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Solvability for some boundary value problems with discrete ϕ-Laplacian operators
- Tianlan Chen^{1}Email author and
- Ruyun Ma^{1}
https://doi.org/10.1186/s13662-015-0473-z
© Chen and Ma; licensee Springer. 2015
- Received: 20 October 2014
- Accepted: 14 April 2015
- Published: 6 May 2015
Abstract
In this paper, we study the existence of solutions of the discrete ϕ-Laplacian equation \(\nabla[\phi(\Delta u_{k})]=\lambda f(k, u_{k}, \Delta u_{k})\), \(k\in[2, n-1]_{\mathbb{Z}}\), with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate of \(\lambda_{0}\) is given such that the problem possesses a solution for any \(|\lambda|<\lambda_{0}\).
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 research. Different mathematical models coupled with the basic theory of this type of equation can be found in the classical monograph by Goldberg [1] and in the book by Lakshmikantham and Trigiante [2]. Besides, they are also natural consequences of the discretization of differential problems.
- (H1)
\(\phi:(-a, a)\rightarrow(-b, b)\) is an increasing homeomorphism with \(\phi(0)=0\) and \(0< a, b\leq\infty\);
- (H2)for all \(k\in[2, n-1]_{\mathbb{Z}}\), \(f(k,\cdot, \cdot):\mathbb{R}\times(-a, a)\rightarrow\mathbb{R}\) with \(\mathbf{f}=(f(2, u, v),\ldots,f(n-1, u, v))\), and for each compact set \(A\subset\mathbb{R}\times(-a, a)\), there exists a bounded function \(\mathbf{h}_{A}=(h_{A}(2),\ldots, h_{A}(n-1))\) defined from \(\mathbb{R}\) to \(\mathbb{R}^{n-2}\) such that$$\bigl\vert f(k, u, v)\bigr\vert \leq h_{A}(k), \quad \text{for } k\in[2, n-1]_{\mathbb{Z}} \text{ and all } (u, v)\in A. $$
- (i)\(a=b=+\infty\) (regular unbounded): we have the p-Laplacian operator$$\phi_{1}(x)=|x|^{p-2}x \quad \text{with } p>1. $$
- (ii)\(a<+\infty\), \(b=+\infty\) (singular unbounded): we have the relativistic operator$$\phi_{2}(x)=\frac{x}{\sqrt{1-x^{2}}}. $$
- (iii)\(a=+\infty\), \(b<+\infty\) (regular bounded): we have the one-dimensional mean curvature operator$$\phi_{3}(x)=\frac{x}{\sqrt{1+x^{2}}}. $$
Among them, the p-Laplacian operator has received a lot of attention and the number of related references is huge (we only mention [3–6] and references therein). For the relativistic operator, it has recently been proved in [7–9] 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 [10, 11]. 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 [10] 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 solutions. Thus, new challenges are faced and innovation is required. In addition, we extend some results of Bereanu and Mawhin in [7]; see Remark 1. 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.
2 The Dirichlet boundary value problem
Lemma 2.1
Proof
By a simple computation, we can get the following result.
Lemma 2.2
Now we are in a position to prove the main result of this section: the solvability of problem (5) for small λ.
Theorem 2.3
Proof
2.1 Unbounded ϕ-Laplacian (\(b=+\infty\))
A consequence of Theorem 2.3 is that, whenever ϕ is unbounded, then (9) is solvable.
Corollary 1
Proof
Remark 1
Corollary 1 applies in particular if ϕ is also singular \(a<+\infty\) and for each \(k\in[2, n-1]_{\mathbb{Z}}\), \(f(k,\cdot,\cdot)\) is continuous on \(\mathbb{R}^{2}\). In this way Corollary 1 improves Theorem 1 in [7].
Example 1
2.2 Bounded ϕ-Laplacian (\(b<+\infty\))
In the case of a bounded ϕ-Laplacian, the ‘universal’ solvability of (9) is no longer true even for a constant nonlinearity \(f(t, u, v)\equiv M\) as we show in the following result.
Proposition 1
Before proving the Proposition 1, we first give the following result, which is a slight modification of Lemma 2 in [7].
Lemma 2.4
Proof of the Proposition 1
Remark 2
Regrettably our method does not determine whether or not there exists a solution of (11) when \(\frac{b}{2(n-2)}\leq|M|<\frac{b}{(n-2)}\), since we cannot guarantee \(u_{n}=0\) (in fact, the difference equation loses the symmetric property about the domain of solution). However, in [10], Proposition 1, the authors prove that the differential problem corresponding to (11) has the complete result.
As a consequence of Theorem 2.3 we obtain the following sufficient condition for the solvability of the Dirichlet problem.
Corollary 2
Proof
3 The mixed boundary value problem
By using the same idea as in Theorem 2.3, we can prove the following result.
Theorem 3.1
Proof
Remark 3
The preceding theorem is sharp in the following sense: when \(\phi(x)=\frac{x}{\sqrt{1+x^{2}}}\) and \(f(k, u, v)\equiv M\), then it is easy to show that problem (13) has a solution if and only if \(|\lambda|<\tilde{\lambda}_{0}=\sup_{0<r<1}\frac{r}{M_{r}}=\frac{1}{M}\).
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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