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Positive Solutions Using Bifurcation Techniques for Boundary Value Problems of Fractional Differential Equations
This paper is concerned with the existence of positive solutions for a class of boundary value problems of fractional differential equations with parameter. The main tools used here are bifurcation techniques and topological degree theory. Finally, an example is worked out to demonstrate the main result.
During the last few decades, fractional calculus and fractional differential equations have been studied extensively since fractional-order models are found to be more adequate than integer-order models in some real-world problems. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth involves derivatives of fractional order. For details and examples, see [1–7] and references therein. Recently, there have been a few papers which deal with the boundary value problem for fractional differential equation. For example, in , Tian and Liu investigated the following singular fractional boundary value problem (BVP, for short) of the form where is Caputo’s fractional derivatives, , , and is continuous; that is, may be singular at and . By constructing a special cone, they obtained that there exist positive numbers and with such that the above system has at least two positive solutions for and no solution for under some suitable assumptions such as the following.(A1) There exists an interval such that uniformly with respect to .
In , Bai and Lü consider the following nonlinear fractional differential equation Dirichlet-type boundary value problem: where is a real number and is the standard Riemann-Liouville differentiation. The corresponding Green function is derived. By means of some fixed point theorems on cone, the existence and multiplicity of positive solutions for BVP (2) were investigated.
In , Jiang and Yuan further investigated BVP (2). Comparing with , they deduced some new properties of the Green function, which extended the results of integer-order Dirichlet boundary value problems. Based on these new properties and Krasnoselskii fixed point theorem, the existence and multiplicity of positive solutions for BVP (2) were considered.
In this paper, by using bifurcation techniques, we consider the following boundary value problem of fractional differential equation: where , is the standard Riemann-Liouville differentiation, is a given constant, and is a given continuous function satisfying some assumptions that will be specified later.
It is remarkable that the method used in references mentioned above was fixed point theorems and the same kind of conditions was used such that the nonlinearity satisfies superlinear or sublinear condition at and , which is similar to (A1). To the best of our knowledge, there is no paper studying such fractional differential equations using bifurcation ideas. As we know, the bifurcation technique is widely used in solving BVP of integer-order differential equations (see, e.g., [11–13] and references therein). In , by virtue of bifurcation ideas, the authors studied a kind of BVP of differential inclusions. The purpose of present paper is to fill this gap. By using bifurcation techniques and topological degree theory, the existence of positive solutions of BVP (3) is investigated. The main features of present paper are as follows. First, the nonlinearity is asymptotically linear at and , not super-linear or sub-linear (see the condition (H1) in Section 2 and example in Section 4). Next, the main method used here is bifurcation techniques and topological degree, not fixed point theorem on cone, which is different from the references.
The paper is organized as described below. At the end of this section, for completeness, we list some results on bifurcation theory from interval and topological degree of completely continuous operators. Section 2 contains background materials and preliminaries. In Section 3, by using bifurcation techniques and topological degree theory, bifurcation results from infinity and trivial solution are established. Then the main results of present paper are given and proved. Finally in Section 4, an example is worked out to demonstrate the main results.
Lemma 1 (Schmitt and Thompson ). Let be a real reflexive Banach space. Let to be completely continuous such that , for all . Let () be such that is an isolated solution of the equation
for and , where , are not bifurcation points of (4). Furthermore, assume that
where is an isolating neighborhood of the trivial solution. Let
Then there exists a connected component of containing in , and either(i) is unbounded in or(ii).
Lemma 2 (Schmitt ). Let be a real reflexive Banach space. Let to be completely continuous, and let () be such that the solution of (4) is, a priori, bounded in for and ; that is, there exists an such that for all with . Furthermore, assume that for sufficiently large. Then there exists a closed connected set of solutions of (4) that is unbounded in , and either(i) is unbounded in direction or(ii)there exists an interval such that and bifurcates from infinity in .
Lemma 3 (Guo ). Let be a bounded open set of real Banach space , and let be completely continuous. If there exists , such that then
2. Background Materials and Preliminaries
For convenience, we present some necessary definitions and results on fractional calculus theory (see ).
Definition 4. The fractional (arbitrary) order integral of the function of order is defined by where is the gamma function. When , we write , where for , and for , and as , where is the delta function.
Definition 5. For a function given on the interval , the th Riemann-Liouville fractional-order derivative of is defined by where .
Lemma 6. Let ; then, the differential equation has solutions , for some , , where is the smallest integer greater than or equal to .
Notice that for all . From Lemma 6, we deduce the following result.
Lemma 7. Assume that with a derivative of order that belongs to . Then for some , , where is the smallest integer greater than or equal to .
For more detailed results of fractional calculus, we refer the reader to .
Now let us list the following assumption satisfied throughout the paper.(H1) There exist two positive numbers with and functions with in any subinterval of such that where , with as uniformly with respect to (), and as uniformly with respect to ().
To solve BVP (3), we first consider the following linear boundary problem of fractional differential equation: where . We cite the following two lemmas from references.
The basic space used in this paper is Obviously, is a Banach space with norm (for all ).
Let It is easy to see that is a cone of . Moreover, from (21), we have, for all ,
For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameter :
A function is said to be a solution of BVP (23) if satisfies (23). In addition, if , for , then is said to be a positive solution of BVP (23). Obviously, if , is a solution of BVP (23), then by (22), we know that is a positive solution of BVP (23), where denotes the zero element of Banach space .
Define Then on . Let By assumption (H1) and using a similar process of the proof of Lemma 4.1 in , we know that is completely continuous.
From Lemma 8, if is the the fixed point of operator , then is the solution of
Let where is the zero element of . From Lemma 9 and the definitions of and the cone , it is easy to see that . Moreover, we have the following conclusion.
Lemma 10. For , if is a nontrivial fixed point of operator , then is a positive solution of BVP (23). Furthermore, is a positive solution of BVP (23) if and only if is a nontrivial solution of BVP (26).
For with in any subinterval of , define the linear operator by where is defined by Lemma 9.
Lemma 11. The operator defined by (28) is completely continuous and has a unique characteristic value , which is positive, real, and simple and the corresponding eigenfunction is of one sign in ; that is, for all .
Notice that the operator can be regarded as . This together with Lemma 11 guarantees that is also the characteristic value of , where is the conjugate operator of . Let denote the nonnegative eigenfunction of corresponding to . Then we have
3. Main Results
The main results of present paper are the following two theorems.
Theorem 12. Suppose that either(i) or(ii).
Then BVP (3) has at least one positive solution.
Theorem 13. Suppose the following.(H2) There exist and such that
In addition, if
then BVP (3) has at least two positive solutions.
Lemma 14. Let be a compact interval with . Then there exists such that
Proof. If this is false, then there exist with such that . Without loss of generality, assume and for all . Notice that . By Lemma 10 and (21), we have in . Set . Then . From the definition of , it is easy to see that is relatively compact in . Taking a subsequence and relabeling if necessary, suppose in . Then and .
On the other hand, from (H1), we know Therefore, by virtue of (25), we know Let and be the positive eigenfunctions of corresponding to and , respectively. Then from (34), it follows that Letting and using condition (H1), we have which implies . Similarly, one can deduce from (35) that .
Consequently, , which contradicts . Therefore, there exists such that
Lemma 15. For , there exists such that
Proof. Notice that . From Lemma 14, there exists such that
Therefore, by the homotopy invariance of topological degree, we have
Lemma 16. For , there exists such that
Proof. First we prove that for , there exists such that
where is the positive eigenfunctions of corresponding to .
If this is false, then there exist with () and such that
By Lemma 10, we have in . From , there exists satisfying . Then condition (H1) guarantees that there exists such that for . Noticing , there exists such that for . Consequently, by virtue of (25) and (45), for , we know Let be the positive eigenfunction of corresponding to . Then
This together with guarantees that which is a contradiction. Therefore, (44) holds. By Lemma 3, for each , there exists such that
Theorem 17. is a bifurcation interval of positive solutions from the trivial solution for BVP (23); that is, there exists an unbounded component of positive solutions of BVP (23), which meets . Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with .
Proof. By virtue of (27) and Lemma 10, we need only to prove that there exists an unbounded component of , which meets , and there exists no bifurcation interval of from the trivial solution which is disjointed with .
For fixed with , by Lemmas 15 and 16 and their proof, there exists such that all of the conditions of Lemma 1 are satisfied with , , and . This together with Lemma 10 guarantees that there exists a closed connected set of containing in . From Lemma 14, the case (ii) of Lemma 1 cannot occur. Thus, bifurcates from and is unbounded in . Moreover, for any closed interval , by Lemma 14, there exists such that the set . Therefore, must be bifurcated from , which implies that can be regarded as . In addition, using Lemma 14 again, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with .
By a process similar to the above, one can obtain the following conclusions.
Lemma 18. Let be a compact interval with . Then there exists such that
Lemma 19. For , there exists such that
Lemma 20. For , there exists such that
Theorem 21. is a bifurcation interval of positive solutions from infinity for BVP (23), and there exists no bifurcation interval of positive solutions from infinity which is disjoint with . More precisely, there exists an unbounded component of solutions of BVP (23) which meets and is unbounded in direction.
Proof of Theorem 12. Obviously the solution of the form of (23) is a positive solution of BVP (3). So by Lemma 10, it is sufficient to show that there is a component of that crosses the hyperplane , where is defined by (27).
Case i . By Theorem 17, there exists an unbounded component of positive solutions of BVP (23), which bifurcates from . Therefore, there exists such that
If there exists some such that , the conclusion follows. Suppose, on the contrary, for all . Since is the only solution of (23) with , by Lemmas 14 and 18, we know . Therefore, for all . Taking a subsequence and relabeling if necessary, suppose as . Then . This together with (53) guarantees that .
Choose for . From Lemma 18, it follows that for each , which means . This is a contradiction.
Case ii . From Theorem 21, there exists an unbounded component of solutions of (23) which bifurcates from and is unbounded in direction.
If , using the fact that and is unbounded in direction, we know that must cross the hyperplane .
If , by and Theorem 17, we know . Therefore, joins to . This together with guarantees that crosses the hyperplane .
Proof of Theorem 13. First we show that there exists such that
where , is defined by (27).
In fact, from assumption (H2), it follows that there exists such that If there is a solution of such that and , then By virtue of (25) and Lemma 9, we have which is a contradiction. Thus, .
Next, from Theorem 17, there exist unbounded components of solutions of (23), which meet . By (54), we know . This together with the fact that is unbounded, , and guarantees that crosses the hyperplane . Then BVP (23) has a positive solution with and . By Lemma 10, is a positive solution of BVP (3).
Similarly, by Theorem 21 and (54), BVP (3) has a positive solution with and . The conclusion follows.
Immediately, from the proof of Theorem 13, we have the following corollary.
Corollary 22. Suppose that assumption (H2) holds. In addition, suppose that one of the following two conditions holds:(i);(ii).
Then BVP (3) has at least one positive solution.
4. An Example
Example 1. Consider the following boundary value problem of fractional differential inclusions:
Then BVP (58) has at least one positive solution as .
Proof. BVP (58) can be regarded as the form (3). Let for ; then, is continuous.
From (59), choose , , , , , , , , .
It is easy to see as and as both uniformly with respect to , .
Therefore, (H1) is satisfied.
By the definition of , it is easy to see , .
As a result, by Theorem 12, BVP (58) has at least one positive solution as .
The author wishes to thank the anonymous referees for their valuable suggestions. Research is supported by the NNSF of China (11171192), the Graduate Educational Innovation Foundation of Shandong Province (SDYY1005), and the Natural Science Foundation of Shandong Province (ZR2013AM005).
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