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

where m0 is the rest mass of the particle and c is the speed of light in the vacuum (see [1–3]). Assume that m0 = c = 1. The existence andmultiplicity of solutions for (2) subjected to Dirichlet, Robin, periodic, orNeumann 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 φ-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 κ = 1 and r = 0, Bereanu and Mawhin [12] proved that the discrete periodic problem with an attractive nonlinearity

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][5][6][7][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][13][14][15][16][17].
In the case  = 1 and r = 0, Bereanu and Mawhin [12] proved that the discrete periodic problem with an attractive nonlinearity has at least one positive solution if and only if e = 1/( − 2) ∑ −1 =2   > 0. When  ≥ 1, they also showed that the repulsive singular periodic problem has at least one positive solution if and only if e < 0.
In the case  = 0, the problem (1) is the classical discrete periodic problem This problem has been studied by  where the existence of positive solution needs a necessary condition   < 0 or 0 <   < 4sin 2 (/2) (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  ̸ = 0 and any forcing term e (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.
The model example is Let  ∈ N with  ≥ 4 be fixed and u = ( 1 ,  2 , . . .,   ) ∈ R  .Then we denote where where R be a continuous function.Then its Nemytskii operator   (u) : R  → R −2 is given by It follows that   is continuous and takes bounded sets into bounded sets.
Let  be the projectors defined by If u ∈ R  , we write ũ = u − u and we will consider the following closed subspaces of R  : Let the vector space  −2 be endowed with the orientation of R  and the norm ‖u‖ ∞ = max 1≤≤ |  |.Its elements can be associated with the coordinates ( 2 , . . .,  −1 ) and correspond to the elements of R  of the form Now, we recall the following technical result given as Lemma 1 from [12].
Lemma 2. Let  : R  → R −2 be a continuous operator which takes bounded sets into bounded sets and consider the abstract discrete periodic problem: for any solution u of (21).
Lemma 4 (see [12,Theorem 3]).If (24) has a lower solution  and an upper solution  such that  ≤ , then (24) has a solution u such that  ≤ u ≤ .Moreover, if  and  are strict, then  < u < , 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.
We have that Then, multiplying both inequalities and using that  ≤ ( + ) 2 /2, for all ,  ∈ R, it follows that and the proof is completed.

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 u such that Proof.Let and define the continuous function  : Let us consider the modified periodic problem and let A  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,  1 = − * − 2 is a lower solution of (37) and  1 =  * + 2 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 Ω, so Ω ̸ = 0. Next, let us consider u ∈ Ω such that A  (u) = u and ‖u‖ ∞ =  * + 2. Notice that one has ‖Δu‖ ∞ < .This implies that there exists In the first case we can assume that This together with  is an increasing homeomorphism implying ∇[(Δ  0 )] ≤ 0. On the other hand, we have that which is a contradiction.If  0 = 1, then from boundary condition  1 =   , Δ 1 = Δ −1 , we can get that Δ 1 ≤ 0 and Δ −1 ≥ 0, which implies that Δ 1 = Δ −1 = 0.This together with ∇[(Δ −1 )] = (Δ −1 ) − (Δ −2 ) = ( − 1,  −1 , 0) +  > 0 implies that   =  −1 <  −2 ; this is a contradiction.Analogously, one can obtain a contradiction in the second case.Consequently, Now, let u ∈ Ω be such that A  (u) = u.It follows from (43) that ‖u‖ ∞ <  * + 2, ‖Δu‖ ∞ <  and u ∈ Ω  1 , ∪ Ω , 1 .We infer that there exists and, consequently, We have divided two cases to discuss.
Case 1. Assume that there exists u ∈ Ω such that A  (u) = u.Using (45), we deduce that ‖u‖ ∞ <  * , implying that u is a solution of ( 24) and (34) holds.Actually, in this case, there exists Case 2. Assume that A  (u) ̸ = u for all u ∈ Ω.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 u ∈ Ω such that A  (u) = u.It follows that there exists Then, using once again the fact that ‖Δu‖ ∞ < , it follows that ‖u‖ ∞ <  * and u is a solution of (24).Moreover, from u ∈ Ω, 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 u satisfying (34).
In particular, As an application of Theorem 7, we deal with singular strong nonlinearities.Consider the following discrete periodic problem: where  : [2,  − 1] Z × R 2 → R and ℎ,  : (0, ∞) → R are continuous functions such that lim and ℎ ≥ 0. Under those assumptions we have the following theorem.
Theorem 9. Assume that (49) has a lower solution  > 0 and an upper solution  > 0. Then (49) has at least one solution u which satisfies (34).
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  1 > 0 and c = ( 2 , . . .,  −1 ) ∈ R −2 such that then (58) has a lower solution  such that Proof.Consider the function  = c + e.We have two cases.
Theorem 12. Assume that (73) holds.If either then problem (72) has at least one solution.
In the case r < 0, there exists  0 < 0 such that (83) has at least two solutions provided that e ≤  0 holds true.In fact, in this case, problem (83) has two strict upper solutions  1 ,  2 > 0 and a strict lower solution  > 0 such that  1 <  <  2 .Thus, the result follows from Lemma 4 and Theorem 9.

Mixed Singularities. Consider the discrete periodic problem
and  * : (/2, ∞) → R,  * : (0, ∞) → R, defined by The following lemma plays a key role to prove the main result in this subsection.
Then it follows that conditions (59) and (60) hold.Thus, from Theorem 10 we infer that (84) has a lower solution  such that  1 ≤  <  1 + /2.On the other hand, using the fact that r < 0, there exists 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 u of (84) is a positive solution since 0 < then the above problem has at least one solution.
In connection with Example 19, if r = 0, then we have the following theorem.