Research Article | Open Access
Multiple Solutions for the Discrete -Laplacian Boundary Value Problems
By employing a critical point theorem, established by Bonanno, we prove the existence of three distinct solutions to boundary value problems of nonlinear difference equations with a discrete p-Laplacian operator. To demonstrate the applicability of our results, we also present an example.
In the present paper we will deal with the following discrete boundary value problem, with homogeneous Dirichlet conditions: where is an integer and denotes the discrete interval . is a real number; for all . Here is the forward difference operator defined by for all , while is a continuous function. is a nonnegative parameter.
Boundary value problems for difference equations have been extensively studied; see the monographs [1–5]. Difference equations represent the discrete counterpart of ordinary equations, and the classical theory of difference equations employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point methods; for an exhaustive description of the subject, we refer the reader to the monographs of Agarwal , Kelley and Peterson , and Lakshmikantham and Trigiante . We remark that, usually, most methods yield existence results for solutions of a difference equation.
The issue of multiplicity of solutions can be investigated through variational methods. Recently, many results have been established by applying variational methods. For example, we may recall here the works of Cabada et al. [7, 8], Cai and Yu , Guo and Yu , and Deng et al. . In all the aforementioned papers, the variational methods are applied to difference equations on discrete intervals and the variational approach represents an important advance as it allows proving multiplicity results as well.
In this paper, under convenient assumptions on the reaction term , we not only give the result that (1) admits at least three distinct solutions but also find the two open intervals which lies in and make estimation of the norm of .
The rest of the paper has the following structure. In Section 2 we introduce the variational framework for problem (1) and transfer the existence of solutions of boundary value problem (1) into the existence of critical points of the corresponding functional. Employing the critical point theorem of Bonanno , we state our main results and give proofs of the main results in Section 3. Finally, we exhibit a simple example to demonstrate the applicability of our results.
2. Variational Framework
This section is devoted to the formulation of a variational framework for (1). We are going to define a suitable Banach space and an energy functional , such that critical points of in are exact solutions of (1).
We define the real vector space and, for every , we denote So is a reflexive Banach space and . We also put, for every , By classical results, the norms and are equivalent on ; then there exists a constant such that
For later use, here we give the estimation of constant .
Lemma 1. The constant defined in (6) can be denoted as , where .
Proof. For every , given a , by Dirichlet conditions and Hölder inequality, we have
Define and by where . Define by
Lemma 2. For any , is continuously Gâteaux differentiable, and, for every , ,
Proof. Clearly ; in what follows we prove (10). Choose , ; let be an arbitrary mapping; we recall the summation by parts formula
Using (11) with and , we get
Besides, we have
Equalities (13) and (14) imply (10).
From Lemma 2, we easily get the following lemma which yields a variational formulation for (1).
Lemma 3. For all , every critical point of is a solution of (1).
Proof. Fix and assume that is a critical point of ; then From Lemma 2, this is equivalent to For any , we define by putting for all , where if and if . If we apply (16) with , we have this is exactly (1). So, solves (1); that is, is in fact a solution of (1).
3. Main Results
In order to present our main results, we introduce some notations at first. Denote ; then write and, for each , Now, we state the following convenient assumptions on the function . There exist two constants , with such that where is defined in (6). There exist constants , , , and such that
Remark 4. If holds, then
In fact, from , we have
it follows that
which means that
With the above preparations, we present our main results.
Theorem 5. Assume that and hold. Then, for each
Equation (1) admits at least three solutions in and, moreover, there exist an open interval and a positive real number such that, for each , (1) admits at least three solutions in whose norms in are less than .
We replace by the following. There exist constants and such that
We have the following.
Corollary 6. If and hold, conclusions in Theorem 5 are true.
Remark 7. Integrate (28); it follows (21). Then conclusions in Corollary 6 are true.
When is independent of , consider the following special case of (1): where .
Denote ; we define and, for each ,
We need the following assumptions. There exist two constants , with such that There exist constants , , , and such that
Remark 8. If holds, then .
Similarly, we obtain the following.
Theorem 9. Assume that and hold. Then, for each Equation (29) admits at least three solutions in and, moreover, there exist an open interval and a positive real number such that, for each , (29) admits at least three solutions in whose norms in are less than .
Corollary 10. Let satisfy and there exist constants and such that Then the conclusions in Theorem 9 also hold.
To give proofs of our results, we need the following three critical point theorems, established by Bonanno .
For the reader’s convenience, we recall the definition of weak closure.
Suppose that . We denote as the weak closure of , that is, , if there exists a sequence such that for every .
Lemma 11 (see ). Let be a separable and reflexive real Banach space; let be a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on ; let be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists such that and that(i) for all ;(ii)there are and such that ;(iii).
Then for each the equation has at least three solutions in , and, for each , there exist an open interval and a positive real number such that, for each , (37) has at least three solutions in whose norms are less than .
Proof of Theorem 5. Let space be the Banach space . It follows from (8) that is a nonnegative Gâteaux differentiable and weakly lower semicontinuous functional, whose Gâteaux derivative admits a continuous inverse on , and let be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Moreover, .
In view of , we have, for any , for each and , Since , it follows that for all . Condition (i) of Lemma 11 is satisfied.
Now, we put It is clear that and In view of , together with (41), it yields which implies that the assumption (ii) of Lemma 11 holds.
Finally, we verify that the assumption (iii) of Lemma 11 is true to complete the proof of Theorem 5.
Note that the estimate implies that for all . By the definition of (see (40)) and (44), we obtain This means that Thus, for all , we have On the other hand, we get which shows that (iii) of Lemma 11 is satisfied.
Note that By , we obtain which implies Making use of this, together with (18) and (19), it follows that . Combining (49) and (50) and applying Lemma 11, we achieve that, for each , (1) admits at least three solutions in .
For each , we note that Then, from Lemma 11, it yields that, for each , there exist an open interval and a real number , such that, for , (1) admits at least three solutions in whose norms in are less than . And the proof of Theorem 5 is completed.
Finally, we exhibit a simple example to demonstrate the applicability of our results.
Example 12. Consider the boundary value problem where It is clear that , , , and . Let , ; it follows that , . Then it can be easily shown that all conditions in Theorem 9 are satisfied. Direct calculations give and , so and . Therefore, (53) has at least three solutions in provided that .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by National Natural Science Foundation of China (no. 11101098), the Foundation of Guangzhou Education Bureau (no. 2012A019), PCSIR (no. IRT1226), and Natural Science Foundation of Guangdong Province (no. S2013010014460).
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Copyright © 2014 Yuhua Long and Haiping Shi. 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.