Solutions for a fractional difference boundary value problem
© Dong et al.; licensee Springer. 2013
Received: 30 July 2013
Accepted: 16 October 2013
Published: 11 November 2013
Using a variational approach and critical point theory, we investigate the existence of solutions for a fractional difference boundary value problem.
MSC:26A33, 35A15, 39A12, 44A55.
Keywordsfractional difference boundary value problem variational approach critical point theory solution
where , and are, respectively, the left fractional difference and the right fractional difference operators, , and is continuous.
Fractional calculus has a long history, and there is renewed interest in the study of both fractional calculus and fractional difference equations. In [1, 2], the authors discussed properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutativity of fractional sums. A number of papers have appeared which build the theoretical foundations of discrete fractional calculus (for more details, we refer the reader to [3–8] and the references therein).
In , the authors used the mountain pass theorem, a linking theorem, and Clark’s theorem to establish the existence of multiple solutions for a fractional difference boundary value problem with a parameter. Under some suitable assumptions, they obtained some results which ensure the existence of a precise interval of parameters for which the problem admits multiple solutions. We note that there are many papers in the literature [9–18] which discuss discrete problems via variational and critical point theory.
and obtained an existence theorem for a nonzero T-periodic solution.
In the literature on discrete problem via critical point theory, the authors are interested in the existence of at least one solution or infinitely many solutions. The existence of a unique solution is not usually studied. In this paper, using Browder’s theorem, first we present a uniqueness result in Section 3. Then a linking theorem is used to establish existence. Finally, assuming an Ambrosetti-Rabinowitz type condition, we show that problem (1.1) has many solutions if the nonlinearity is odd.
For convenience, throughout this paper, we arrange for . We present some definitions and lemmas for discrete fractional operators.
For any integer β, let and , where t and ν are determined by (1.1). We also appeal to the convention that if is a pole of the gamma function and is not a pole, then .
We also define the ν th fractional difference for by , where and is chosen so that .
Definition 2.4 (see [, p.303])
Let X be a reflexive real Banach space and its dual. The operator is said to be demicontinuous if L maps strongly convergent sequences in X to weakly convergent sequences in .
Lemma 2.5 (Browder theorem, see [, Theorem 5.3.22])
L is bounded and demicontinuous,
L is coercive, i.e., ,
- (iii)L is monotone on the space X, i.e., for all , we have(2.5)
Then the equation has at least one solution for every . If, moreover, the inequality (2.5) is strict for all , , then the equation has precisely one solution for all .
Let X be a real Banach space, and . We say that I satisfies the (PS) c condition if any sequence such that and as has a convergent subsequence.
then c is a critical point of I.
Let X be a real Banach space and . We say that I satisfies the Cerami condition ((C) condition for short) if any sequence such that is bounded and as , there exists a subsequence of which is convergent in X.
there are two positive constants ρ, η and a closed linear subspace of X such that codim and , where is an open ball of radius ρ with center θ;
- (ii)there is a subspace with , , such that
Then I possesses at least distinct pairs of nontrivial critical points.
for all x belonging to X (or its subspace).
Therefore, in order to obtain the existence of solutions for (1.1), we only need to study the existence of critical points of the energy functional I on X.
By direct verification, we see that A is a positive definite matrix. Let be the orthonormal eigenvectors corresponding to the eigenvalues of A, where . Clearly, . Let , , for . Then .
3 Main results
, and there is a constant c such that , .
There exist a constant and such that .
for all .
- (H4)There exist and such that
There is a constant such that .
- (H6)There is such that
Theorem 3.1 Let (H1) hold. Then (1.1) has precisely one solution for .
Therefore, is bounded and continuous, as required. Hence, L is bounded and continuous, so demicontinuous.
Therefore, , i.e., L is coercive on X.
All the conditions of Lemma 2.5 are satisfied, as claimed. Hence, (1.1) has precisely one solution. This completes the proof. □
Theorem 3.2 Let (H2)-(H4) hold. Then (1.1) has at least one solution.
Since and , we see that is bounded.
Thus the functional I satisfies all the conditions of Lemma 2.7, and then I has a critical point, and (1.1) has at least one solution. This completes the proof. □
Theorem 3.3 Let (H2), (H5)-(H7) hold. Then (1.1) has at least solutions.
Thus (i) of Lemma 2.9 holds true.
Since , as , . Thus (ii) of Lemma 2.9 holds true.
Let , and we get a contradiction.
It is easy to see that I is even and . Thus all the conditions of Lemma 2.9 are satisfied, and (1.1) has at least solutions. The proof is complete. □
Let , where and . Clearly, (H1) holds.
- 2.Let . Then . Thus, (H2) and (H3) hold automatically. For , , we see
Therefore, (H4) holds.
- 3.Let . Then and (H2), (H7) hold. Choose , , and we see
Hence (H5) and (H6) hold.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. Research is supported by the NNSF-China (10971046 and 11371117), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007 and A2012402036), GIIFSDU (yzc12063) and IIFSDU (2012TS020).
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