- Open Access
Existence and multiplicity of difference ϕ-Laplacian boundary value problems
© Bai and Xu; licensee Springer 2013
Received: 5 June 2013
Accepted: 22 August 2013
Published: 8 September 2013
Concerned are the difference ϕ-Laplacian boundary value problems. The multiplicity result based on the lower and upper solutions method associated with Brouwer degree is applied to a difference ϕ-Laplacian eigenvalue problem. An existence result of at least three positive solutions is established for the eigenvalue problem with the parameter belonging to an explicit open interval. In addition, an example is given to illustrate the three solutions result.
Recently, Kim  studied a one-dimensional differential p-Laplacian boundary value problem with a positive parameter and established an existence result of three positive solutions by the lower and upper solutions method associated with Leray-Schauder degree theory. Kim and Shi  studied the global continuum and multiple positive solutions of a p-Laplacian boundary value problem. Motivated by the methods in [1, 2], we consider difference ϕ-Laplacian boundary value problems.
where is a given positive integer, , and
(A1) is an odd and strictly increasing function;
(A2) is continuous.
where λ is a positive parameter. Under some suitable assumptions imposed on g, we establish the existence of three positive solutions of (2) with λ belonging to an explicit open interval.
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 [3–10] 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 [12–19] and the references therein.
For discrete ϕ-Laplacian problems, there are fewer study results than for differential ϕ-Laplacian problems. See Cabda , Cabada and Espinar  and Bondar . To the best of our knowledge, there are no results on the existence and multiplicity of positive solutions for difference ϕ-Laplacian problems.
The remaining part of this paper is organized as follows. In Section 2, we show the lower and upper solutions method and establish the existence and multiplicity of solutions of (1). In Section 3, we establish the existence of three positive solutions of (2). Finally, we give an example to illustrate our main results.
2 The upper and lower solutions method
In this section, we establish the existence and multiplicity results of solutions for problem (1) by lower and upper solutions method associated with Brouwer degree.
Let with the norm .
if for all , .
if for all , and , .
If the first inequality above is strict, then α is called a strict lower solution of (1).
In the same way, we define the upper solution and the strict upper solution of (1) by reversing the inequalities above.
has the unique solution .
Similarly, if , then and , , which implies that , which is a contradiction. The proof is complete. □
Assume that there exist α and β, respectively lower and upper solutions of (1) such that . Then problem (1) has at least one solution u with .
- (ii)Assume that problem (1) has two pairs of lower and upper solutions and with and being strict, satisfying that
Remark We denote that the result (i) has been proved in  by Brouwer fixed point theorem. Here, for the convenience of the proof of (ii), it is proven by Brouwer degree theory. The proof of (ii) is motivated by the idea in .
- (ii)First, we show that if α and β are strict lower and upper solutions, respectively, such that , then , where . By the arguments above, each solution u of (5) satisfies that . We claim that . In fact, if it is not true, then there exists an such that . Since , , we have by the monotonicity of ϕ that
Then , and . Thus by the additivity property of Brouwer degree, we have . Therefore, problem (7) has three solutions , and with , and . By the facts that all solutions of (7) satisfy and are solution of (1), the proof is complete. □
3 Three positive solutions of eigenvalue problems
with , . Then for all , and for , for , where satisfies .
we have by the monotonicity of that for , for , which implies that holds for all by the boundary conditions , . □
Remark If inequality (8) is strict, then for , and there exists such that , and for , , and for .
In the following arguments, we assume that
(B1) is an odd and strictly increasing homeomorphism.
- (i)if () is odd, then , and the solution u of (9) can be expressed as
- (ii)if () is even, then , and the solution u of (9) can be expressed as
- (i)Assume that () is odd. Since , by the symmetry of h and (14), we have
- (ii)If () is even, then (14) and the symmetry of h imply that
Clearly, . The proof is complete. □
We make the following assumptions.
(B4) and ;
We denote that condition (B4) implies that (see , Lemma 2.8). Clearly, is nondecreasing on .
Thus by Theorem 2.1, problem (2) has three positive solutions for . □
Remark If g is nondecreasing on , then we take and .
4 An example
Thus by Theorem 3.1, problem (15) has at least three positive solutions for .
The authors are 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).
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