New generalizations of Darbo’s fixed point theorem
- Longsheng Cai^{1} and
- Jin Liang^{1}Email author
https://doi.org/10.1186/s13663-015-0406-2
© Cai and Liang 2015
Received: 17 May 2015
Accepted: 13 August 2015
Published: 28 August 2015
Abstract
In this paper, we propose a new idea to investigate the important basic problem: how to extend Darbo’s fixed point theorem? We establish some new generalizations of Darbo’s fixed point theorem by using ‘integral conditions’. Our fixed point theorems extend the existing results on the problem above.
1 Introduction
It is well known that the Schauder fixed point theorem plays an important role in nonlinear analysis. In 1955, Darbo [1] proved a fixed point property for α-set contraction on a closed, bounded and convex subset of Banach spaces in terms of the measure of noncompactness, which was first defined by Kuratowski [2]. Darbo’s fixed point theorem is a significant extension of the Schauder fixed point theorem, and it also plays a key role in nonlinear analysis especially in proving the existence of solutions for a lot of classes of nonlinear equations. Since then, some generalizations of Darbo’s fixed point theorem have appeared. For example, we refer the reader to [3–6] and the references therein.
In this paper, we propose a new idea to investigate the important basic problem: how to extend Darbo’s fixed point theorem? We establish some new generalizations of Darbo’s fixed point theorem by using ‘integral conditions’. Our fixed point theorems extend the existing results on the problem above.
First of all, let us recall some basic concepts, notations and known results which will be used in the sequel. In this paper, we let E be a Banach space with the norm \(\| \cdot\|\) and 0 be the zero element in E. The closed ball centered at x with radius r is denoted by \(B(x,r)\), by simply \(B_{r}\) if \(x=0\). If X is a nonempty subset of E, then we denote by X̄ and \(\operatorname{co}(X)\) the closure and closed convex hull of X, respectively. Moreover, let \(\mathfrak{M}_{E}\) be the family of all nonempty compact subsets of E.
We use the following definition of the measure of noncompactness, which is in the form of an axiomatic way.
Definition 1.1
- (1)the subfamilyis nonempty and \(\operatorname{ker}\mu\subset\mathfrak{M}_{E}\);$$\operatorname{ker}\mu=\bigl\{ X\in\mathfrak{M}_{E}:\mu(X)=0\bigr\} $$
- (2)
if \(X\subset Y\), then \(\mu(X)\leq\mu(Y)\);
- (3)
\(\mu(\bar{X})=\mu(X)\);
- (4)
\(\mu(\operatorname{co}(X))=\mu(X)\);
- (5)
\(\mu(\lambda X+(1-\lambda)Y)\leq\lambda\mu(X)+(1-\lambda )\mu(Y)\) for \(\lambda\in[0,1]\);
- (6)
if \(\{X_{n}\}\) is a nested sequence of closed sets in \(\mathfrak{M}_{E}\) and \(\lim_{n\to\infty}\mu(X_{n})=0\), then the intersection set \(X_{\infty}=\bigcap^{\infty}_{n=1}X_{n}\) is nonempty.
Definition 1.2
- (1)
For \(u,v\in[0,+\infty)\) if \(\Psi(u)\leq\Phi(v)\), then \(u\leq v\).
- (2)For \(u_{n},v_{n}\in[0,+\infty)\) withif \(\Psi(u_{n})\leq\Phi(v_{n})\) for all n, then \(w=0\).$$\lim_{n\to\infty}u_{n}=\lim_{n\to\infty}v_{n}=w, $$
Example 1.3
Theorem 1.4
(Schauder’s fixed point theorem)
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E. Then each continuous and compact map \(F:\Omega\to\Omega\) has at least one fixed point in Ω.
As a generalization of the Schauder fixed point theorem, we have the following fixed point theorem.
Theorem 1.5
(Darbo’s fixed point theorem)
2 Main results
In this section, we present and prove our new generalizations of Darbo’s fixed point theorem.
Theorem 2.1
Proof
Now since \(\Omega_{n}\) is a nested sequence, in view of (6) of Definition 1.1, we conclude that \(\Omega_{\infty}=\bigcap^{\infty}_{n=1}\Omega_{n}\) is a nonempty, closed and convex subset of Ω. Moreover, we know that \(\Omega_{\infty}\) belongs to kerμ. So \(\Omega_{\infty}\) is compact and invariant under the mapping T. Consequently, Theorem 1.4 implies that T has a fixed point in \(\Omega_{\infty}\). Since \(\Omega_{\infty}\subset\Omega\), the proof is completed then. □
Remark 2.2
Remark 2.3
Remark 2.4
Remark 2.5
It is easy to verify that Ψ and Φ constructed above are a pair of shifting distance functions.
Corollary 2.6
Proof
Now, motivated by the contractive condition in Theorem 3.1 of [7], we present another generalization of Darbo’s fixed point theorem as follows.
Theorem 2.7
Proof
Remark 2.8
When \(\varphi(t)=1\), we get Theorem 2.1 of [4].
Theorem 2.9
- (1)
\(\Psi(s)\leq\Phi(t)\Rightarrow s\leq t\);
- (2)
\(\theta(t)=0\Leftrightarrow t=0\) and \(\theta\geq0\);
- (3)for any sequence \(\{s_{n}\}\) in \(R_{+}\) with \(s_{n}\to t>0\),$$\Psi(t)-\lim_{n\to\infty}\sup\Phi(s_{n})+\lim _{n\to\infty}\inf\theta(s_{n})>0. $$
Proof
Remark 2.10
When \(\varphi(t)=1\), we get Theorem 9 of [6].
Remark 2.11
Let \(\Psi(t)=t\) and \(\theta(t)=0\) in the above theorem, we have the following corollary.
Corollary 2.12
Declarations
Acknowledgements
The authors acknowledge support from NSFC (No. 11171210).
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
References
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