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A note on some fundamental results in complete gauge spaces and application
Fixed Point Theory and Applications volume 2015, Article number: 62 (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.
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])
Let \((X,d)\) be a complete metric space and \(\varphi: X \to[0, + \infty]\) be a proper and lower semicontinuous function. Then, for all \(c > 0\), \(\delta> 0\), and \(x_{0} \in X\) such that \(\varphi(x_{0}) \leq\inf_{x \in X} \varphi(x) + c \delta\), there exists \(x^{*} \in X\) such that

(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])
Let \((X, d)\) be a complete metric space and \(f :X \to X\) be a mapping not necessarily continuous. Assume that there exists a function \(\varphi:X \to[0, + \infty)\), which is lower semicontinuous, such that
Then f has a fixed point z, that is, \(z = fz\).
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 settheoretic 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 socalled 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.
Preliminaries
We collect some preliminaries on gauge spaces and basic definitions.
Definition 2.1
Let X be a nonempty set. A function \(d:X\times X \rightarrow [0,+\infty)\) is called a pseudometric in X whenever

(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
Let X be a nonempty set endowed with a pseudometric d. The dball of radius \(\varepsilon>0\) centered at \(x\in X\) is the set
Definition 2.3
A family \(\mathcal{F}=\{d_{\lambda}\mid \lambda\in\mathcal{A}\}\) of pseudometrics 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
Let X be a nonempty set and \(\mathcal{F}=\{d_{\lambda}\mid \lambda\in \mathcal{A}\}\) be a family of pseudometrics on X. The topology \(\mathcal{T}(\mathcal{F})\) having as a subbasis the family
of balls is called the topology in X induced by the family \(\mathcal {F}\). The pair \((X,\mathcal{T}(\mathcal{F}))\) is called a gauge space. Notice that \((X,\mathcal{T}(\mathcal{F}))\) is Hausdorff if we require that \(\mathcal{F}\) is separating.
Definition 2.5
Let \((X,\mathcal{T}(\mathcal{F}))\) be a gauge space with respect to the family \(\mathcal{F}=\{d_{\lambda}\mid \lambda\in\mathcal{A}\}\) of pseudometrics on X. Let \(\{x_{n}\}\) be a sequence in X and \(x\in X\). Then

(a)
the sequence \(\{x_{n}\}\) converges to x if and only if
$$\mbox{for all } \lambda\in\mathcal{A} \mbox{ and } \varepsilon>0, \mbox{there exists } N\in\mathbb{N} \mbox{ such that } d_{\lambda}(x_{n},x)< \varepsilon\mbox{ for all } n\geq N. $$In this case, we denote \(x_{n}\xrightarrow{\mathcal{F}}x\).

(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 pseudometric space. Also, if a pseudometric 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 pseudometrics; however, pseudometrics arise in a natural way in functional analysis and in the theory of hyperbolic complex manifolds [17].
Frigon in 2011, see [10], proved useful generalizations of the Ekeland variational principle and Caristi’s fixed point theorem in complete gauge spaces. However, in establishing her results, she does not require that the family \(\mathcal{F}\) is separating, but she uses a gauge structure \(\{d_{n}\mid n \in\mathbb{N} \}\) satisfying the following condition:
Theorem 2.1
([10])
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in\mathbb{N} \}\) satisfying condition (1). For every \(n \in\mathbb{N}\), let \(\varphi_{n} : X \to[0, +\infty]\) be a proper and lower semicontinuous function. Then, for all sequences of positive numbers \(\{c _{n}\}\), \(\{\delta_{n} \} \), and \(x_{0} \in X\) such that \(\varphi_{n}(x_{0}) \leq\inf_{x \in X} \varphi _{n} (x) + c_{n} \delta_{n}\), there exists \(x^{*} \in X\) such that

(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])
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let \(f: X \to X\) be a mapping. For every \(n \in\mathbb{N}\), let \(\varphi_{n} : X \to[0, +\infty)\) be a lower semicontinuous function such that
Then f has a fixed point z, that is, \(z = fz\).
Main results
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
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let \(T,f: X \to X\) be two mappings with T continuous. For every \(n \in\mathbb{N}\), let \(r_{n}\) be a negative real number such that
Then f has a fixed point z, that is, \(z = fz\).
Proof
The continuity of T implies that the function \(\varphi_{n} : X \to[0, +\infty)\) defined by
is lower semicontinuous. From (2), we get
Thus, by Theorem 2.2, f has a fixed point. □
Theorem 3.2
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let \(f: X \to X\) be a mapping. For every \(n \in\mathbb{N}\), let \(k_{n} \in[0,1)\) be such that
If one of the following conditions holds:

(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;
then f has a fixed point in X.
Proof
Note that (ii) implies (i). In fact, let \(x \in X\) and \(\{x_{m} \} \subset X\) such that \(x_{m} \to x\) as \(m \to+ \infty\) and assume that f is continuous. From
we get
Now, we prove that f has a fixed point in X if (i) holds. By (3), we have
This implies that
where \(\varphi_{n}:X \to[0,+\infty)\) is defined by
Now, by (i), the function \(\varphi_{n}\) is lower semicontinuous for all \(n \in\mathbb{N}\). Thus, the existence of a fixed point follows by an application of Theorem 2.2. □
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
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let \(f: X \to X\) be a mapping. For every \(n \in\mathbb{N}\), let \(k_{n} \in[0,1)\) be such that
Then f has a fixed point in X.
Theorem 3.4
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let \(f: X \to X\) be a mapping. For every \(n \in\mathbb{N}\), let \(k_{n} \in[0,1)\) be such that
If one of the following conditions holds:

(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;
then f has a fixed point in X.
Example 3.1
Let \(X=\mathbb{R}\) and, for any \(n \in\mathbb{N}\), define
Clearly, \(\{d_{n}\mid n \in\mathbb{N} \}\) is a complete gauge structure satisfying condition (1). Also, define \(f: X \to X\) by \(fx=\frac{x}{2}\) for all \(x \in X\). Now, for every \(n \in\mathbb{N}\), let \(k_{n} \in[\frac{1}{4},1)\) so that condition (4) is satisfied for all \(x,y \in X\). Therefore, f has a fixed point in X; here 0 is a unique fixed point of f.
Results for cyclic mappings
In [20], Kirk et al. obtained extensions of wellknown fixed point theorems for cyclic mappings, by considering, for instance, a cyclical contractive condition as given by the next theorem.
Definition 3.1
Let A, B be two nonempty subsets of a metric space \((X,d)\). Then \(f:X\rightarrow X\) is called a cyclic mapping associated to \((A,B)\) if the following conditions hold:

(i)
\(X=A \cup B\);

(ii)
\(f(A)\subseteq B \) and \(f(B)\subseteq A\).
Theorem 3.5
([20])
Let A, B be two nonempty closed subsets of a metric space \((X,d)\) and \(f:X\rightarrow X\) be a cyclic mapping associated to \((A,B)\). Let \(k \in(0,1)\) be such that
Then f has a unique fixed point in \(A \cap B\).
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
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let A, B be two nonempty closed subsets of X and \(f: A \cup B \to A \cup B\) be a cyclic mapping associated to \((A,B)\). For every \(n \in\mathbb{N}\), let \(k_{n} \in[0,1)\) be such that
Then f has a fixed point in \(A \cap B\).
Now, we prove the following theorem.
Theorem 3.7
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let A, B be two nonempty closed subsets of X and \(f: A \cup B \to A \cup B\) be a cyclic mapping associated to \((A,B)\). For every \(n \in\mathbb{N}\), let \(\varphi^{1}_{n}:A \to[0,+\infty)\) and \(\varphi^{2}_{n} : B \to[0,+\infty)\) be lower semicontinuous functions such that
and
Then f has a fixed point in \(A \cap B\).
Proof
Let \(x_{1} \in A\) and let \(x_{m+1}=fx_{m}\) for all \(m \in\mathbb{N}\). From (5) and (6) we get
This implies that the sequences \(\{\varphi^{1}_{n}(x_{2m1})\}\) and \(\{ \varphi^{2}_{n}(x_{2m})\}\) are nonincreasing and have the same limit, say \(r \geq0\). Let \(p>m\). Then
Since \(d_{n}(x_{m},x_{m+1}) \to0\) as \(m \to+\infty\), we see that \(\{x_{m} \} \) is a Cauchy sequence and \(A \cap B \neq\emptyset\). Now, we have the following:
Thus,
where \(\varphi_{n} : A \cap B \to[0,+\infty)\) is defined by \(\varphi _{n}(x):=\frac{1}{2}(\varphi^{1}_{n}(x)+ \varphi^{2}_{n}(x))\) for all \(x \in A \cap B\). Clearly, \(\varphi_{n}\) is lower semicontinuous and, hence, the conclusion follows from Theorem 2.2. □
Example 3.2
Let \(A=B=X=[0,+\infty)\) and define
and, for any \(n \in\mathbb{N} \setminus\{1\}\),
Clearly, \(\{d_{n}\mid n \in\mathbb{N} \}\) is a complete gauge structure satisfying condition (1). Also, let \(f:X \to X\) be defined by
It follows that
and
Notice that \(\varphi: X \to X\), defined by
is a lower semicontinuous function such that \(d_{n}(x,fx) \leq\varphi(x)\varphi(fx)\), for all \(x \in X\). Thus, we can apply Theorem 3.7, with \(\varphi^{1}_{n}=\varphi^{2}_{n}=\varphi\), to conclude that f has a fixed point; here 0 and 1 are fixed points of f.
The following result uses a nice contractive condition introduced by Geraghty in 1973; see [24].
Theorem 3.8
Let X be endowed with a complete gauge structure \(\{d_{n}\mid n \in \mathbb{N} \}\) satisfying condition (1). Let A, B be two nonempty closed subsets of X and \(f: A \cup B \to A \cup B\) be a cyclic mapping associated to \((A,B)\). For every \(n \in\mathbb{N}\), let \(\alpha_{n} : [0,+\infty) \to[0,1)\) be such that \(\alpha_{n}(t_{m}) \to1\) implies \(t_{m} \to0\). Assume that, for every \(n \in\mathbb{N}\), the following condition holds:
Then f has a fixed point in \(A \cap B\).
Proof
Let \(x_{1} \in A\) and let \(x_{m+1}=fx_{m}\) for all \(m \in\mathbb{N}\). From (7), we deduce that
and so the sequence \(\{d_{n}(x_{m},x_{m+1})\}\) is nonincreasing and bounded from below. This implies that there exists \(r_{n} \geq0\) such that \(d_{n}(x_{m},x_{m+1}) \to r_{n}\) as \(m \to+\infty\). If \(r_{n} >0\). Then, by (7), we obtain
Letting \(m \to+\infty\), we deduce that \(\alpha_{n}(d_{n}(x_{m},x_{m+1})) \to1\) and so \(d_{n}(x_{m},x_{m+1}) \to0\). To show that the sequence \(\{x_{m} \}\) is Cauchy, we suppose the contrary. Assume that, given \(k \in\mathbb{N}\), there exist \(m >p>k\) such that
From
we get
Letting \(p,m \to+ \infty\), we deduce that \(\alpha _{n}(d_{n}(x_{2p1},x_{2m})) \to1\) and so \(d_{n}(x_{2p1},x_{2m}) \to0\). This implies that the sequence \(\{x_{m} \}\) is Cauchy. Then \(A \cap B \neq\emptyset\) and \(x_{m} \to z \in A \cap B\). Since \(f: A \cap B \to A \cap B\) is continuous, we get \(z=fz\), that is, z is a fixed point of f in \(A \cap B\). □
Ordered sets and oriented graphs
In this section, we adapt the ideas in [25] to get further theorems in complete gauge spaces.
Fixed points of monotone nondecreasing mappings
Let \((X,\preceq)\) be a partially ordered set and \(f:X \to X\) be a mapping. Then f is said to be a monotone nondecreasing mapping if the following condition holds:
Also, two elements \(x,y \in X\) such that \(x \preceq y\) are said to be comparable.
Theorem 4.1
Let \((X, \preceq)\) be a partially ordered set endowed with a complete gauge structure \(\{d_{n}\mid n \in\mathbb{N} \}\) satisfying condition (1). Let \(f: X \to X\) be a continuous and monotone nondecreasing mapping. For every \(n \in\mathbb{N}\), let \(\varphi_{n} : X \to[0, +\infty)\) be a lower semicontinuous function such that
Then f has a fixed point if and only if there exists \(x_{0} \in X\) with \(fx_{0} \preceq x_{0}\).
Proof
Let \(x_{0} \in X\) with \(fx_{0} \preceq x_{0}\) and let \(x_{m}=fx_{m1}\) for all \(m \in\mathbb{N}\). Since f is monotone nondecreasing, then \(x_{m+1} \preceq x_{m}\) for every \(m \in\mathbb{N}\). Therefore, for every \(m,n \in\mathbb{N}\), we have
This implies that the sequence \(\{\varphi_{n}(x_{m})\}\) is nonincreasing and so there exists \(r_{n} \geq0\) such that \(\varphi_{n}(x_{m}) \to r_{n}\) as \(m \to+\infty\). For every \(m,n,p \in\mathbb{N}\), we get
This implies that \(\{x_{m}\}\) is a Cauchy sequence in X. Now, by completeness, there exists \(z \in X\) such that \(x_{m} \to z\) as \(m \to +\infty\). Finally, by continuity of f we conclude that \(fz=z\), that is, z is a fixed point of f.
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.
Fixed points of Gedge 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 Gdirected if for every \(x, y \in D\), there exists \(z \in V\) such that \((x, z), (y, z) \in E\).
Example 4.1
Let \(V = F([0,1],\mathbb{R})\) be the set of functions \(u : [0,1]\to\mathbb{R}\) and define \(E \subseteq V \times V\) by
Then \(G = (V,E)\) is a graph. Let \(D = M([0,1],\mathbb{R})\) be the set of measurable functions \(u : [0,1] \to\mathbb{R}\). Then D is Gdirected. Indeed, for every \(u, v \in D\), the function \(z = \max\{u, v\}\) satisfies \((u, z), (v, z) \in E\).
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 Gmonotone 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 Gmonotone.
Remark 4.2
If \(q= 1\) (\(G =G_{1}\)), we say that f is Gedge 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
Let X be a complete gauge structure \(\{d_{n}\mid n \in\mathbb{N} \}\) satisfying condition (1). Let G be an oriented graph on X such that \((x,x) \in E(G)\) for all \(x \in X\). Let \(f: X \to X\) be a continuous and Gedge preserving mapping. For every \(n \in\mathbb {N}\), let \(\varphi_{n} : X \to[0, +\infty)\) be a lower semicontinuous function such that
Then f has a fixed point if and only if there exists \(x_{0} \in X\) with \((fx_{0}, x_{0}) \in E(G)\).
Proof
Let \(x_{0} \in X\) such that \((fx_{0}, x_{0}) \in E(G)\) and let \(x_{m}=fx_{m1}\) for all \(m \in\mathbb{N}\). Since f is Gedge preserving, by Definition 4.3 and in view of Remark 4.2, we deduce that \((x_{m+1}, x_{m}) \in E(G)\) for every \(m \in\mathbb {N}\). Then, for every \(m,n \in\mathbb{N}\), we get
This implies that the sequence \(\{\varphi_{n}(x_{m})\}\) is nonincreasing and so there exists \(r_{n} \geq0\) such that \(\phi_{n}(x_{m}) \to r_{n}\) as \(m \to+\infty\). For every \(m,n,p \in\mathbb{N}\), we have
This implies that \(\{x_{m}\}\) is a Cauchy sequence in X. Now by completeness, there exists \(z \in X\) such that \(x_{m} \to z\) as \(m \to +\infty\). Finally, by continuity of f we conclude that \(fz=z\), that is, z is a fixed point of f.
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. □
Application to ordinary differential equation
A typical application of fixed point methods is in establishing sufficient conditions for the existence of solution of integrodifferential problems. Referring to [14], we consider the following second order nonlinear initial value problem:
where \(k: [0,+\infty)\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a continuous function. It is well known that the above problem is equivalent to the following integral equation:
Let \(X=C([0,+\infty),\mathbb{R}^{n})\) be the set of continuous functions defined on \([0,+\infty)\). Then, for every \(n \in\mathbb{N}\), we consider the seminorm \(\\cdot\_{n}: X\rightarrow[0,+\infty)\) given by
where \(\cdot\) denote the norm in \(\mathbb{R}^{n}\). Also, for every \(n \in\mathbb{N}\), let
Clearly, \(\mathcal{F}=\{d_{n}\mid n \in\mathbb{N}\}\) is a family of pseudometrics on X satisfying condition (1). Also, \((X,\mathcal{T}(\mathcal{F}))\) is a complete gauge space.
We shall prove the following theorem.
Theorem 5.1
For every \(n \in\mathbb{N}\), assume that the following condition holds:
and \(\gamma: [0,+\infty)\rightarrow[0,+\infty)\) is such that the function \(t\mapsto\int_{0}^{t} (ts) \gamma(s)\,ds\) is bounded on \([0,+\infty)\) and
Then the second order nonlinear initial value problem (8) has a solution \(x^{*}\in C([0,+\infty), \mathbb{R}^{n})\).
Proof
Consider the operator \(f: X\rightarrow X\) defined by
which is well defined, since k is a continuous function.
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.
Let \(n \in\mathbb{N}\) and let \(x,y \in X\), then for all \(t\in[0,n]\) we write
Then, for all \(n \in\mathbb{N}\), we get
which further gives us
Notice that, for every \(n \in\mathbb{N}\), the function \(\alpha_{n} : [0,+\infty) \to[0,1)\) given by
is such that \(\alpha_{n}(t_{m}) \to1\) implies \(t_{m} \to0\), as \(m \to +\infty\). Thus, by an application of Theorem 3.8 with \(A=B=X\), we see that f has a fixed point \(x^{*}\in X\), that is, \(x^{*}\in C([0,+\infty),\mathbb {R}^{n})\) is a solution of (8). □
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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. IRG1404.
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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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C Vetro is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F Vetro is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Jleli, M., Samet, B., Vetro, C. et al. A note on some fundamental results in complete gauge spaces and application. Fixed Point Theory Appl 2015, 62 (2015). https://doi.org/10.1186/s1366301503118
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DOI: https://doi.org/10.1186/s1366301503118
MSC
 47H10
 54A20
 34L30
Keywords
 gauge structure
 fixed point
 monotone operator
 ordinary differential equation