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Existence of solutions for a multi-point boundary value problem with a \(p(r)\)-Laplacian
- Zhiguo Luo^{1} and
- Jinfang Liang^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1846-x
© The Author(s) 2018
- Received: 2 June 2018
- Accepted: 15 October 2018
- Published: 7 December 2018
Abstract
In this paper, we consider the existence of solutions to the \(p(r)\)-Laplacian equation with multi-point boundary conditions. Under some new criteria and by utilizing degree methods and also the Leray–Schauder fixed point theorem, the new existence results of the solutions have been established. Some results in the literature can be generalized and improved. And as an application, two examples are provided to demonstrate the effectiveness of our theoretical results.
Keywords
- \(p(r)\)-Laplacian
- Boundary condition
- Fixed point theorem
- Leray–Schauder degree method
1 Introduction
In recent years, there has been extensive interest in boundary value problems (BVPs) with variable exponent in a Banach space, see [1–9]. Such problems usually arise in the study of image processing, elastic mechanics, electrorheological fluids dynamics, etc. (see [10–18]).
- (H1)
\(f \in C([0, 1]\times \mathbb {R}\times \mathbb {R}, \mathbb {R})\), \(p \in C([0, 1], \mathbb {R})\), \(p(r)>1\), \(a \in C((0, 1), \mathbb {R})\) is possibly singular at \(r=0\) or \(r=1\) and satisfies \(0<\int_{0}^{1}|a(r)|\mathrm {\,d}r<+\infty\);
- (H2)
\(\alpha,\beta_{i}>0\), \(0<\xi<\eta_{1}<\eta_{2}<\cdots <\eta_{m-3}<1\).
- (1)
f is the nonlinear term and \(a(r)\) is allowed to be singular at \(r=0\) or \(r=1\). Additionally, compared to two-point or three-point BVPs, which have been extensively studied, we discuss a multi-point BVP in this article.
- (2)
The model we are concerned with is more generalized, some ones in the articles [19–22] are the special cases of it. \(p(r)\) is a general function, which is more complicated than the case when p is a fixed constant. That is to say, the comprehensive model is originally considered in the present paper.
- (3)
An innovative approach based on degree methods and the Leray–Schauder fixed point theorem are utilized to obtain the existence of solutions for the addressed equations (1.1). The results established are essentially new.
The following article is organized as follows: In Sect. 2, we introduce some necessary notations and important lemmas, while Sect. 3 is devoted to establishing the existence of solutions for problem (1.1) by a fixed point theorem and degree methods, and then we come up with the main theorems. To explain the results clearly, we finally give two examples in Sect. 4.
2 Preliminaries
In this section, we are going to present some basic notations and lemmas which are used throughout this paper.
Obviously, \(\varphi^{-1}(r,\cdot)\) is continuous and sends a bounded set into a boundary set. Aiming to obtain the existence of solutions to problem (1.1), we need the following lemmas. The proofs are standard, thus some details can be omitted.
Lemma 2.1
([31])
Lemma 2.2
- (S1)
T is a compact map;
- (S2)
For any \(u \in U\), \(T(u,0)=0\);
- (S3)
If one has \(u=T(u,\lambda)\) for some \(\lambda\in [0, 1]\), then there exists \(M>0\) such that \(\|u\|_{1}\leq M\) for any \(u \in U\). Then \(T(u,1)\) has a fixed point in U.
Lemma 2.3
Lemma 2.4
Proof
Since \(h \in C[0,1]\), and let \(R_{0}=2\|h\|\). It is easy to see that if \(|\rho|>R_{0}\), then for any \(r\in[0, 1]\) we have \(( \rho-\int _{0}^{r} h(s)\mathrm {\,d}s )\cdot\rho> 2\|h\|^{2}\).
Lemma 2.5
Proof
3 Existence of solutions
In this section, we will show that under some suitable conditions solutions to problem (1.1) do exist.
Theorem 3.1
Proof
From the continuity of \(f, \varphi^{-1}\) and also the definition of a, it is easy to see that u is a solution of problem (1.1) if and only if u is a fixed point of the integral operator T when \(\lambda=1\). In order to apply Lemma 2.2, the proof includes three steps:
(1) T is a compact map.
Applying the Ascoli–Arzelà theorem, there exists a convergent subsequence of \(\{T(u_{n},\lambda_{n})\}\) in \(C[0,1]\). Without loss of generality, we denote the convergent subsequence again by \(\{ T(u_{n},\lambda_{n})\}\).
From the above, we know that T is a compact operator, which implies that condition (S1) in Lemma 2.2 holds.
(2) Evidently, \(T(u,0)=0\) for \(u\in U\), so condition (S2) is satisfied.
(3) Now, we verify condition (S3) in Lemma 2.2.
We can conclude that \(\{(u_{n},\lambda_{n})\}\) is bounded, which leads to a contradiction. Therefore, condition (S3) in Lemma 2.2 holds.
Applying Lemma 2.2, we can obtain that \(T(u,1)\) has a fixed point in U, that is to say, problem (1.1) has at least one solution. This completes the proof. □
Furthermore, we prove the existence of solutions to problem (1.1) under other innovative conditions.
Theorem 3.2
Proof
- (i)
for any \(\lambda\in[0,1)\), \(u=T(u,\lambda)\) has no solution on \(\partial\Omega_{t}\);
- (ii)
\(\deg(I-T(u,0), \Omega_{t}, 0)\neq0\).
Therefore, upon an application of Leray–Schauder degree method, we obtain that problem (1.1) has at least one solution. This completes the proof. □
4 Example
In the section, we will present the following two examples to illustrate our main results.
Example 4.1
Conclusion
Problem (4.1) has at least one solution.
Proof
Example 4.2
Conclusion
Problem (4.2) has at least one solution.
Proof
5 Conclusions
In this paper, we are concerned with a class of differential equations involving a \(p(r)\)-Laplacian operator. The addressed equation with the multi-point boundary value is quite different from the related references discussed in the literature [26–28, 32]. The nonlinear differential system studied in the present paper is more generalized and more practical. By applying the degree methods (see Lemma 2.4, Theorem 3.2) and the fixed point theorem (see Theorems 3.1, 3.2), we employ innovative arguments, and easily verifiable sufficient conditions have been provided to determine the existence of the solutions to the considered equation. Consequently, this paper shows theoretically that some related references known in the literature can be enriched and complemented.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11471109).
Authors’ contributions
All authors have equally contributed to obtaining new results in this article and also read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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