Abstract
In this paper, we obtain the existence and multiplicity of solutions for discrete Neumann boundary value problem with singular ϕ-Laplacian operator , , by using upper and lower solutions method and Brouwer degree theory, where is a constant, , and f is a continuous function. We also give some examples to illustrate the main results.
MSC:34B10, 34B18.
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1 Introduction
In this paper we present some existence and multiplicity results for the discrete Neumann boundary value problem with singular ϕ-Laplacian operator
where is a constant, Δ is the forward difference operator defined by , ∇ is the backward difference operator defined by , , is a continuous function and with is an integer.
This problem originated from the study of hypersurfaces in the Lorentz-Minkowski space with coordinates and the metric leads to partial differential equations (PDE) of the type
where Ω is a domain in () and is a nonlinearity prescribing the mean curvature of the hypersurface. A first essential result concerning the above PDE was proved by Calabi [1] in the case and . This was later extended to arbitrary dimension by Cheng and Yau in [2]. On the other hand, if and , then Treibergs [3] obtained an existence result about entire solutions for (1.2) in the presence of a pair of well-ordered upper and lower solutions, and (1.2) coupled with the Neumann boundary conditions has been considered by López [4] and Bereanu et al. [5–7]. For existence and multiplicity results concerning (positive) solutions of the classical case (), see for example [8, 9], and for other results concerning the Neumann boundary value problems, see [10] and their references.
This paper addresses a question of interest regarding the discrete Neumann problem (1.1):
Under what conditions does the discrete Neumann problem (1.1) have at least one solution?
Particular significance in the above question lies in the fact that strange and interesting distinctions can occur between the theory of differential equations and the theory of difference equations. For example, properties such as existence, uniqueness, and multiplicity of solutions may not be shared between the theory of differential equations and the theory of difference equations [11, 12], even though the right-hand side of the equations under consideration may be the same. Moreover, when investigating difference equations, as opposed to differential equations, basic ideas from calculus are not necessarily available, such as the intermediate value theorem, the mean value theorem, and the Rolle theorem. Thus, one faces new challenges and innovation is required.
It is worth to point out that corresponding results for the discrete Neumann problem (1.1) with and have been proved in [13, 14]. The classical case has been studied by [15, 16]. It is interesting to remark that, in contrast to the classical case, the discrete Neumann problem with relativistic acceleration
has at least one solution for any and any forcing term e (see [[14], Corollary 2 and Remark 2]).
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 [13–18], we consider the discrete Neumann problem (1.1) under the nonlinearity satisfying some suitable conditions and obtain the existence and multiplicity of solutions of (1.1). We shall show that if and f is bounded, then (1.1) has at least one solution; see Theorem 3.1. Moreover, suppose that f does not depend on in (1.1) and , then (1.1) has at least one solution if either f is superlinear at zero and sublinear at infinity (Corollary 3.1) or f is sublinear at zero and superlinear at infinity and (Corollary 3.2).
On the other hand, Bereanu and Mawhin [14] dealt with the Ambrosetti-Prodi type results for the problem (1.1) with , , they obtain the result that there exists () such that problem (1.1) has zero, at least one or at least two solutions according to , or (, or ) if (), as uniformly for ; see [[14], Theorem 6, Theorem 7 and Remark 9]. We note that these results also hold for the problem (1.1) by the same argument in [[14], Theorem 6, Theorem 7]. Naturally we can ask: what would happen if f is null at infinity? Theorem 3.4 will give the existence, multiplicity, and nonexistence of solutions of (1.1) when f is null at infinity.
The rest of the paper is organized as follows. In Section 2, we introduce some notations, auxiliary results and present the method of lower and upper solutions. In addition, we also introduce the method to construct lower and upper solutions. In Section 3 we give some applications to deal with the discrete Neumann problem with various nonlinearities such as the nonlinearity is bounded and super-sub linear perturbations, the nonlinearity is null at infinity and the nonlinearity is singular. We also give some examples to illustrate the main results.
2 Some notations and the method of lower and upper solutions
In the sequel, let us introduce some notations. Let with , we denote . In addition, we denote with and with .
For , set , . If , we write (resp. ) if (resp. ) for all . The following assumption upon ϕ (called singular) is made throughout the paper:
() () is an increasing homeomorphism with .
The model example is
Let with be fixed and . Then we denote
by for and if , define
by 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 P, Q be the projectors defined by
If , we write and we shall consider the following closed subspaces of :
Let the vector space be endowed with the orientation of and the norm . Its elements can be associated to the coordinates and correspond to the elements of of the form
For , we set () and, for brevity, we shall write instead of .
Now, we recall the following technical result given as Proposition 4 and Proposition 6 in [14].
Lemma 2.1 Let be a continuous operator which takes bounded sets into bounded sets and consider the abstract discrete Neumann problem
A function u is a solution of (2.1) if and only if is a fixed point of the continuous operator defined by , where satisfying
Furthermore, for all and
for any solution u of (2.1).
Let us consider the discrete Neumann problem
Obviously, from Lemma 2.1, the fixed point operator associated to (2.3) is
In what follows, we present the method of lower and upper solutions for difference equations (see [[14], Theorem 3]) to the Neumann boundary value problem (2.3).
Definition 2.1 A function (resp. ) is called a lower solution (resp. an upper solution) for (2.3) if (resp. ) and
Such a lower or an upper solution is called strict if the inequality (2.4) is strict.
We need the following result, which can be proved by the strategy of the proof of Theorem 3 in [14]; see [[14], Remark 8].
Lemma 2.2 If (2.3) has a lower solution and an upper solution such that , then (2.3) has a solution u such that . Moreover, if α and β are strict, then , and
where .
Notice that Lemma 2.2 proved that the problem (2.3) has at least one solution if it has a lower solution α and an upper solution β with . In the following result we prove some additional results concerning the location of the solution. In particular we have a posteriori estimations which will be very useful in the sequel (Remark 2.1).
Theorem 2.1 Assume that (2.3) has a lower solution α and an upper solution β such that
Then (2.3) has at least one solution u such that
Proof Let
and define the continuous function by
Let us consider the modified Neumann problem
and let be the fixed point operator associated to (2.8).
It is not difficult to verify that α is a lower solution and β is an upper solution of the problem (2.8). Moreover, by computation, is a lower solution of (2.8) and is an upper solution of (2.8). Notice that
which, together with (2.6), implies that
So, we can consider the open bounded set
It follows that
and
Clearly, any constant function between and is contained in Ω, 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 , then , . This, together with ϕ is an increasing homeomorphism, implies . On the other hand, we have
which is a contradiction. Analogously, one can obtain a contradiction in the second case. Consequently,
Now, let be such that . It follows from (2.9) that , and . We infer that there exists such that or , implying that . Then
and, consequently,
We have distinguished two cases to discuss.
Case 1. Assume that there exists such that . Using (2.10), we deduce that , implying that u is a solution of (2.3) and (2.7) holds. Actually, in this case there exists such that or .
Case 2. Assume that for all . Then, from Lemma 2.2 applied to g, 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, implies that there exists such that . It follows that there exist such that and . Then, using once again that , it follows that , and u is a solution of (2.3). Moreover, from it follows that (2.7) is true. □
Remark 2.1 Assume that (2.3) has a lower solution α and an upper solution β. From Lemma 2.2 and Theorem 2.1, we deduce that (2.3) has at least one solution u satisfying (2.7). In particular,
Remark 2.2 The corresponding result for second-order continuous periodic problems has been proved in Theorem 1 of [17] by the proof using the same strategy as above.
The following result is a particular case of [[14], Lemma 6 and Remark 8] for the discrete Neumann boundary value problem.
Lemma 2.3 Let . Then the discrete Neumann problem
has at least one solution for all .
The next result is an elementary estimation of the function .
Lemma 2.4 Let . Then
Proof Let be such that and be such that . If , then . If , then
If , then
Therefore, it follows that
and the proof is completed. □
In the following, we give a method to construct the lower solution and upper solution of the discrete Neumann problem
where is a continuous singular nonlinearity and .
The following result gives a method to construct a lower solution to (2.14), getting also control over its localization.
Theorem 2.2 Suppose that there exist and such that
If
then (2.14) has a lower solution α such that
Proof Consider the function . We have two cases.
Case 1. Assume that . Taking and, using , it follows from (2.15) that α is a lower solution of (2.14).
Case 2. Assume that . Let . Then using
and [[14], Proposition 6], it follows that there exists such that
Let us take and for . Then we define
Let , , then . On the other hand, we have
Since , Lemma 2.4 implies (2.17). Now, using (2.16), it follows that , implying that
From (2.15) and (2.17), we deduce that
Consequently,
□
By a similar argument, it is easy to prove the following theorem.
Theorem 2.3 Suppose that there exist and such that
If
then (2.14) has an upper solution β such that
3 Some applications
3.1 Bounded and super-sub linear perturbations
In this section we will study the discrete Neumann problem
where and is a continuous function.
In the following theorem we prove that if and f is bounded on , then (3.1) has at least one solution. So, resonance occurs only when .
Theorem 3.1 If and f is bounded on , then (3.1) has at least one solution.
Proof Let be a constant such that
For any , let us consider the discrete Neumann problem
Let be the fixed point operator associated to (3.2) by Lemma 2.1. Notice that if is such that , then (3.2) is satisfied and
implying that
So, one has
Then, for any sufficiently large, one has
The invariance under homotopy of the Brouwer degree implies that
Notice that from it follows that
So, the range of the operator is contained in the space of constant functions which is isomorphic to ℝ. Hence, using the reduction property of the Brouwer degree we deduce that, for ρ sufficiently large,
which, together with the fact that f is bounded and
implies that
We infer that
and the existence property of the Brouwer degree implies that has at least one fixed point u which is also a solution of (3.1). □
Example 3.1 Consider the discrete Neumann problem with attractive singularity
where is a constant, , and is a continuous function. If , then the above problem has at least one solution. In fact, let , . Then
So, the result follows from Theorem 3.1.
In the following theorem we assume that f is superlinear at zero and sublinear at infinity and we prove that (3.1) has at least one nontrivial solution if .
Theorem 3.2 Assume that f does not depends on in (3.1). If one has and
then (3.1) has at least one nontrivial solution.
Proof First of all, our assumption implies that there exists such that
This means that β is an upper solution of (3.1).
On the other hand, from (3.4), there exist and such that
We will apply Theorem 2.2 with and
Notice that
implying that (2.15) holds. Next, we have
Hence, from Theorem 2.2 we deduce that (3.1) has a lower solution α such that . In particular , and using Theorem 2.1, we infer that (3.1) has at least one solution u such that , for some , which is also a nontrivial solution. □
Corollary 3.1 If and
then (3.1) has at least one nontrivial solution.
Example 3.2 Consider the discrete Neumann problem with attractive singularity
where is a constant, and . If and , then the above problem has at least one solution.
The following dual result also holds, that is, f is superlinear at infinity and sublinear at zero and we prove that (3.1) has at least one nontrivial solution if .
Theorem 3.3 Assume that f does not depend on in (3.1). If one has and
then (3.1) has at least one nontrivial solution.
Proof Obviously, the assumption (3.5) implies that there exists such that
This means that α is an upper solution of (3.1).
On the other hand, it follows from (3.5) that there exist and such that
We will apply Theorem 2.3 with and
Notice that
implying that (2.18) holds. Next, we have
Hence, from Theorem 2.3 we deduce that (3.1) has an upper solution β such that . In particular , and, using Lemma 2.2, we infer that (3.1) has at least one solution u such that , which is also a nontrivial solution. □
Corollary 3.2 If and
then (3.1) has at least one nontrivial solution.
Example 3.3 Consider the discrete Neumann problem with attractive singularity
where is a constant, and . If and , then the above problem has at least one solution.
3.2 Nonlinearities null at infinity
In this section, we deal with nonlinearities null at infinity. This type of nonlinearities has been introduced in [18] and studied in [19, 20]. We consider the discrete Neumann problem
where is a continuous function, with and is a parameter. We have the following theorem.
Theorem 3.4 Assume that
and there exists with such that
Then there exist such that (3.6) has no solutions if and at least one solution if . Moreover, if and , then (3.6) has at least two solutions.
Proof For any fixed integer , let us consider the discrete Neumann problem
Then, taking into account that , it follows from Lemma 2.3 that (3.9) has at least one solution, . Notice that is a solution of (3.6) for . So, in particular, there exists at least one such that (3.6) has at least one solution.
Next, let us define
and , . Using that f is bounded on and for any solution u of (3.6), we infer that , are finite.
Now, we will prove that . It suffices to prove that there exists such that . One has
Suppose on the contrary that
Using (3.7), (3.8), and the fact that for all , it follows that there exists such that
and
for all . It follows that
which is a contradiction with the assumption . So, (3.10) holds true. This implies that is a lower solution of (3.6) for all . Analogously, it follows that there exists such that is an upper solution of (3.6) for all . Then , just taking δ sufficiently small and applying Theorem 4 and Remark 8 of [14].
Next, let us prove that . Consider . It follows that there exists such that , so, (3.6) has at least one solution α for . Then α is a strict lower solution of (3.6). Using once again (3.7) and the fact that for all , it follows that there exists sufficiently large such that and
It follows that , are strict upper solution for (3.6). Then from Lemma 2.2 we infer that (3.6) has a solution such that . On the other hand, from Theorem 2.1, it follows that (3.6) has a solution such that for some . Hence, and . Consider a sequence in converging to and a solution of (3.6) with . Notice that
which, together with for all , and (3.7), implies that is a bounded sequence. Consequently, is a bounded sequence in . Subsequently, there exists a subsequence of converging uniformly to some which is a solution of (3.6) with . Analogously, one has . □
Example 3.4 Consider the discrete Neumann problem
where is a constant, and is a parameter. From Theorem 3.4, there exist such that (3.11) has no solutions if and at least one solution if . Moreover, if and , then (3.11) has at least two solutions.
Remark 3.1 It is interesting to note that in [14], the authors deal with nonlinearities at infinity for the discrete periodic problem, see [[14], Theorem 6 and Theorem 7], which also hold for the discrete Neumann problem ([[14], Remark 9]).
3.3 Singular perturbations problem
In the following we will apply Theorem 3.1 to study the singular Neumann problem
where and .
Theorem 3.5 Assume that , with and
Then (3.12) has at least one positive solution.
Proof Let us define the auxiliary increasing functions
From (3.13) it follows that , and there exists such that
Now, consider the continuous function given by
and consider the modified Neumann problem
Using that g is bounded and , it follows from Theorem 3.1 that (3.15) has a solution .
We will show that . Summing (3.15) from to we deduce that
which, together with Lemma 2.4, implies that
On the other hand, using , one has
Let us assume that . Then, using (3.17) and , we infer that
contradicting (3.14) and (3.18). So, .
Next, using (3.17), (3.13), Lemma 2.4, and , it follows that
which, together with , implies that .
From this, together with (3.14) and (3.16), we deduce that
implying that . This, together with Lemma 2.4, implies that , and our claim is proved. Consequently, u is also a solution of (3.12). □
Remark 3.2 It is not difficult to show that the results proved in this paper also hold for the discrete periodic boundary value problem.
Example 3.5 Consider the discrete Neumann problem with attractive singularity
where is a constant, and . By using Theorem 3.5, the above problem has at least one solution if .
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Acknowledgements
Research was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Fundamental Research Science Foundation of China (No. 1208RJZA258).
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YL and RM completed the main study, carried out the results of this article and drafted the manuscript, YL checked the proofs and verified the calculation. All the authors read and approved the manuscript.
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Lu, Y., Ma, R. Existence and multiplicity of solutions of second-order discrete Neumann problem with singular ϕ-Laplacian operator. Adv Differ Equ 2014, 227 (2014). https://doi.org/10.1186/1687-1847-2014-227
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DOI: https://doi.org/10.1186/1687-1847-2014-227