- Research
- Open Access
A note on some fundamental results in complete gauge spaces and application
- Mohamed Jleli^{1},
- Bessem Samet^{1}Email author,
- Calogero Vetro^{2} and
- Francesca Vetro^{3}
https://doi.org/10.1186/s13663-015-0311-8
© Jleli et al.; licensee Springer. 2015
- Received: 3 January 2015
- Accepted: 17 April 2015
- Published: 29 April 2015
Abstract
We discuss the extension of some fundamental results in nonlinear analysis to the setting of gauge spaces. In particular, we establish Ekeland type and Caristi type results under suitable hypotheses for mappings and cyclic mappings. Our theorems generalize and complement some analogous results in the literature, also in the sense of ordered sets and oriented graphs. We apply our results to establishing the existence of solution to a second order nonlinear initial value problem.
Keywords
- gauge structure
- fixed point
- monotone operator
- ordinary differential equation
MSC
- 47H10
- 54A20
- 34L30
1 Introduction
The variational principle established by Ekeland in 1972, see [1, 2], is one of the most discussed and applied result in the context of nonlinear analysis. This principle plays a crucial role in establishing many theoretical results. Here, we refer to the statement below.
Definition 1.1
Let \((X,d)\) be a metric space. A function \(\varphi:X \to[0, + \infty)\) is lower semicontinuous at \(x \in X\) if and only if, for every sequence \(\{x_{n}\}\) in X with \(x_{n} \to x\) as \(n \to+ \infty\), \(\liminf_{n \to+\infty} \varphi(x_{n}) \geq \varphi(x)\). Also, φ is lower semicontinuous if and only if it is lower semicontinuous at every \(x \in X\).
Now, \(L(y):=\{x \in X : \varphi(x) \leq y \}\) is called the lower counter set defined by a point \(y \in[0, + \infty)\). Then the following results hold true.
Proposition 1.1
Let \((X,d)\) be a metric space. Let \(\varphi:X \to[0, + \infty)\) be a function. Then ϕ is lower semicontinuous if and only if \(L(y)\) is closed for every \(y \in[0, + \infty)\).
Theorem 1.1
([2])
- (i)
\(\varphi(x^{*}) \leq\varphi(x_{0})\);
- (ii)
\(d(x_{0}, x^{*}) \leq\delta\);
- (iii)
\(\varphi(x^{*}) < \varphi(x) + c d(x, x^{*})\) for all \(x \neq x^{*}\).
Of course, in the above statement we can, alternatively, consider a function \(\varphi: X \to(-\infty, + \infty]\), by adding the hypothesis that it is bounded from below. The other fundamental theorem that we would like to discuss is the following fixed point result established by Caristi in 1976; see [3].
Theorem 1.2
([3])
Also, f is called a Caristi mapping on \((X, d)\). The above theorems are strongly related each other: it is well known that the results of Ekeland and Caristi are equivalent.
On the other hand, we notice that most of the spaces studied in mathematical analysis, share many algebraic and topological properties as well as metric properties. Consequently, there is no line separating clearly metric theory from the other topological or set-theoretic branches. In view of this fact, many authors considered the problem of establishing theoretic results of nonlinear analysis in a metric space (see, for instance, [4–6]). On the other hand, since several notions and theorems in the literature do not require that all the properties of a metric hold true, various definitions of generalized metrics were introduced (see, for example, [7, 8]). Here we are interested in the so-called gauge spaces that are characterized by the fact that the distance between two points of the space may be zero even if the two points are distinct. For instance, Frigon [9, 10], Chiş and Precup [11] gave generalizations of fixed point theorems and Ekeland’s variational principle on gauge spaces (see also [12–15]). Consistent with this line of research, our aim is to further discuss the above fundamental theorems, by establishing new results under modified conditions in complete gauge spaces. In particular, we deal with ordered sets and oriented graphs. Then, to illustrate the usefulness of our theory, we apply our results to establishing the existence of a solution to a second order nonlinear initial value problem.
2 Preliminaries
We collect some preliminaries on gauge spaces and basic definitions.
Definition 2.1
- (i)
\(d(x,x)=0\) for all \(x\in X\);
- (ii)
\(d(x,y)=d(y,x)\) for all \(x,y \in X\);
- (iii)
\(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z \in X\).
Definition 2.2
Definition 2.3
A family \(\mathcal{F}=\{d_{\lambda}\mid \lambda\in\mathcal{A}\}\) of pseudo-metrics is called separating if for each pair \((x,y)\) with \(x\neq y\), there is a \(d_{\lambda}\in \mathcal{F}\) such that \(d_{\lambda}(x,y)\neq0\).
Definition 2.4
Definition 2.5
- (a)the sequence \(\{x_{n}\}\) converges to x if and only ifIn this case, we denote \(x_{n}\xrightarrow{\mathcal{F}}x\).$$\mbox{for all } \lambda\in\mathcal{A} \mbox{ and } \varepsilon>0, \mbox{there exists } N\in\mathbb{N} \mbox{ such that } d_{\lambda}(x_{n},x)< \varepsilon\mbox{ for all } n\geq N. $$
- (b)The sequence \(\{x_{n}\}\) is Cauchy if and only if$$\mbox{for all } \lambda\in\mathcal{A} \mbox{ and } \varepsilon>0, \mbox{there exists } N\in\mathbb{N} \mbox{ such that } d_{\lambda}(x_{n+p},x_{n})< \varepsilon, \forall n\geq N, p\in\mathbb{N}. $$
- (c)
\((X,\mathcal{T}(\mathcal{F}))\) is complete if and only if any Cauchy sequence in \((X,\mathcal{T}(\mathcal{F}))\) is convergent to an element of X.
- (d)
A subset of X is said to be closed if it contains the limit of any convergent sequence of its elements.
For complete reading on gauge spaces we suggest [16]. Notice that every metric space is a pseudo-metric space. Also, if a pseudo-metric d is not a metric, it is because there are at least two points \(x \neq y\) for which \(d(x,y)=0\). In most situations this does not happen, which means that metrics come up in mathematics more often than pseudo-metrics; however, pseudo-metrics arise in a natural way in functional analysis and in the theory of hyperbolic complex manifolds [17].
Theorem 2.1
([10])
- (i)
\(\varphi_{n}(x^{*}) \leq\varphi_{n}(x_{0})\) for all \(n \in\mathbb{N}\);
- (ii)
\(d_{n}(x_{0}, x^{*}) \leq\delta_{n}\) for all \(n \in\mathbb{N}\);
- (iii)
for all \(x \neq x^{*}\), there exists \(n \in\mathbb{N}\) such that \(\varphi_{n}(x^{*}) < \varphi_{n} (x) + c_{n} d_{n}(x, x^{*})\).
Theorem 2.2
([10])
3 Main results
3.1 Some consequences of Frigon’s theorems
Inspired by the significant work of Frigon [10], we give some consequences of Theorem 2.2.
Theorem 3.1
Proof
Theorem 3.2
- (i)
the function \(h_{n} : X \to[0, +\infty)\) defined by \(h_{n}(x):=d_{n}(x,fx)\) is lower semicontinuous;
- (ii)
the mapping f is continuous;
Proof
As consequences of Theorem 3.2, we give the following results, without proof. For the origin of Theorem 3.4 and different contractive conditions, see [18, 19].
Theorem 3.3
Theorem 3.4
- (i)
the function \(h_{n} : X \to[0, +\infty)\) defined by \(h_{n}(x):=d_{n}(x,fx)\) is lower semicontinuous;
- (ii)
the mapping f is continuous;
Example 3.1
3.2 Results for cyclic mappings
In [20], Kirk et al. obtained extensions of well-known fixed point theorems for cyclic mappings, by considering, for instance, a cyclical contractive condition as given by the next theorem.
Definition 3.1
- (i)
\(X=A \cup B\);
- (ii)
\(f(A)\subseteq B \) and \(f(B)\subseteq A\).
Theorem 3.5
([20])
Inspired by this result, other fixed point theorems with cyclical contractive conditions were obtained (see, for instance, [21–23]). Our aim in this section is to prove some fixed point theorems for cyclic mappings in complete gauge spaces. First, we state the extension of Theorems 3.3 and 3.5 for a cyclic mapping and a complete gauge structure.
Theorem 3.6
Now, we prove the following theorem.
Theorem 3.7
Proof
Example 3.2
The following result uses a nice contractive condition introduced by Geraghty in 1973; see [24].
Theorem 3.8
Proof
4 Ordered sets and oriented graphs
In this section, we adapt the ideas in [25] to get further theorems in complete gauge spaces.
4.1 Fixed points of monotone non-decreasing mappings
Theorem 4.1
Proof
On the other hand, if \(x_{0}\) is a fixed point of f, then \(x_{0}=fx_{0}\) and so the order relation \(fx_{0} \preceq x_{0}\) is trivially satisfied. This completes the proof. □
Remark 4.1
The novelty of the last theorem over the corresponding theorem without ordering is due to the fact that the contractive behavior of f is restricted to the elements \(x \in X\) which are comparable to fx.
4.2 Fixed points of G-edge preserving mappings
Definition 4.1
A graph G is an ordered pair \((V,E)\), where V is a set and \(E \subseteq V \times V\) is a binary relation. We say that V is the vertex set and E is the edge set.
We refer the reader to [26] for a more detailed background on this topic.
Definition 4.2
Let \(G = (V,E)\) be a graph and D be a subset of V. We say that D is G-directed if for every \(x, y \in D\), there exists \(z \in V\) such that \((x, z), (y, z) \in E\).
Example 4.1
Let \((V,d)\) be a metric space. We consider a family \(G = \{G_{i}: 1 \leq i \leq q \}\) of \(q \geq1\) graphs such that \(G_{i} = (V,E_{i} )\), \(E_{i} \subseteq V \times V\), \(i = 1, 2, \ldots,q\).
Definition 4.3
Let \(f : V \to V\) be a given mapping. We say that f is G-monotone if for all \(i = 1, 2, \ldots,q\), we see that \((x, y) \in E_{i}\) implies that \((f x,f y) \in E_{i+1}\), with \(E_{q+1} = E_{1}\). Consequently, \((f^{kq}x,f^{kq}y) \in E_{i}\) for each nonnegative integer number k if f is G-monotone.
Remark 4.2
If \(q= 1\) (\(G =G_{1}\)), we say that f is G-edge preserving; see [27].
Building on Theorem 4.1, we give a further generalization of Caristi type, by substituting the ordering relation with an oriented graph.
Theorem 4.2
Proof
On the other hand, if \(x_{0}\) is a fixed point of f, then \((fx_{0},x_{0}) \in E(G)\), since by hypothesis \((x,x) \in E(G)\) for all \(x \in X\). This completes the proof. □
5 Application to ordinary differential equation
We shall prove the following theorem.
Theorem 5.1
Proof
It is immediate that \(x^{*}\) is a solution of (8) if and only if \(x^{*}\) is a fixed point of f. Then we need to show that Theorem 3.8 is applicable to the operator f to conclude the proof of Theorem 5.1.
Declarations
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project no. IRG14-04.
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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