A global result for discrete ϕ-Laplacian eigenvalue problems
© Bai; licensee Springer 2013
Received: 11 June 2013
Accepted: 15 August 2013
Published: 29 August 2013
Concerned is the existence, nonexistence and multiplicity of positive solutions for discrete ϕ-Laplacian eigenvalue problems. By using lower and upper solutions method and fixed point index theory, a global result with respect to parameter is established.
where is a given positive integer, , λ is a positive parameter, and is continuous. We assume that
(A1) is an odd and strictly increasing homeomorphism;
The function covers two important cases: and (). If , then problem (1) is the classical second order difference Dirichlet boundary value problem. For the case that , problem (1) is the well-known discrete p-Laplacian problem. The two cases have been widely studied. To name a few, see [1–8] and the references therein.
which rises from the study of radial solutions for p-Laplacian equations () on an annular domain (see , and references therein). Recently, the differential ϕ-Laplacian problems have been widely studied in many different papers. We refer the readers to [10–17] and the references therein.
For discrete ϕ-Laplacian problems, there are less study results than differential ϕ-Laplacian problems. See Cabada , Cabada and Espinar  and Bondar . In [18, 19], some existence results were established by the upper and lower solutions method. In , the existence and uniqueness of solutions were discussed for mixed and Dirichlet boundary value problems by the fixed point theory of contraction mapping. To the best of our knowledge, there are no results on the existence and multiplicity of positive solutions for difference ϕ-Laplacian problems. Therefore, the purpose of this paper is to establish a global result of positive solutions of (1). We state our main result as follows.
Theorem 1.1 Let (A1) and (A2) hold. Then there exists such that problem (1) has at least two positive solutions for , at least one positive solution for and no solution for .
The result is motivated mainly by the ideas in [11, 21], in which some global results of positive solutions were established for boundary value problems of p-Laplacian differential systems and ϕ-Laplacian differential systems, respectively.
Generally, in order to make priori estimations on possible positive solutions of ϕ-Laplacian problems, the function ϕ satisfies not only condition (A1), but also other additional conditions. For example, Wang  used the following condition.
In , a more general condition was given.
(A1∗∗) For , there exists a constant such that for all , .
In our discussion for the single discrete problem (1), we only assume that ϕ satisfies condition (A1). In addition, in the discussion of nonlinear differential systems in [11, 21], monotonicity conditions were imposed on nonlinear terms. In this paper g does not have to satisfy monotonicity conditions.
The remaining part of this paper is organized as follows. In Section 2, we show some lemmas for the later use. In Section 3, we show the proof of Theorem 1.1. Our proofs are mainly based on the upper and lower solutions technique arguments and the fixed-point index theory for cones.
2 Some lemmas
First, we introduce an existence result of solutions based on lower and upper solutions method for discrete ϕ-Laplacian boundary value problems, which has been proved by Cabada .
(B1) is an odd and strictly increasing function;
(B2) is continuous.
Let with the norm . Given , we say that if holds for all .
In a same way, we define the upper solution of (3) by reversing the above inequalities.
Lemma 2.1 
Let (B1) and (B2) hold. Assume that there exist α and β, respectively lower and upper solutions of (3) such that . Then problem (3) has at least one solution u with .
A function u of integer variable is said to be concave if . This is the same as saying that the first difference is non-increasing. If , then u is said to be strictly concave.
for all , and for , for , where satisfies .
- (ii)For each ( denotes the integer part of ),
Specially, , .
- (ii)For with , let and denote the closed and open interval on R, respectively. For any given , define
which implies the expected results. □
Remark If is strictly concave on , and , . Then for , and there exists such that , for , , and for .
with , . Then is nonnegative concave on . Specially, if inequality (4) is strict, then is strictly concave on .
Proof Since , , we have by the monotonicity of ϕ that , , which implies that is concave on and for by Lemma 2.2(i). □
Lemmas 2.2 and 2.3 yield the following result.
Lemma 2.4 Let (B1) hold. Then each solution u of (1) is strictly concave and for all .
By Lemmas 2.2 and 2.3, the following result holds.
Then and is continuous. We know that u is a positive solution of (1) if and only if on E.
Lemma 2.6 Let (A1) and (A2) hold, ℜ be a compact subset in . Then there exists a constant such that for all and all possible positive solutions u of (1) at λ, one has .
where satisfies . The proof is completed. □
Lemma 2.7 Let (B1) hold. Then there exists such that (1) has a positive solution at .
Therefore, β is an upper solution of (1) at . By Lemma 2.1, (1) has a positive solution u at , and the proof is done. □
Lemma 2.8 Let (B1) hold. If the problem (1) has a positive solution at , then (1) also has a positive solution at λ for all .
Proof Let be a positive solution of (1) at and λ satisfy . Then is an upper solution of (1) at λ and is a lower solution. Thus, Lemma 2.1 implies that (1) has a positive solution at . □
Lemma 2.9 Let (A1) and (A2) hold. Then Λ is bounded.
Lemma 2.10 Let (A1) and (A2) hold. Then there exists such that .
which implies that solves problem (1). By Lemma 2.4, is positive on and . The proof is completed. □
In the succeeding arguments, we need the following well-known fixed point index theorem on cones. For proof and details, see Guo .
3 The proof of Theorem 1.1
Therefore, problem (1) has one positive solution in and another in . The proof is completed.
The author worked on the results independently.
The author is very grateful to the referees for their helpful comments. This research is supported partially by the Research Funds for the Doctoral Program of Higher Education of China (No. 20104410120001, 20114410110002), PCSIRT of China (No. IRT1226) and the Natural Science Fund of China (No. 11171078).
- Agarwal RP, O’Regan D: Boundary value problems for discrete equations. Appl. Math. Lett. 1997, 10(4):83–89. 10.1016/S0893-9659(97)00064-5MathSciNetView ArticleGoogle Scholar
- Henderson J, Thompson HB: Existence of multiple solutions for second-order discrete boundary value problems. Comput. Math. Appl. 2002, 43: 1239–1248.MathSciNetView ArticleGoogle Scholar
- Rachunkova I, Rachunek L: Solvability of discrete Dirichlet problem via lower and upper functions method. J. Differ. Equ. Appl. 2007, 13(5):423–429. 10.1080/10236190601143302MathSciNetView ArticleGoogle Scholar
- Bai D, Xu Y: Nontrivial solutions of boundary value problems of second order difference equations. J. Math. Anal. Appl. 2007, 326(1):297–302. 10.1016/j.jmaa.2006.02.091MathSciNetView ArticleGoogle Scholar
- Bereanu C, Mawhin J, Neuve L: Existence and multiplicity results for nonlinear second order difference equations with Dirichlet value conditions. Math. Bohem. 2006, 2: 145–160.Google Scholar
- Agarwal RP, Perera K, O’Regan D: Multiple positive solutions of singular discrete p -Laplacian problems via variational methods. Adv. Differ. Equ. 2005, 2: 93–99.MathSciNetGoogle Scholar
- Cabada A, Iannizzotto A, Tersianc S: Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 2009, 356: 418–428. 10.1016/j.jmaa.2009.02.038MathSciNetView ArticleGoogle Scholar
- Jiang D, Pang P, Agarwal RP: Upper and lower solutions method and a superlinear singular discrete boundary value problem. Dyn. Syst. Appl. 2007, 16: 743–754.MathSciNetGoogle Scholar
- Dang H, Oppenheimer S: Existence and uniqueness results for some nonlinear boundary value problems. J. Math. Anal. Appl. 1996, 198: 35–48. 10.1006/jmaa.1996.0066MathSciNetView ArticleGoogle Scholar
- Wang H: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 2003, 281: 287–306. 10.1016/S0022-247X(03)00100-8MathSciNetView ArticleGoogle Scholar
- Lee E, Lee Y: A global multiplicity result for two-point boundary value problems of p -Laplacian systems. Sci. China Math. 2010, 53(4):967–984. 10.1007/s11425-010-0088-5MathSciNetView ArticleGoogle Scholar
- Cabada A, Habets P, Pouso RL: Optimal existence conditions for ϕ -Laplacian equations with upper and lower solutions in the reversed order. J. Differ. Equ. 2000, 166: 385–401. 10.1006/jdeq.2000.3803MathSciNetView ArticleGoogle Scholar
- Cabada A, Pouso RL:Existence results for the problem with nonlinear boundary conditions. Nonlinear Anal. 1999, 35: 221–231. 10.1016/S0362-546X(98)00009-1MathSciNetView ArticleGoogle Scholar
- Arrázola E, Ubilla P: Positive solutions for the 1-dimensional generalized p -Laplacian involving a real parameter. Proyecciones 1998, 17: 189–200.MathSciNetGoogle Scholar
- Henderson J, Wang H: An eigenvalue problem for quasilinear systems. Rocky Mt. J. Math. 2007, 37: 215–228. 10.1216/rmjm/1181069327MathSciNetView ArticleGoogle Scholar
- Lian W, Wong F: Existence of positive solutions for higher order generalized p -Laplacian BVPs. Appl. Math. Lett. 2000, 13: 35–43.MathSciNetView ArticleGoogle Scholar
- Bai D, Chen Y: Three positive solutions for a generalized Laplacian boundary value problem with a parameter. Appl. Math. Comput. 2013, 219: 4782–4788. 10.1016/j.amc.2012.10.100MathSciNetView ArticleGoogle Scholar
- Cabada A: Extremal solutions for the difference ϕ -Laplacian problem with nonlinear functional boundary conditions. Comput. Math. Appl. 2001, 42: 593–601.MathSciNetView ArticleGoogle Scholar
- Cabada A, Otero-Espinar V: Existence and comparison results for difference ϕ -Laplacian boundary value problems with lower and upper solutions in reverse order. J. Math. Anal. Appl. 2002, 267: 501–521. 10.1006/jmaa.2001.7783MathSciNetView ArticleGoogle Scholar
- Bondar K, Borkar V, Patil S: Existence and uniqueness results for difference ϕ -Laplacian boundary value problems. ITB J. Sci. 2011, 43A(1):51–58.MathSciNetView ArticleGoogle Scholar
- Lee E, Lee Y: A multiplicity result for generalized Laplacian system with multiparameters. Nonlinear Anal. 2009, 71: e366-e376. 10.1016/j.na.2008.11.001View ArticleGoogle Scholar
- Guo D: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan; 2002. (in Chinese)Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.