain. And in  Agarwal et al. investigate the existence of positive solutions for the singular Riemann-Liouville fractional Dirichlet boundary value problem: where , , and satisfying the Carathéodory conditions and being singular at . The proofs are based on a fixed point theorem on a cone, regularization, and sequential techniques.
It is well known that the method of upper and lower solutions is a powerful tool for proving the existence and multiplicity results of solutions for nonlinear differential equations. Using this method and monotone iterative technique, the authors in [10–12] investigate some nonlinear fractional differential equations with nonlinear boundary conditions and establish some fractional comparison principles and further obtain the existence results of solutions, including extremal solutions, yet, mainly focus on the case of order .
Nagumo conditions play an important role in the boundary value problems with nonlinear term involved in the derivative, since as it is known, for instance, for second order differential equations, the existence of upper and lower solutions, by itself, is not sufficient to ensure the existence of solutions. The studies dealing with the Nagumo conditions are well established by applying the method of upper and lower solutions combined with fixed point theorem or topological degree theory for the case of integer order (see [15–17]). To the best of our knowledge, no work has been done concerning the existence of solutions for fractional boundary value problem with nonlinear terms involving fractional derivative under Nagumo conditions.
Inspirited by the papers mentioned above, in this paper, under Nagumo conditions we aim to apply the method of upper and lower solutions combined with fixed point theorems to discuss the existence of solutions for the following Riemann-Liouville fractional boundary value problem (FBVP for short): where is continuous and is the Riemann-Liouville fractional derivative of order , . Our results extend some classical results for second order differential equations to the case of fractional order .
This paper is organized as follows. In Section 2, some notations, definitions, and lemmas are presented. We establish some results concerning the Riemann-Liouville fractional derivatives at extreme points under weaker conditions than those in . In Section 3, sufficient conditions are given for the existence of at least one solution for FBVP (3). In Section 4, an explicit example is given to illustrate our main results.
In this section, we introduce some definitions and lemmas, which are used throughout this paper.
A function is Hölder continuous, if there exist nonnegative constant and exponent , such that A function is, especially, Lipschitz continuous, if the above inequality holds for .
Lemma 3 (see ). Suppose that exists and is integrable on , ; then
Lemma 4. Suppose that and that , , then .
Proof. For all , , we let
By Lemma 3, (8), and , one gets
Together with Property 1(3) and the continuity of , taking the limit , we obtain
From (11), (12), and the uniform convergence of , it follows that
Using (9) and Property 1(1)-(2) we obtain that exists and belongs to , satisfying
To the end, it suffices to show that , . Obviously, it follows from (9) and (14) that For the homogeneous Abel integral equation (15), we observe that the integrand belongs to . Then by Lemma 2.5 in  Taking the limit in (16), together with (13) we have Hence , . The proof is complete.
Lemma 6. Suppose that and , , then , .
Lemma 7. If , then the following fractional boundary value problem has a unique solution: where
The proof is standard; we omit it here.
Remark 8. Obviously, and , .
In , Al-Refai obtained the following interesting result concerning the Riemann-Liouville fractional derivative at extreme points, where there is a little mistake. Now we state it correctly without proof.
Theorem 9 (see ). Let attain its global minimum at ; then
Remark 10. In the sense of Riemann-Liouville fractional derivative of order , in general the assumption that is difficult to meet due to the fact that the fundamental solution of certain corresponding homogeneous differential equations possesses a singularity at . Hence, we hope to weaken the conditions of the above theorem.
Lemma 11. Assume that satisfies the following conditions: (i) exist, , for ;(ii)there exists constant , such that is Hölder continuous with exponent ;(iii) attains its global minimum at .Then, Moreover, if , then .
Proof. Let , . Obviously, , satisfying the conditions (i)–(iii). It follows that
Since , we know that . Then for , it follows from proofs of Lemma 2.1 in  that
At this point, we choose enough small constant , such that . Since exists, . We consider It obviously follows from that the second integration in (26) converges, and then the first integration in (26) also converges. For the latter, applying the integration by parts, together with (23), we have due to the fact , , and .
Hence (26) yields that
From the Hölder continuity of on for some , it follows that there exists constant , such that For , we have . Thus, Dividing by on both sides of (30) and taking the limit , one gets Together with (24), we obtain The proof is complete.
Remark 12. Lemma 11 is an essential improvement of Theorem 9 and crucial for our main theorems. By applying the above results on , analogous results for Riemann-Liouville fractional derivatives at global maximum points are derived. It is worth mentioning that the weaker requirement in (ii) seems to be unsatisfactory as well, since in general the exponent may be guaranteed only up to . To solve the difficulty, the ideas of reducing the order and approach method are employed in our main theorems.
Spontaneously for the case , Lakshmikantham and Vatsala in  established the following result concerning Riemann-Liouville fractional derivative.
Theorem 13 (see ). Let be locally Hölder continuous such that for any , we have Then it follows that
Remark 14. As the literature  points out, in general, the function containing term is not locally Hölder continuous of any order. For this reason, in [20, 21] the authors attempt to weaken the locally Hölder continuity to continuity on ; nevertheless, their arguments seem to be flawed as well. Similarl to the proof of Lemma 11, we get the following lemma.
Lemma 15. Assume that satisfies the following conditions: (i) exist, , for ;(ii)there exists constant , such that is Hölder continuous with exponent ;(iii) attains its global minimum at .Then, Moreover, if , then .
Recently, some results, for instance, Lemma 2.13 in  and Property 4  concerning Hölder continuity for Riemann-Liouville fractional integral operator have been obtained. Therein Bourdin in  has proved that fractional integral operator , maps functions to Hölder continuous functions, the exponent of which depends on and . In the following, we give some other results about Hölder continuity.
Lemma 17. Let , . Then is Hölder continuous with exponent on .
Lemma 18 (see ). Suppose that , ; then,
Remark 19. Obviously, Lemma 18 is valid under the assumptions , .
Lemma 20. Let be Hölder continuous with exponent on . Then , is Hölder continuous with exponent on .
Proof. From the Hölder continuity of on , it obviously follows that there exist some , such that and Without loss of generality, let . Then, If , using Lemma 18 we have If , using the mean value theorem we have The proof is complete.
Corollary 21. Let , . Then , is Hölder continuous with exponent on .
Now we introduce the upper and lower solutions of FBVP (3).
Definition 22. A function , satisfying with , , is called a lower solution of FBVP (3), if it satisfies Analogously, a function , satisfying with , , is called an upper solution of FBVP (3), if it satisfies (41)-(42) with reversed inequalities.
Definition 23. Given a pair of functions satisfying . A function is said to satisfy the Nagumo condition with respect to and , if there exists a function such that for all , and
3. Main Results
In this section, we will apply the method of upper and lower solutions combined with fixed point theorem to consider the existence of solutions of FBVP (3).
Denote by the set consisting of the Hölder continuous functions with exponent on and .
The main results in this paper are the following.
Theorem 24. Assume that the following conditions hold: FBVP (3) has a pair of upper and lower solutions , respectively, with is nonincreasing with respect to the third variable; satisfies the Nagumo condition with respect to and .Then FBVP (3) has at least one solution with , such that
Proof. From the assumptions and , we know
due to the monotonicity of Riemann-Liouville fractional integral operator . And choose constant , such that
It is easy to obtain that . We let and consider the following modified fractional boundary value problem: where Obviously, is bounded; that is, there exists positive constant , such that
To the end, it is sufficient to show that the modified FBVP (51)-(52) has at least one solution , satisfying
We divide the proof into three steps.
Step 1. FBVP (51)-(52) has at least one solution with .
Firstly, we define the operator by From the continuity of and , it is not difficult to verify that the operator is well defined and continuous.
By Lemma 7 we can see that the fixed points of coincide with the solutions of FBVP (51)-(52). In the following, we prove that has a fixed point in .
Secondly, since , , is bounded by , and , together with (56) we easily obtain that is uniformly bounded (here, is a bounded subset of ).
For , , without loss of generality, let . We have That is, is equicontinuous. According to the Ascoli-Arzela theorem, we know that is completely continuous.
By the Schauder fixed point theorem, we can easily obtain that has at least one fixed point with .
Step 2. The function satisfies , .
Suppose that on is not true; then has a negative minimum at some ; that is, .
If , then . From (42) and (52) we have the contradiction .
If , that is, . Obviously, , , which deduces . Choosing enough small , by Lemma 4 we obtain that exists, and
Now, denote , for brevity, by . From (56) it follows that Choosing in Lemma 11, then Obviously, the first term on the right side of (60) is Lipschitz continuous in . For the second term on the right side of (60), by the continuity of and Corollary 21 we obtain . Thus, it is deduced that .
On account of and , we have and is Hölder continuous with exponent . At this point, by Lemma 11 we obtain Taking the limit in (61), it follows by (58) and that
On the other hand, firstly we claim that it holds that . It obviously follows that . Then by Lemma 6 we know Analogously with above arguments for (62), by Lemma 15 and (63) it is not hard to obtain that That is, . The claim is proved. Again together with , , and , for , we have the following two cases.
Case 1. When , which contradicts (62).
Case 2. When , which contradicts (62).
Thus, we know that the minimum point satisfying does not occur on .
If , that is, . By the boundary conditions (42) and (52), we have
However, analogously with above arguments for (64) we obtain which is a contradiction.
Then it holds that , . Analogously we can also obtain that , . Hence, we have , .
Step 3. We prove that on .
We only need to show that on . Similarly we can show that on .
Suppose that on is not true; then there exists , satisfying . Due to , and , , we know that , and by the mean value theorem, there exists , such that Since , there exists an interval (or ) such that Thus, by we have for , Then, However, by (49)-(50) and Property 1(4) we have which is a contradiction with (72). So there holds that , . Hence, we have that , .
Consequently, combining Step 2 and Step 3, we obtain that That is to say, the solution is a solution of FBVP (3). Then FBVP (3) has at least one solution with , such that , .
Remark 25. Observe that the validity of the first inequality in (62) can be guaranteed by means of applying approach method. With this idea, combining Lemma 3, Remark 5, Lemma 11, and Corollary 21, under certain stronger conditions instead of (i)-(ii) in Lemma 11, in the end we have two more concise conclusions as follows.
Theorem 26. Assume that satisfies the following conditions: (i), for ;(ii) attains its global minimum at .Then, Moreover, if , then .
Remark 27. In fact, with implies . We emphasize that this result may help to establish some fractional comparison principles for the case of order lying in , which play a very important role in studying Riemann-Liouville fractional differential equations by means of monotone iterative method. Some related studies will be given in a future paper.
Theorem 28. Assume that satisfies the following conditions: (i), for ;(ii) attains its global minimum at .Then, Moreover, if , then .
Example 29. Consider the following fractional boundary value problem:
Let , . Obviously, , and , . It follows that is nonincreasing with respect to . Choose , , then , , and , , which deduces that . And it is not difficult to check out that is a lower solution of FBVP (77). Analogously, , is an upper solution of FBVP (77).
Meanwhile, if we choose , then when , , , it holds that