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Bifurcation of Positive Solutions for a Class of Boundary Value Problems of Fractional Differential Inclusions
Using Krein-Rutman theorem, topological degree theory, and bifurcation techniques, this paper investigates the existence of positive solutions for a class of boundary value problems of fractional differential inclusions.
Fractional differential equations have been of great interest recently. Engineers and scientists have developed new models that involve fractional differential equations. These models have been applied successfully, for example, in mechanics (theory of viscoelasticity and viscoplasticity), (bio)chemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), and so forth. For details, see [1–7] and references therein. For example, in , Qiu and Bai considered the existence of positive solutions to BVP of the nonlinear fractional differential equation where , , and is the Caputo’s fractional derivatives. They obtained the existence of at least one positive solution by using Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone.
In , Tian and Liu investigated the following singular fractional boundary value problem (BVP, for short) of the form where , , and is continuous; that is, may be singular at and . By constructing a special cone, under some suitable assumptions, 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 .
In this paper, we consider the following boundary value problem of fractional differential inclusions of the form where , , is the Caputo’s fractional derivatives, and .
As mentioned in , the field of differential inclusions is a versatile and general area of mathematics that provides a framework for modelling physical processes that feature discontinuities. Examples of such phenomena include mechanical systems with Coulomb friction modeled as a force proportional to the sign of a velocity and systems whose control laws have discontinuities . In addition, differential inclusions are a useful format for treating differential equations where the right-hand side may be inaccurately known . Differential inclusions are also employed in the dynamic modelling of economic processes and game theory , control theory, optimization, partial differential equations, and the study of general evolution processes . The types of the aforementioned applications naturally motivate a deeper theoretical analysis of the subject.
Also there are some papers concerned with initial or boundary value problems of fractional differential inclusions (see, for instance, [9, 14–20] and references therein). The method used in these references is fixed point theorem. However, to the best of our knowledge, there is no paper studying such problems using bifurcation ideas. As we know, the bifurcation technique is widely used in solving boundary value problems (see, for instance, [21–24] and references therein). The purpose of present paper is to fill this gap. By using Krein-Rutman theorem, topological degree theory, and bifurcation techniques, the existence of positive solutions of BVP (3) is investigated.
The paper is organized as follows. Section 2 contains some preliminaries. In Section 3, by using bifurcation techniques, Krein-Rutman theorem, and topological degree theory, bifurcation results from infinity and trivial solution are established. Finally, in Section 4, the main results of the present paper are given and proved.
For convenience, we present some necessary definitions and results from fractional calculus theory (see ).
Definition 1. 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 2. For a function given on the interval , the th Caputo fractional-order derivative of is defined by Here is the smallest integer greater than or equal .
Lemma 3. Let , then the differential equation has solutions , for some , , where is the smallest integer greater than or equal to .
Lemma 4. Assume that with a derivative of order that belongs to . Then for some , , where is the smallest integer greater than or equal .
Lemma 5. The relation is valid in the following case:
For more detailed results of fractional calculus, we refer the reader to . In addition, we need the following preliminaries on multivalued operators.
Let be a Banach space. Then a multivalued map is convex (closed) valued if is convex (closed) for all . is bounded if is bounded in for any bounded set of .
is said to be lower semicontinuous, l.s.c. for short, if is open in whenever is open.
Let be a multivalued map and a single-valued function; if , , then is called a selection function of . If in addition is continuous, then is called a continuous selection.
The following lemmas are crucial in the proof of our main result.
Lemma 6 (25, Lemma 2.1, page 14). Let be a subset of a Banach space , and a l.s.c. with closed convex values. Then, given , has a continuous selection such that .
For more details on multivalued maps, see the books of Deimling .
Finally in this section, we list the following results on topological degree of completely operators.
Lemma 7 (Schmitt and Thompson ). Let be a real reflexive Banach space. Let to be completely continuous such that . Let be such that is an isolated solution of the equation
for and , where , are not bifurcation points of (10). 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 8 (Schmitt ). Let be a real reflexive Banach space. Let to be completely continuous, and let be such that the solution of (10) 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 (10) 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 9 (Guo ). Let be a bounded open set of infinite-dimensional real Banach space , and let be completely continuous. Suppose that(i);(ii), .
3. Bifurcation Results
3.1. Assumptions and Conversion of BVP (3)
Suppose that the following two assumptions hold throughout the paper.
(H1) Let be a nonempty, closed and convex multivalued map such that is l.s.c., where .
(H2) There exist functions with in any subinterval of such that for , where with as uniformly with respect to , , and as uniformly with respect to , .
The basic space used in this paper is . Obviously, is a Banach space with norm (). Let It is easy to see that is a cone of . Moreover, from (17), we have for all ,
We first consider the following linear boundary problem of fractional differential equation: where .
For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameters:
A function is said to be a solution of BVP (24) if satisfies (24). In addition, if , for , then is said to be a positive solution of BVP (24). Obviously, if , is a solution of BVP (24), then by (18) we know that is a positive solution of BVP (24), where denotes the zero element of Banach space .
For with in any subinterval of , define the linear operator by where is defined by (21).
Lemma 12. The operator defined by (25) has a unique characteristic value , which is positive, real, and simple and the corresponding eigenfunction is of one sign in ; that is, we have .
Notice that the operator can be regarded as . This together with Lemma 12 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
Define Then on . From Lemma 10, the solution of is equivalent to the fixed point of operator
Lemma 13. For , is a positive solution of BVP (27) if and only if is a nontrivial solution of BVP (29); that is, is a nontrivial fixed point of operator in . Therefore, the closure of the set of nontrivial solutions of BVP (29) in is exactly .
3.2. Bifurcation from Infinity and Trivial Solution
Lemma 14. Let be a compact interval with . Then there exists such that
Proof. Suppose, on the contrary, that there exist with such that . Without loss of generality, assume . Notice that . By Lemma 13, (17), and (18), we have in . Set . Then . From the continuity 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 (H2) we know Therefore, by virtue of (30), we know Let and be the positive eigenfunctions of , corresponding to and , respectively. Then from (33), it follows that Letting and using condition (H2), we have which implies . Similarly, one can deduce from (34) that .
To sum up, , which contradicts with . The conclusion of this lemma follows.
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. We first prove that for , there exists such that
Suppose, on the contrary, that there exist with such that .
By Lemma 13, in . Set ; that is, . Without loss of generality, assume . First we show . From (32) and the continuity of , it is easy to see that is relatively compact in . Suppose . Notice that and . Therefore, for . Consequently, From (H2) and Lemma 11, it is easy to see . If , letting in the above inequality, we can obtain a contradiction. So and it is reasonable to suppose (relabeling if necessary) in . By virtue of (32), we know
Letting and using condition (H2), we obtain that which implies . This is a contradiction. Therefore, (42) holds. By Lemma 9, for each , there exists such that
The conclusion of this lemma follows.
Theorem 17. is a bifurcation interval of positive solutions from infinity for BVP (27), 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 (27) which meets and is unbounded in direction.
Proof. From Lemma 13, we need only to prove that the conclusion holds for (29).
For fixed with , by Lemmas 15, 16, and their proof, there exists such that all of the conditions of Lemma 8 are satisfied with , , and . So, there exists a closed connected set of solutions of (29), which is unbounded in . From Lemma 14, the case (ii) of Lemma 8 cannot occur. Thus, bifurcates from infinity in and is unbounded in direction. In addition, for any closed interval , by Lemma 14, the set is bounded in . Therefore, must be bifurcated from infinity in , which implies that can be regarded as . Consequently, is unbounded in direction.
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 the trivial solution for BVP (27); that is, there exists an unbounded component of positive solutions of BVP (27), which meets . Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with .
4. Main Results
The main results of this paper are the following two conclusions.
Theorem 22. Suppose that (H1) and (H2) hold. In addition, suppose either(i) or(ii).
Then BVP (3) has at least one positive solution.
Proof. We need only to prove that there is a component of that crosses the hyperplane , where is the closure of the set of positive solutions of BVP (27). Notice that is the only solution of (27) with . By Lemmas 14 and 18, for any component of , we have .
Case (i). Consider .
From Theorem 17, there exists an unbounded component of solutions of (27), which meets and is unbounded in direction.
If , by and Theorem 17, we know that must cross the hyperplane .
If , by Theorem 21, we know . Therefore, joins to . This together with guarantees that crosses the hyperplane .
Case (ii). Consider .
From Theorem 21, there exists an unbounded component of positive solutions of BVP (27), which meets . Moreover, there exists no bifurcation interval of positive solutions from the trivial solution, which is disjointed with .
We show that must cross the hyperplane . Suppose, on the contrary, . From , we know . Notice that is unbounded. Then must joint . By Theorem 17, it is a contradiction with . Thus the result follows.
Theorem 23. Suppose that (H1), (H2), and the following assumption holds.(H3)There exist and such that for ,
In addition, suppose
Then BVP (3) has at least two positive solutions.
Proof. From Theorems 17 and 21, there exist two unbounded components and of solutions of (27), which meet and , respectively. It is sufficient to show that and are disjoint in and both cross the hyperplane .
For this sake, from assumption (H3), there exists such that Now we show , where . Suppose that, on the contrary, is a solution of (27) such that and . Then by Lemma 13, we know . Therefore, for . From (H3), (30), and Lemma 13, it follows that which is a contradiction. Thus, , which implies
Immediately, and are disjoint in .
Notice that and are both unbounded. Moreover, , , and is unbounded in direction. So and both cross the hyperplane . This means that there exist and with and .
Consequently, BVP (3) has at least two positive solutions.
5. An Example
Example 24. Consider the following boundary value problem of fractional differential inclusions
Then BVP (55) has at least one positive solution.
Proof. BVP (55) can be regarded as the form (3). From (56), one can see that (H1) and (H2) are satisfied with , , , , and .
By the definition of , it is easy to see .
Therefore, by Theorem 22, BVP (55) has at least one positive solution.
The research is supported by NNSF of China (11171192) and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF025).
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Copyright © 2013 Yansheng Liu and Huimin Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.