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Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory
This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: , , where , with , and stand for the left and right Riemann-Liouville fractional derivatives of order , respectively, and is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.
Consider the following nonlinear fractional boundary value problem (BVP for short):where , with , and stand for the left and right Riemann-Liouville fractional derivatives of order , respectively, and is continuous.
Fractional calculus has numerous applications in various fields, including signal processing and control, fractal theory, neural network model, mechanics and engineering, and chemical physics. For details, see [1–5] and the references therein. Fractional differential operators have attracted many researchers’ attention because of their applications in modeling physical phenomena exhibiting anomalous diffusion. Recently, the existence of solutions for nonlinear fractional differential equations was obtained by using nonlinear analysis, including some fixed theorems, coincidence degree theory, the monotone iterative method, and the method of upper-lower solutions. It is worth mentioning that the methods above cannot be used to deal with BVP (1) because it is very difficult to find the equivalent integral equation.
On the other hand, variational methods have proved to be a very effective approach in dealing with the existence and multiple solutions for fractional boundary value problem including both left and right fractional derivatives; see [6–16].
For example, in , by using mountain pass theorem, Jiao and Zhou studied the existence of at least one nontrivial solution for the following fractional boundary value problem:where , is a suitable given function, and is the gradient of with respect to
In , by means of variational methods and critical point theory due to Bonanno and Marano, the existence of three distinct weak solutions is obtained for the following fractional boundary value problem:where is a positive real parameter, with and , is a function such that is continuous in , and is a function.
In , by using fountain theorem and dual fountain theorem, the existence of infinitely many solutions is established for the following fractional boundary value problem:where and are the left and right Riemann-Liouville fractional integrals of order , respectively.
To the best of our knowledge, there are fewer results on multiple solutions for system (1). The purpose of this paper is to fill this gap. Inspired by the above references, we apply variant fountain theorems to study the existence of infinitely many small or high energy solutions for BVP (1). It should be noted that the (P.S.) condition is not needed, which is an important condition assumed in [7, 9]. As is well known, the so-called Ambrosetti-Rabinowitz conditionfor some is often assumed to deduce the boundedness of the (P.S.) sequences. By taking advantage of the variant fountain theorems, the Ambrosetti-Rabinowitz condition is not needed.
The rest of this paper is organized as follows. Section 2 contains preliminary results. In Section 3, we apply variant fountain theorems to prove the existence of infinitely many small or high energy solutions for BVP (1). In Section 4, two examples are presented to illustrate the main results.
To obtain multiple solutions for BVP (1), it is necessary to construct a suitable function space and introduce some definitions, preliminary lemmas.
Definition 1 (see ). Let be a function defined on The left and right Riemann-Liouville fractional derivatives of order for function , denoted by and , are defined by where , , and .
Lemma 2 (see ). If , , , or , , , thenDenote
Definition 3 (Let ). The fractional derivative space is defined by the closure of with respect to the norm
Remark 4. (i) The fractional derivative space is the space of functions having an -order Riemann-Liouville fractional derivative , with , and . (ii) is a reflexive and separable Banach space.
Lemma 5 (see , let ). For any , one has
Remark 6. (i)By (10), we can get (ii)According to (11) and (12), for , we can obtain(iii)From (11), it is easy to see that the norm of defined in (9) is equivalent to the norm (iv) is a separable Hilbert space and the corresponding inner product is
Lemma 7 (see , let ). If the sequence converges weakly to in , that is, , then in , that is, as .
In the following, we denote for convenience.
Definition 8. A function is a weak solution of BVP (1) if
The energy functional associated with BVP (1) is defined by where . It is easily seen that .
For any , one has Moreover, the critical points of correspond to the weak solutions of BVP (1).
To prove our main results, we need the following two variant fountain theorems in .
Let be a Banach space with the norm and with for each . Set , , , , where .
Consider a family of functions defined by
Lemma 9. Assume that the functional defined above satisfies(A1) maps bounded sets into bounded sets uniformly for , and for all ;(A2) for all or as ; or(A3) for all , as ;(A4)There exist such that Then where . Moreover, for almost every , there exists a sequence such that
Lemma 10. Assume that the functional defined above satisfies(B1) maps bounded sets into bounded sets uniformly for , and for all ;(B2) for all , as on any finite dimensional subspace of ;(B3)There exist such that Then there exist , such thatwhere . In particular, if has a convergent subsequence for every , then has infinitely many nontrivial critical points satisfying as .
Since is a separable Hilbert space in terms of Remark 6. Let be an orthonormal basis of and define , , . In order to apply Lemmas 9 and 10 to prove the existence of infinitely many solutions of BVP (1), we define , , and on fractional derivative space by
For convenience, we list the following assumptions.(H1)There exist constants , , and such that (H2)There exist and such that (H3).(H4), .(H5)There exist and with such that (H6)There exist and such that (V), and .(), and , where
3. Main Results
Theorem 11. Assume that (H1)–(H4), ( hold. Then BVP (1) possesses infinitely many high energy solutions satisfying
Proof. For any , it follows from (H1) that there exist and such that Combining (32) and (13), it is easily seen that maps bounded sets into bounded sets uniformly for By (H4), for all . Thus, condition (A1) holds. Assumption (H3) implies that . Condition (A2) holds for the fact that as and .
In what follows, we verify condition (A4). For this sake, we need to prove that there exist two sequences such that First, we prove that (33) is true.
Let with . We claim that as . Indeed, if not, then there exist and with such that and , where as . For any , we may choose such that as . Thus, in By Lemma 7, we have in . Consequently, . This contradicts the fact that .
For , by (32), we have Choose . Then as . Then, for any with , we obtain Therefore,
Next, we prove that (34) is true.
We first prove that, for any finite dimensional subspace of , there exists a constant such that If not, there exists a sequence such that Set . Then , and On account of the boundedness of and , passing to a subsequence if necessary, we may assume that in for some . According to the equivalence of any norm in finite dimensional space, we get Since , there exists a constant such that Set , , and . Then for large enough, it follows from (41) and (43) that Consequently, This contradicts (42). Hence, (39) holds.
Since is a finite dimensional subspace of , from (39), for any , there exists such that where . In view of (H2), for any , there exists a constant such that By (46), for any with , we get for all . Therefore, If we take , we get
Until now, all the conditions of Lemma 9 hold. Hence, for , there exists a sequence such that Furthermore,Thus,
Choose a sequence such that (50) holds. Now we show that possesses a strong convergent subsequence. In fact, the sequence is bounded. Then there exists such that as . By Lemma 7, in as . From , we know Also from in , we get
Moreover, by the continuity of , it follows that Hence, Thus, we conclude that in as . By (50) and (52), we can get
Next, we shall prove that is bounded.
By (H2), (), (11), and (30), we haveTherefore, is bounded in . Similar arguments show that in as . Then is a critical point of with . According to as , we know that BVP (1) has infinitely many nontrivial high energy solutions.
Theorem 12. Assume that (H3)–(H6) and (V) hold. Then BVP (1) possesses infinitely many small energy solutions satisfying
Proof. By (H5), we can getCombining (60) and (13), it is easily seen that maps bounded sets into bounded sets uniformly for . It follows from (H4) that for all . Thus, condition (B1) holds. In the following, we verify condition (B2). By (H3), is obvious. According to (H6), there exists such that Suppose is a finite dimensional subspace of . As in the proof of Theorem 11, there exists such that (39) holds. Then where , . Hence, for any with , from (61), we have