Periodic Solutions of Second Order Nonlinear Difference Equations with Singular -Laplacian Operator

Abstract

We obtain some existence results of solutions for discrete periodic boundary value problems with singular -Laplacian operator , , , and by using the upper and lower solutions method and Brouwer degree theory, where is a constant, , , , and is a parameter. We also give some examples with singular nonlinearities to illustrate our main results.

1. Introduction

In this paper we present some existence results for the discrete periodic boundary value problems with singular -Laplacian operator where is a constant, , , , is a parameter, and with is an integer.

These problems are originated in the study that a particle moves on a straight line, subject to a restoring force with relativistic effects, which can be described by the differential equation of motion: where is the rest mass of the particle and is the speed of light in the vacuum (see [1–3]). Assume that . The existence and multiplicity of solutions for (2) subjected to Dirichlet, Robin, periodic, or Neumann boundary conditions have been studied by various methods, such as the method of lower and upper solutions, topological degree theory, and critical point theory; see [4–8] and the references therein.

An interesting question is which techniques and theorems regarding the continuous differential equations can be adapted for difference equations (see Kelly and Peterson [9], Agarwal [10], and Bereanu and Mawhin [11, 12]). The purpose of this paper is to show that some known existence and multiplicity results of periodic solution for singular perturbations of the singular -Laplacian operator also hold for the corresponding difference equation and develop some results for the singular difference equation boundary value problems; see [12–17].

In the case and , Bereanu and Mawhin [12] proved that the discrete periodic problem with an attractive nonlinearity has at least one positive solution if and only if . When , they also showed that the repulsive singular periodic problem has at least one positive solution if and only if .

In the case , the problem (1) is the classical discrete periodic problem This problem has been studied by Ma and Lu [15–17] where the existence of positive solution needs a necessary condition or (see [14]). For other results concerning the existence of solutions for singular nonlinear difference equation boundary value problems, see, for example, [9, 10, 18].

It is interesting to remark that, in contrast to the classical case, the periodic problem with discrete relativistic acceleration has at least one solution for any and any forcing term (see [12, Corollary  2]). Note that, for this type of problems, in some sense, the same situation occurs also if we add a singular nonlinearity.

In order to explain the main result, let us introduce some notation. For any , we write and . For , we put and note that .

Motivated by the above results from [12, 14–17], we consider the discrete periodic problem where , , and . If either or and , then we prove that the above problem has at least one solution (see Example 14). In the case , we show that the periodic problem with repulsive singular term with and , is solvable (see Example 18) provided that On the other hand, we also consider the periodic problem with attractive singularity where , , and with and . If either or and in both cases , then the problem (12) has at least one solution (see Example 19). Moreover, in the pure attractive case, that is, , it follows that (12) is solvable if either or and (Theorem 21).

The rest of the paper is organized as follows. In Section 2, we introduce some notations and auxiliary results. In Section 3 we establish the method of non-well-ordered lower and upper solutions and give an application on the discrete periodic problem with the strong repulsive nonlinearities. In addition, we introduce some methods to construct lower and upper solutions. Finally, in Section 4 we give some applications to deal with the singular perturbations periodic problems.

2. Some Notations and Auxiliary Results

In this section, we first introduce some notations. Let with ; we denote .

For , set , . If , we write (resp., ) if (resp., ) for all . The following assumption upon (called singular) is made throughout the paper:

() is an increasing homeomorphism with .

The model example is

Let with be fixed and . Then we denote where for and if , define where for .

Let be a continuous function. Then its Nemytskii operator is given by It follows that is continuous and takes bounded sets into bounded sets.

Let be the projectors defined by If , we write and we will consider the following closed subspaces of :

Let the vector space be endowed with the orientation of and the norm . Its elements can be associated with the coordinates and correspond to the elements of of the form

Now, we recall the following technical result given as Lemma 1 from [12].

Lemma 1. For each , there exists a unique such that Moreover, the function is continuous.

Lemma 2. Let be a continuous operator which takes bounded sets into bounded sets and consider the abstract discrete periodic problem: A function is a solution of (21) if and only if is a fixed point of the continuous operator defined by , where satisfying Furthermore, for all and for any solution of (21).

Let us consider the discrete periodic problem Obviously, from Lemma 2, the fixed point operator associated with (24) is

Now, we state the method of upper and lower solutions for discrete periodic problem (24) according to Bereanu and Mawhin [12].

Definition 3. A function (resp., ) is called a lower solution (resp., an upper solution) for (24) if (resp., ) and Such a lower or an upper solution is called strict if the inequality (26) is strict.

Lemma 4 (see [12, Theorem  3]). If (24) has a lower solution and an upper solution such that , then (24) has a solution such that . Moreover, if and are strict, then , and where .

An easy adaption of the proof of [12, Theorem  3] provides the following useful result.

Lemma 5. Assume that (24) has a lower solution and an upper solution such that , and Then

The next result is an elementary estimation of the oscillation of a periodic function.

Lemma 6. If is a -periodic function, then

Proof. Let be such that , and let be such that . We have that Then, multiplying both inequalities and using that , for all , it follows that and the proof is completed.

3. The Method of Lower and Upper Solutions and Application

In 2008, Bereanu and Mawhin [12] proved that problem (24) has at least one solution if it has a lower solution and an upper solution with . In the following result we prove some additional concerning the location of the solution. In particular, we have a posteriori estimations which will be very useful in the sequel (Remark 8).

Theorem 7. Assume that (24) has a lower solution and an upper solution such that Then (24) has at least one solution such that

Proof. Let and define the continuous function by
Let us consider the modified periodic problem and let be the fixed point operator associated with (37).
It is not difficult to verify that is a lower solution and is an upper solution of the problem (37). Moreover, by computation, is a lower solution of (37) and is an upper solution of (37). Notice that which together with (33) imply that So, we can consider the open bounded set It follows that Clearly, any constant function between and is contained in , so .
Next, let us consider such that and . Notice that one has . This implies that there exists such that or . In the first case we can assume that . If , then , . This together with is an increasing homeomorphism implying . On the other hand, we have that which is a contradiction. If , then from boundary condition , , we can get that and , which implies that . This together with implies that ; this is a contradiction. Analogously, one can obtain a contradiction in the second case. Consequently,
Now, let be such that . It follows from (43) that and . We infer that there exists such that or , implying that . Then, and, consequently,
We have divided two cases to discuss.
Case  1. Assume that there exists such that . Using (45), we deduce that , implying that is a solution of (24) and (34) holds. Actually, in this case, there exists such that or .
Case  2. Assume that for all . Then, from Lemma 5 applied to , it follows that This together with the additivity property of the Brouwer degree implies that which together with the existence property of the Brouwer degree imply that there exists such that . It follows that there exists such that and . Then, using once again the fact that , it follows that and is a solution of (24). Moreover, from , it follows that (34) is true.

Remark 8. Assume that (24) has a lower solution and an upper solution . From Lemma 4 and Theorem 7, we deduce that (24) has at least one solution satisfying (34). In particular,

As an application of Theorem 7, we deal with singular strong nonlinearities. Consider the following discrete periodic problem: where and are continuous functions such that and . Under those assumptions we have the following theorem.

Theorem 9. Assume that (49) has a lower solution and an upper solution . Then (49) has at least one solution which satisfies (34).

Proof. First, we define some notations as follows: From (50), there exists such that where , .
Let , be the continuous functions given by and consider the auxiliary periodic problem From , it follows that and are lower and upper solutions of (54), respectively.
If , then (54) has a solution satisfying from Lemma 4 and [12, Remark  3] (without any additional assumption). If condition (33) holds, then (54) has a solution satisfying (34). Obviously, the solution satisfies Next, we will prove that . From (55), there exists such that . Suppose on the contrary that , summing from to for (54); then we have
This together with (52) implies that which is a contradiction. Hence, , implying that is also a solution of (49).

Now, we give a method to construct the lower solution and the upper solution of the following discrete periodic problem: where is a continuous singular nonlinearity and .

The following result gives a method to construct a lower solution to (58), getting also control on its localization.

Theorem 10. Suppose that there exist and such that If then (58) has a lower solution such that

Proof. Consider the function . We have two cases.
Case  1. Assume that . Taking and using that , it follows from (59) that is a lower solution of (58).
Case  2. Assume that . Let . Then using and [12, Proposition  3], it follows that there exists such that Let us take and for . Then we define Let ; then . On the other hand, we have that Since , Lemma 6 implies (61). Now, we will show that is the lower solution of (58). By using (60), it follows that ; this together with the definitions of , , and implies that From (59) and (61), we deduce that Consequently,

By a similar argument, it is easy to prove the following theorem.

Theorem 11. Suppose that there exist and such that If then (58) has an upper solution such that

4. Some Applications for Singular Perturbations Problems

4.1. Strong Repulsive Perturbations

Consider the discrete periodic problem where , , and is continuous and satisfies

The main result of this subsection is the following theorem.

Theorem 12. Assume that (73) holds. If either then problem (72) has at least one solution.

Proof. Notice that from (73) it follows that there exists a constant sufficiently small such that which means that is an upper solution of (72).
Now we construct a lower solution of (72) by applying Theorem 10. Consider the continuous function defined by given by and defined by
Case  1. Assume that . This together with (73) implies that so there exists such that . In order to apply Theorem 10, define It follows that and , meaning that condition (60) is fulfilled. One has that for all . So, condition (59) holds. Then, from Theorem 10 we infer that (72) has a lower solution . Therefore, from Theorem 9, we can obtain the result.
Case  2. Assume that and . It follows that Then, there exists such that , since . The result follows by a similar argument to that used in Case  1.

Remark 13. Theorem  9 in [12] follows from Theorem 12 just taking .

Example 14. Consider the repulsive singular periodic problem where , , and . If either or , , then (83) has at least one solution.

In the case , there exists such that (83) has at least two solutions provided that holds true. In fact, in this case, problem (83) has two strict upper solutions , and a strict lower solution such that . Thus, the result follows from Lemma 4 and Theorem 9.

4.2. Mixed Singularities

Consider the discrete periodic problem where , , and is continuous and satisfies Let the continuous function defined by , given by and , , defined by

The following lemma plays a key role to prove the main result in this subsection.

Lemma 15. Let (85) hold and . If and , then (84) has at least one solution.

Proof. Since , there exists such that . Let us take and by Then it follows that conditions (59) and (60) hold. Thus, from Theorem 10 we infer that (84) has a lower solution such that .
On the other hand, using the fact that , there exists such that . Consider by Then, it follows that conditions (69) and (70) hold. Therefore, from Theorem 11 we can get that (84) has an upper solution such that . The result follows from Lemma 4.

Remark 16. From Lemma 15, the solution of (84) is a positive solution since .

Let us consider the discrete periodic problem where , , , and . We have the following theorem.

Theorem 17. If    and then (91) has at least one solution.

Proof. We have divided two cases.
Case  1. Assume that . In this case one has that implying that . So, (92) becomes , and the result follows from Lemma 15.
Case  2. Assume that . Notice that the minimum of is attained in and It is not difficult to verify that by using (92). Hence, , and the result follows from Lemma 15.

Example 18. Consider the discrete periodic problem with repulsive singularity: where , , and with and . If and then the above problem has at least one solution.

Example 19. Consider the periodic problem with attractive singularity where , , and with and . If and then the above problem has at least one solution.

In connection with Example 19, if , then we have the following theorem.

Theorem 20. Consider the discrete periodic problem with attractive singularity where , with and . If then (99) has at least one solution.

Proof. We will use the same strategy as in the proof of Lemma 15. In this case one has that .
Clearly, it follows from (100) that there exists such that . Let us define and by Then, it follows that conditions (59) and (60) hold. Therefore, from Theorem 10 we infer that (99) has a lower solution such that .
Using again (100), it follows that there exists such that . Let us define by . Then it follows that conditions (69) and (70) are true. Hence, from Theorem 11 we infer that (99) has an upper solution such that .
Consequently, (99) has a lower solution and an upper solution such that . The result follows from Lemma 4.

In the “pure” attractive case we have the following result concerning (84).

Theorem 21. Assume that (85) and hold. Then (84) has at least one solution provided that either

Proof. Notice that from (102) it follows that any sufficiently small positive constant is a lower solution of (84). The construction of an upper solution for (84) is similar as in Lemma 15. Then the result follows from Lemma 4.

Remark 22. Theorem 21 follows from [12] taking in Theorem 21.

Example 23. Let us consider again problem (97), with the condition . If either or and , then (97) has at least one solution.

Example 24. Let , with and consider the discrete periodic problem where with . It is not difficult to prove that the above problem has at least one solution for any by using Theorem 9.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Ruyun Ma completed the main study, carried out the results of this paper, and drafted the paper. Yanqiong Lu checked the proofs and verified the calculation. All the authors read and approved the final paper.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 11361054), SRFDP (no. 20126203110004), and Gansu provincial National Science Foundation of China (no. 1208RJZA258).