Solvability for some boundary value problems with discrete ϕ Laplacian operators

(1)

R E S E A R C H

Open Access

Solvability for some boundary value

problems with discrete

φ

-Laplacian operators

Tianlan Chen

*

_{and Ruyun Ma}

*_{Correspondence:}

[email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China

Abstract

In this paper, we study the existence of solutions of the discrete

φ

-Laplacian equation

∇[

φ

(

uk)] =

λ

f(k,uk,

uk),k∈[2,n– 1]Z, with Dirichlet or mixed boundary

conditions. Under general conditions, an explicit estimate of

λ0

is given such that the problem possesses a solution for any|

λ

|<

λ

0.

Keywords:

φ

-Laplacian; existence; Schauder’s ﬁxed point theorem; mixed boundary value problem; Dirichlet boundary value problem

1 Introduction

The study of difference equations represents a very important field in mathematical re-search. Different mathematical models coupled with the basic theory of this type of equation can be found in the classical monograph by Goldberg [] and in the book by Lakshmikantham and Trigiante []. Besides, they are also natural consequences of the discretization of differential problems.

In this paper, we consider the existence of solutions of discreteφ-Laplacian problems

∇φ(uk)

=λf(k,uk,uk), k∈[,n– ]Z, ()

with Dirichlet boundary conditions

u=  =un ()

or mixed boundary conditions

u=  =un– ()

under the following assumptions:

(H) φ: (–a,a)→(–b,b)is an increasing homeomorphism withφ() = and  <a,b≤ ∞;

(H) for allk∈[,n– ]Z,f(k,·,·) :R×(–a,a)→Rwithf= (f(,u,v), . . . ,f(n– ,u,v)),

and for each compact setA⊂R×(–a,a), there exists a bounded function

hA= (hA(), . . . ,hA(n– ))deﬁned fromRtoRn–such that

f(k,u,v)≤hA(k), fork∈[,n– ]Zand all(u,v)∈A.

(2)

The study of theφ-Laplacian equation is a classical topic that has attracted the atten-tion of many researchers because of its interest for applicaatten-tions. Usually, aφ-Laplacian operator is said to besingularwhen the domain ofφis ﬁnite (that is,a< +∞); in the con-trary case, the operator is said to beregular. On the other hand we say thatφisbounded if its range is ﬁnite (that is, b< +∞) andunboundedin the other case. There are three paradigmatic models in this context:

(i) a=b= +∞(regular unbounded): we have thep-Laplacian operator

φ(x) =|x|p–x withp> .

(ii) a< +∞,b= +∞(singular unbounded): we have the relativistic operator

φ(x) =

x √

 –x.

(iii) a= +∞,b< +∞(regular bounded): we have the one-dimensional mean curvature operator

φ(x) =

x √

 +x.

Among them, thep-Laplacian operator has received a lot of attention and the number of related references is huge (we only mention [–] and references therein). For the rela-tivistic operator, it has recently been proved in [–] that the Dirichlet problem is always solvable. This is a striking result closely related to the ‘a priori’ bound of the derivatives of the solutions. For the curvature operator, this is no longer true, but other results as regards the existence of solutions of diﬀerential problems can be found in [, ]. To the best of our knowledge, the discrete problem has received almost no attention. In this article, we will discuss it in detail.

The purpose of this paper is to show that analogs of the existence results of solutions for differential problems proved in [] hold for the corresponding difference equations. However, some basic ideas from differential calculus are not necessarily available in the field of difference equations, such as Rolle’s theorem and symmetry of the domain of so-lutions. Thus, new challenges are faced and innovation is required. In addition, we extend some results of Bereanu and Mawhin in []; see Remark . The proof is elementary and relies on Schauder’s fixed point theorem after a suitable reduction of the problem to a first-order summing-difference equation.

We end this section with some notations. For x∈Rp_set

|x|∞= max

≤k≤p|xk|, |x|= p

k=

|xk|.

For everyl,m∈Nwithm>l, we setl_k₌_mxk= , and [l,m]Z:={l,l+ , . . . ,m}. Letn∈N,n≥ be ﬁxed and x = (x, . . . ,xn)∈Rn. Deﬁne

x= (x, . . . ,xn–)∈Rn–,

as

(3)

and if|x|_∞<a, deﬁne

2 The Dirichlet boundary value problem Let us consider the boundary value problem

∇φ(uk)

=λf(k,uk,uk), k∈[,n– ]Z, u=  =un, ()

whereλ∈Ris a parameter and (H) and (H) hold. Let us deﬁne the space

H= y∈Rn–y= (y, . . . ,yn–) with|y|∞<b

modiﬁcation of Lemma  in [], but we include the proof for the sake of completeness.

Lemma . For anyy∈H,there exists a unique constantγ:=Pφ[y]such that

Proof From the properties ofφ, it is clear that

n–

Becauseφis a homeomorphism such thatφ() = , it follows thatφ–is strictly monotone and satisﬁes (), so there exists a uniqueγ :=Pφ[y] verifying () withγ ≤ |y|∞. Note that

Pφtakes bounded sets into bounded sets. So the continuity ofPφfollows easily.

(4)

By a simple computation, we can get the following result.

Lemma . Ify= (y, . . . ,yn–)is a solution of the problem()with|y|∞<b,then

uk= k–

i=

φ–Pφ[y] +yi

is a solution of().

Now we are in a position to prove the main result of this section: the solvability of prob-lem () for smallλ.

Theorem . For each <r<b

,let Mrbe deﬁned by().If

|λ|<λ:= sup <r<b_

r Mr

,

then problem()has a solution.

Proof Let  <r<b_ be such that|λ| ≤_Mr__r

 and consider the ball

Br=

y∈Rn|y= (y, . . . ,yn) with|y|∞≤r

.

For each y∈Br, we deﬁne the operator

Tyk=λ k

j=

f

j, j–

i=

φ–Pφ[y] +yi

,φ–Pφ[y] +yj

.

It is easy to show thatTis completely continuous. Moreover, by our assumptions and the choice ofrwe have

|Ty|_∞≤ |λ|Mr≤r,

which implies thatT(Br)⊂Br. Thus Schauder’s ﬁxed point theorem yields a ﬁxed point

forT which is a solution of (), and therefore by Lemma . it is also a solution of

prob-lem ().

Now we are going to apply Theorem . to study the solvability of the Dirichlet problem

∇φ(uk)

=f(k,uk,uk), k∈[,n– ]Z, u=  =un. ()

We point out that problem () presents interesting diﬀerent features depending on the bounded or unbounded behavior ofφ.

2.1 Unbounded

φ

-Laplacian (b= +∞)

(5)

Corollary  Assume thatφ is unbounded(that is,b= +∞)and there exists a bounded functionh= (h(), . . . ,h(n– ))such that

f(k,u,v)≤h(k), k∈[,n– ]Z,u∈–(n– )a, (n– )a,v∈(–a,a). ()

Then the Dirichlet problem()has at least one solution.

Proof By conditions () it is clear that

Mr=|h|, for eachr> .

Therefore

λ= sup <r<+∞

r Mr

= +∞,

and then Theorem . ensures us that the problem () has a solution for eachλ∈R, and

in particular forλ= .

Remark  Corollary  applies in particular ifφis also singulara< +∞and for eachk∈ [,n– ]Z,f(k,·,·) is continuous onR. In this way Corollary  improves Theorem  in [].

Example  The assumptions of Corollary  are satisﬁed forφ(v) =√v

–v and for allk∈

[,n– ]Z,

f(k,u,v) =

, u∈[, (n– )a),v∈(–a,a), –, u∈(–(n– )a, ),v∈(–a,a).

Thus, by Corollary , the problem () has a solution.

2.2 Bounded

φ

-Laplacian (b< +∞)

In the case of a boundedφ-Laplacian, the ‘universal’ solvability of () is no longer true even for a constant nonlinearityf(t,u,v)≡Mas we show in the following result.

Proposition  Assume thatφis bounded(that is,b< +∞),let M∈Rand consider the Dirichlet problem

∇φ(uk)

=M, k∈[,n– ]Z, u=  =un. ()

Then the following claims hold: (i) If|M| ≥ b

n–,then()has no solution.

(ii) If|M|< b

(n–),then()has a solution.

Before proving the Proposition , we ﬁrst give the following result, which is a slight mod-iﬁcation of Lemma  in [].

Lemma . For anyy∈E,

E:= y∈Rn–y= (y, . . . ,yn–)with|y|<

b 

(6)

there exists a unique constantγ :=Qφ[y]such that

In this case u given by () would not be well deﬁned since the domain ofφ–is the interval (–b,b), and thus a solution of () cannot exist.

(ii) If|M|<_(_nb_–), then the function u given by () is well deﬁned forτ= , and satisﬁes

Remark  Regrettably our method does not determine whether or not there exists a so-lution of () when _(_nb_–)≤ |M|<₍_nb_–), since we cannot guaranteeun=  (in fact, the dif-ference equation loses the symmetric property about the domain of solution). However, in [], Proposition , the authors prove that the diﬀerential problem corresponding to () has the complete result.

(7)

Corollary  Assume thatφis bounded(that is,b< +∞)and there exists a bounded

func-and thus Theorem . implies the existence of a solution for problem () withλ= .

Example  Letn∈N,n≥ be ﬁxed. We takeφ(v) =√v

+v andf(k,u,v) = 

nsin(k+u– v)π. Thus, by a simple computation, we show by Corollary  that problem () has a solu-tion.

3 The mixed boundary value problem

Compared with the Dirichlet problem, the mixed boundary value problem

∇φ(uk)

=λf(k,uk,uk), k∈[,n– ]Z, u=  =un– ()

is less studied in the related literature. In this case, by means of the change of variable yk=φ(uk),k= , . . . ,n– , we see that a solution u = (u, . . . ,un) of () is equivalent to a

By using the same idea as in Theorem ., we can prove the following result.

Theorem . If

then problem()has a solution.

Proof Let  <r<bbe such thatλ≤_Mrr

and apply Schauder’s ﬁxed point theorem to the completely continuous operatorT:Br→

(8)

Remark  The preceding theorem is sharp in the following sense: whenφ(x) = √x +x and

f(k,u,v)≡M, then it is easy to show that problem () has a solution if and only if|λ|< ˜

λ=sup<r<Mrr =



M.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have achieved equal contributions to each part of this paper. All the authors read and approved the ﬁnal manuscript.

Acknowledgements

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).

Received: 20 October 2014 Accepted: 14 April 2015

References

1. Goldberg, S: Introduction to Diﬀerence Equations. Wiley, New York (1960)

2. Lakshmikantham, V, Trigiante, D: Theory of Diﬀerence Equations: Numerical Methods and Applications. Mathematics in Science and Engineering, vol. 181. Academic Press, Boston (1988)

3. Cabada, A, Otero-Espinar, V: Existence and comparison results for diﬀerenceφ-Laplacian boundary value problems with lower and upper solutions in reverse order. J. Math. Anal. Appl.267, 501-521 (2002)

4. Wang, D, Guan, G: Three positive solutions of boundary value problems forp-Laplacian diﬀerence equations. Comput. Math. Appl.55, 1943-1949 (2008)

5. Coster, C: Pairs of positive solutions for the one dimensionalp-Laplacian. Nonlinear Anal.23, 669-681 (1994) 6. Jiang, L, Zhou, Z: Three solutions to Dirichlet boundary value problems forp-Laplacian diﬀerence equations. Adv.

Diﬀer. Equ.2008, Article ID 345916 (2008)

7. Bereanu, C, Mawhin, J: Boundary value problem for second-order nonlinear diﬀerence equations with discrete φ-Laplacian and singularφ. J. Diﬀer. Equ. Appl.14, 1099-1118 (2008)

8. Bereanu, C, Thompson, H: Periodic solutions of second order nonlinear diﬀerence equation with discrete φ-Laplacian. J. Math. Anal. Appl.330, 1002-1015 (2007)

9. Bereanu, C, Mawhin, J: Existence and multiplicity results for some nonlinear problems with singularφ-Laplacian. J. Diﬀer. Equ.243, 536-557 (2007)

10. Cid, J, Torres, P: Solvability for some boundary value problems withφ-Laplacian operators. Discrete Contin. Dyn. Syst.

23, 727-732 (2009)