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Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach
By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.
Variational methods for dealing with difference equations have appeared as early as 1985 in  in which the positive definiteness of quadratic forms (which are functionals) is related to the existence of “nodes” of solutions (or positive solutions satisfying “conjugate” boundary conditions) of linear self-adjoint second-order difference equations of the form where is real and positive for and is real for . Later there are interests in solutions of nonlinear difference equations under various types of boundary or subsidiary conditions, and more sophisticated methods such as the mountain pass theorems are needed to handle the existence problem (see, e.g., [2–11]).
Recently, fractional differential and difference “operators” are found themselves in concrete applications, and hence attention has to be paid to associated fractional difference and differential equations under various boundary or side conditions. For example, a recent paper by Atici and Eloe  explores some of the theories of a discrete conjugate fractional BVP. Similarly, in , a discrete right-focal fractional BVP is analyzed. Other recent advances in the theory of the discrete fractional calculus may be found in [14, 15]. In particular, an interesting recent paper by Atici and Sengül  addressed the use of fractional difference equations in tumor growth modeling. Thus, it seems that there exists some promise in using fractional difference equations as mathematical models for describing physical problems in more accurate manners.
In order to handle the existence problem for fractional BVPs, various methods (among which are some standard fixed-point theorems) can be used. In this paper, however, we show that variational methods can also be applied. A good reason for picking such an approach is that, in Atici and Sengül , some basic fractional calculuses are developed and a simple variational problem is demonstrated, and hence advantage can be taken in obvious manners. We remark, however, that fractional difference operators can be approached in different manners and one by means of operator convolution rings can be found in the book by Cheng [17, Chapter 3] published in 2003.
More specifically, in this paper, we are interested in the existence of multiple solutions for the following -order fractional difference boundary value problem where , and are, respectively, left fractional difference and the right fractional difference operators (which will be explained in more detail later), , is continuous, and is a positive parameter.
By establishing the corresponding variational framework and using critical point theory, we will establish various existence results (which naturally depend on , , and ).
For convenience, throughout this paper, we arrange , for .
First, for any integer , we let . We define , for any and for which the right-hand side is defined. We also appeal to the convention that, if is a pole of the Gamma function and is not a pole, then .
Definition 2.1. The fractional sum of for is defined by for . We also define the fractional difference for by , where and is chosen so that .
Definition 2.2. Let be any real-valued function and . The left discrete fractional difference and the right discrete fractional difference operators are, respectively, defined as
Definition 2.3. For , we say satisfies the Palais-Smale condition (henceforth denoted by (PS) condition) if any sequence for which is bounded and as possesses a convergent subsequence.
Lemma 2.4 (see ). A real symmetric matrix is positive definite if there exists a real nonsingular matrix such that , where is the transpose.
Lemma 2.5 (see : linking theorem). Let be a real Banach space, and satisfies (PS) condition and is bounded from below. Suppose has a local linking at the origin , namely, there is a decomposition and a positive number such that for with ; for with . Then has at least three critical points.
Lemma 2.6 (see ). Let be a real reflexive Banach space, and let the functional be weakly lower (upper) semicontinuous and coercive, that is, (resp., anticoercive, i.e., ). Then there exists such that (resp., ). Moreover, if , then is a critical point of functional .
Recall that, in the finite dimensional setting, it is well known that a coercive functional satisfies the (PS) condition.
Let denote the open ball in a real Banach space of radius about , and let denote its boundary. Now some critical point theorems needed later can be stated.
Lemma 2.7 (mountain pass theorem ). Let be a real Banach space and , satisfying (PS) condition. Suppose and(I1) there are constants such that ,(I2) there is such that .
Then possesses a critical value . Moreover can be characterized as where
Lemma 2.8 (see ). Let be a reflexive Banach space and with . Suppose that is an even functional satisfying (PS) condition and the following conditions:(I3) there are constants and a closed linear subspace of such that codim and ,(I4) there is a finite dimensional subspace of with , , such that as , . Then possesses at least distinct pairs of nontrivial critical points.
Lemma 2.9 (the Clark theorem ). Let be a real Banach space, with even, bounded from below, and satisfying (PS) condition. Suppose , there is a set such that is homeomorphic to (the dimensional unit sphere) by an odd map and . Then possesses at least distinct pairs of critical points.
3. Main Results
Firstly, we establish variational framework. Let be the -dimensional Hilbert space with the usual inner product and the usual norm For , we recall the -norm on : . We also recall the standard fact that there exist positive constants and , such that
Define a functional on by for , where Obviously, . Let Then by (1.3) it is easy to see that is isomorphic to . In the following, when we say , we always imply that can be extended to if it is necessary. Now we claim that if , is a critical point of , then , is precisely a solution of BVP (1.2) and (1.3). Indeed, since can be viewed as a continuously differentiable functional defined on the finite dimensional Hilbert space , the Frechet derivative is zero if and only if for all .
Next, observe by Definition 2.2 that, for ,
We let then that is, , where , , , :
By Lemma 2.4, is a positive definite matrix. Let and denote, respectively, the minimum and the maximum eigenvalues of .
Since , we may easily see that
Then if and only if . Next, let
By direct verifications, we may find that is a positive definite matrix. Let be the orthonormal eigenvectors corresponding to the eigenvalues of , where .
For convenience, we list the following assumptions.(C1) There exists such that for , where is a constant.(C2) There is a constant such that for .(C3) There exists a constant such that for .(C4) satisfies for , where is a constant.(C5) is odd with respect to , that is, , for , and .(C6) There is a positive constant such that for .
Proof. By (C1), we obtain
where is some sufficiently large numbers and . Thus, by the continuity of on , there exists such that
Combining with (3.3)–(3.15), we have
So, in view of our assumption , we see that, for ,?? as , that is, is a coercive map. In view of Lemma 2.6, we know that there exists at least one such that ; hence BVP (1.2), (1.3) has at least one solution. The proof is completed.
Remark 3.2. If and , from the proof of Theorem 3.1, we can get that, for , our functional is also coercive.
Proof. Similar to the proof Theorem 3.1, we have
By (C2), there exists and such that for with , so where . Since is continuous on , through (3.17), we obtain Thus as for . That is, is an anticoercive. In view of Lemma 2.6, we know that there exists at least one such that ; hence BVP (1.2),??(1.3) has at least one solution. The proof is completed.
Proof. First, we know from Theorem 3.3 that as . Clearly, is a real reflexive finite dimensional Banach space and , so functional is weakly upper semicontinuous. By Lemma 2.6, there exists such that and . Set . Let , such that there exists and for . By (C2) and (3.19), we may see that
In view of , we see that is bounded, and hence is bounded. Since is finite dimensional, there is a subsequence of , which is convergent in . Therefore, the (PS) condition is verified.
By (C3), there exists for , . Thus, for with , we have
For , we choose and . Then we have , so that the condition (I1) in Lemma 2.7 holds.
Since as , we can find with sufficiently large norm such that . Hence (I2) in Lemma 2.7 is satisfied. Thus, functional has one critical value where . If , the proof is completed. It suffices to consider the case . Then that is, for each .
Similarly, we can also choose such that . Applying Lemma 2.7 again, we obtain another critical value of the functional , where . If , then the proof is completed. It suffices to consider the case where . Then for each . By the definitions of and , we may choose and such that . Therefore, we get the maximum of the functional on and , respectively, that is, we find two distinct nontrivial critical points of the functional . Therefore, our BVP (1.2), (1.3) possesses at least two nontrivial solutions.
Proof. By (C1) and Theorem 3.1, we obtain , thus functional is bounded from below. Similar to the proof of (PS) condition in Theorem 3.4, we can verify that functional satisfies (PS) condition in our hypothesis. In order to apply linking theorem, we prove functional is local linking at origin as follows. Clearly, ??. Let ????, then .
By (C4), for , there exists , such that
So, for with , such that
Since , we have
Thus, for , we have for with .
Similarly, for with , then for , we have for with . So, by Lemma 2.5, for , if , functional possesses at least three critical points. By the arbitrariness of , we get for , the problem (1.2), (1.3) possesses at least three solutions.
Proof. By (C5), functional is even, and based on the proof of Theorem 3.4, we know that satisfies (PS) condition. In order to obtain our result, we need to verify (I3) and (I4) of Lemma 2.8.
First, in view of (C3), there exists such that
For , if we choose , then .
So for with , since , we have
Thus, for , , where , (I3) of Lemma 2.8 holds.
Next if we choose , then for , in view of (C2) and Theorem 3.3, we get as . (I4) of Lemma 2.8 is satisfied.
Therefore, for , functional possesses at least pair of critical points in , and problem (1.2), (1.3) has at least pairs of solutions.
Remark 3.7. In Theorem 3.6, if we choose , then for , the BVP (1.2), (1.3) possesses at least pairs of solutions.
Obviously, compared with Theorem 3.4, the even condition (C5) ensures that the problem (1.2), (1.3) possesses more solutions.
Proof. is an even functional on by (C5). From (C1), we obtain as , so it is clear that is bounded from below on and satisfies the (PS) condition. For , if we choose and set , then is homeomorphic to by an odd map. By (C6), there exists such that for , . So for ,
For , we have . Therefore, by Lemma 2.9, functional has at least pairs of nontrivial solutions.
Remark 3.9. From Theorem 3.5, it is easy to see that, when is odd about the second variable, we can obtain more solutions of the problem (1.2), (1.3), and the number of solutions depends on where lies.
In the final section, we apply the results developed in Section 3 to some examples.
Example 4.1. Consider the following problem where . Choose and in (C2) and (C3). Since we see that (C2) and (C3) hold. Thus, by Theorem 3.4, when , problem (4.1) has at least two nontrivial solutions.
Example 4.2. Consider the problem Suppose there exists such that . If we choose in (C1) and (C4), then for , we have and hence (C1) and (C4) are satisfied. So, in view of Theorem 3.5, for , problem (4.3) has at least three solutions.
Example 4.3. Consider the problem Condition (C5) is satisfied. If we choose and in (C2) and (C3), then by some simple calculation, we may show that the hypotheses (C2) and (C3) are fulfilled. Therefore, by Theorem 3.6, for any and , problem (4.5) has at least pairs of solutions.
Example 4.4. Finally, consider the problem where . Let and . Then it is easy to verify that (C1), (C5), and (C6) hold. Thus, by Theorem 3.8, for each and , problem (4.6) has at least pairs of solutions.
This Project was supported by the National Natural Science Foundation of China (11161049).
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