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Solutions for a fractional difference boundary value problem
Advances in Difference Equations volume 2013, Article number: 319 (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.
In this work, using variational methods and critical point theory, we study the fractional difference boundary value problem
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).
Atici and Eloe  considered the existence of positive solutions for the following two-point boundary value problem for a nonlinear finite fractional difference equation:
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.
In , Tian and Henderson studied the 2n th order nonlinear difference equation
and established some existence results for anti-periodic solutions under various assumptions on the nonlinearity. In , Ye and Tang considered the second-order discrete Hamiltonian system
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 .
The ν th fractional sum of f for is defined by
We also define the ν th fractional difference for by , where and is chosen so that .
Let f 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 Suppose that X is a Banach space and is a functional defined on X. For given , assume that
exists. Then I is Gateaux differentiable at x, the limit in (2.3) is called the Gateaux differential of I at x in direction y, and is denoted by , i.e.,
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])
Let X be a reflexive real Banach space. Moreover, let be an operator satisfying the conditions
L is bounded and demicontinuous,
L is coercive, i.e., ,
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.
Lemma 2.7 (Linking theorem, Rabinowitz, see [20–22])
Let be a Banach space with Z closed in X and . Let , and let be such that . Define
Let be such that
If I satisfies the (PS) c condition with
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.
Lemma 2.9 (Mountain pass theorem, see [20–22])
Let X be a real Banach space, and let satisfy the (C) condition. If and the following conditions hold:
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;
there is a subspace with , , such that
Then I possesses at least distinct pairs of nontrivial critical points.
In what follows, we establish the variational framework for (1.1). Let
Then X is the -dimensional Hilbert space with the usual inner product and the usual norm
For , we define the α-norm on X: . Since , we see that there exist , such that
for all x belonging to X (or its subspace).
In view of [, (3.4)], we can define an energy functional on X by
Clearly, . Let
Then, from the boundary conditions of (1.1), it is easy to see that E is isomorphic to X. In the following, when we say , we always imply that x can be extended to if it is necessary. Now we claim that if is a critical point of I, then is precisely a solution of (1.1). Indeed, since I can be viewed as a continuously differentiable functional defined on the finite dimensional Hilbert space X, the Fréchet derivative is zero if and only if for all . From the relation between the Fréchet derivative and the Gateaux derivative, we obtain
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.
Next, noting Definition 2.2, for , we let
Then we have
i.e., , where , ,
Clearly, is a positive definite matrix. All the eigenvalues of are positive. Let and denote respectively the minimum and the maximum eigenvalues of . Since , we have
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
Now we state our main results and give their proof. For convenience, we list assumptions on f and F:
, and there is a constant c such that , .
There exist a constant and such that .
for all .
There exist and such that
There is a constant such that .
There is such that
Theorem 3.1 Let (H1) hold. Then (1.1) has precisely one solution for .
Proof We shall apply Lemma 2.5 to prove the result. From (2.11), we define the operator
Clearly, if for all , there exists such that , then is a solution of (1.1). Let
We sketch the properties of and . It is clear that is a linear operator, and furthermore, is bounded. Indeed, the Cauchy-Schwarz inequality enables us to obtain, notice (2.12) and (2.13),
Consequently, is continuous on X. Next, we show that is bounded and continuous. Let in (H1) and . Then we have from (H1)
From (3.3), the definition of , and the Cauchy-Schwarz inequality, we obtain
Therefore, is bounded and continuous, as required. Hence, L is bounded and continuous, so demicontinuous.
From (3.3), notice (2.12) and (2.13), we see
Therefore, , i.e., L is coercive on X.
Finally, we prove that L is strictly monotone. Indeed, from (H1), we have
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.
Proof From (H2), there exists with
Thus, for with , it follows from the Hölder inequality that
By virtue of the inequality , there exists such that
From (H3), for , we see
From (H4), we see that there exist such that
Hence, for , we find
Set with is given in (3.10). For , (2.8) holds true. This, together with (3.13), implies
Since , we obtain
Since and then , (3.11) and (3.14) guarantee that there is such that
It remains to prove that I satisfies the (PS) c condition. This will be the case if we show that any sequence such that
contains a convergent subsequence. Note that , so we only need to show the boundedness of . Take such that for n large enough, and (H4), (3.12) and (2.8) enable us to obtain
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.
Proof We shall utilize Lemma 2.9 to prove the result. If , we see codim . From (H2), noting (3.9), we can take so that , where . Therefore,
Thus (i) of Lemma 2.9 holds true.
Choose , where , and . From (H5), we see that there exist such that
Therefore, from (3.13) and (2.8), we arrive at
Since , as , . Thus (ii) of Lemma 2.9 holds true.
Finally, we prove that I satisfies the (C) condition. Let be such that for some ,
We claim that is bounded. Otherwise, suppose that as . It is easy to see that for any , there exists such that
On the other hand, from (H6), there exist such that
Consequently, from (2.8),
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.
Let . Then . Thus, (H2) and (H3) hold automatically. For , , we see
Therefore, (H4) holds.
Let . Then and (H2), (H7) hold. Choose , , and we see
Hence (H5) and (H6) hold.
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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).
The authors declare that they have no competing interests.
WD and JX carried out the main results of this article and drafted the manuscript. DO directed the study and helped with the inspection. All the authors read and approved the final manuscript.
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Dong, W., Xu, J. & O’Regan, D. Solutions for a fractional difference boundary value problem. Adv Differ Equ 2013, 319 (2013). https://doi.org/10.1186/1687-1847-2013-319
- fractional difference boundary value problem
- variational approach
- critical point theory