Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 293101, 8 pages

http://dx.doi.org/10.1155/2013/293101

## Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

^{1}Department of Mathematics, Faculty of Sciences of Tunis, University Campus, 2092 Tunis, Tunisia^{2}Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia^{3}Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy

Received 3 May 2013; Accepted 10 August 2013

Academic Editor: Anna Kamińska

Copyright © 2013 Meryam Cherichi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

#### 1. Introduction

In the context of fixed-point theory, the metric fixed-point theory is the branch where metric conditions on the involved spaces and mappings play a crucial role in establishing theoretical results. In a certain sense, this theory is a far-reaching outgrowth of a well-known theorem of Banach [1] which states that if is a complete metric space and if is a mapping which satisfies the condition where is a constant, then has a unique fixed point. Since most of the spaces studied in mathematical analysis share many algebraic and topological properties as well as metric properties, there is no clear line separating the metric fixed-point theory from the topological or the set-theoretic branch of the theory. Consequently, several authors considered the problem of existence (and uniqueness) of a fixed point for generalized contractions in a metric space (see, e.g., [2–9]). On the other hand, many definitions and theorems in the literature do not require that all of the properties of a metric hold true. For this reason, in the last decades, various concepts of generalized metrics were introduced (see, e.g., [10, 11]). 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 [12] and Chiş and Precup [13] gave generalizations of the Banach contraction principle on gauge spaces (see also [14–16]). Consistent with this line of research, the aim of this paper is to present some fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Then, to illustrate the usefulness of our theorems, we apply our results to the study of existence and uniqueness of solutions to a second-order nonlinear initial-value problem.

#### 2. Preliminaries

In this section, we recall some preliminaries on gauge spaces and introduce some basic definitions.

*Definition 1. *Let be a nonempty set. A function is called a pseudometric in whenever the following hold:(i) for all ,(ii) for all ,(iii) for all .

*Definition 2. *Let be a nonempty set endowed with a pseudometric . The -ball of radius centered at is the set

*Definition 3. *A family of pseudometrics is called separating if, for each pair with , there is a such that .

*Definition 4. *Let be a nonempty set, and let be a separating family of pseudometrics on . The topology having for a subbasis the family
of balls is called the topology in induced by the family . The pair is called a gauge space. Note that is Hausdorff because we require to be separating.

*Definition 5. *Let be a gauge space with respect to the family of pseudometrics on . Let be a sequence in , and . Then, the following are considered.(a)The sequence converges to if and only if
In this case, we denote that .(b)The sequence is Cauchy if and only if
(c) is complete if and only if any Cauchy sequence in is convergent to an element of . (d)A subset of is said to be closed if it contains the limit of any convergent sequence of its elements.

For more details on gauge spaces, we refer the reader to [17]. Obviously, every metric space is automatically a pseudometric space. On the contrary, if a pseudometric is not a metric, it is because there are at least two points for which . In most situations, this does not happen; indeed, 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 [18].

#### 3. Main Results

Let denote the set of all functions which satisfy the following conditions. (i) is continuous and nondecreasing.(ii) if and only if .Similarly, let denote the set of all functions which satisfy the following conditions. (i) is lower semicontinuous. (ii) if and only if .

*Definition 6. *Let be a gauge space, and let be a mapping. For , let denote the orbit of ; that is, .

Our first result is the following theorem.

Theorem 7. *Let be a complete gauge space, and let be a mapping satisfying the following conditions.*(i)*For all , for all , for all , and for all .*(ii)*For all and for all , there exist and such that
**Then, has a unique fixed point.*

*Proof. *First, we will remark that
In fact, let , and let such that ; by taking and , we get, by applying assumption (ii) of the theorem, the existence of and such that
Since , we have , and so we get
Using the monotony of , we obtain the desired result (7). Moreover, assumption (i) implies that

Now, let us take , and let be the sequence in given by
Thus, we complete the proof in the following four steps.*Step **1.* We will prove that
Let . Since , by applying (10), we obtain
It follows that is a decreasing sequence of nonnegative real numbers, and, hence, there is such that
Suppose that ; this implies that . Now, taking and , by applying (ii), we get the existence of and such that
Letting in the above inequality, using the continuity of and the lower semicontinuity of , we get
which implies that . This is a contradiction, and, therefore, our claim (12) holds.*Step **2.* We will prove that is a Cauchy sequence in . Suppose that is not a Cauchy sequence. Then, there exists for which we can find two sequences of positive integers and such that, for all positive integers ,
Using (17) and the triangular inequality, we get
Thus, we have
Letting in the above inequality and using (12), we obtain
By the triangular inequality, we have
Letting in the above inequality, using (12) and (20), we get
On the other hand, from (20), there exists a positive integer such that
Applying the hypothesis of the theorem by taking and , we get that there exist and such that
that is,
Letting in the above inequality, from (20), (22), the continuity of , and the lower semicontinuity of , we obtain
which implies that , which leads to the contradiction . Finally, we deduce that is a Cauchy sequence.*Step **3* (existence of fixed point). As is a Cauchy sequence in the complete gauge space , then there exists such that
This implies that . On the other hand, by (10), for all , we get
Now, from
for all , letting , we obtain , for all .

In the virtue of the separating structure of , we deduce that , and, hence, the existence of the fixed point is proved.*Step **4* (uniqueness). Suppose that there exist such that , , and . Then, there exists such that , and so, by using (7), we get a contradiction.

*Remark 8. *Theorem 7 is a generalization of the main result of [6].

By taking in Theorem 7, we get the following corollary.

Corollary 9. *Let be a complete gauge space, and let be a mapping satisfying the following conditions.*(i)*For all , for all , for all , and for all .*(ii)*For all and for all , there exists such that
**Then, has a unique fixed point.*

By taking in Corollary 9, we get the following result.

Corollary 10. *Let be a complete gauge space, and let be a mapping satisfying the following conditions.*(i)*For all , for all , for all , and for all .*(ii)*For all and for all , there exists such that
**Then, has a unique fixed point. *

Next, we give other consequences of Theorem 7.

Corollary 11. *Let be a complete gauge space, and let be a mapping satisfying the following condition.**For all , there exist and such that
**
Then, has a unique fixed point.*

*Proof. *Condition (i) of Theorem 7 is satisfied from inequality (32), and taking and , we get Corollary 11.

By taking in Corollary 11, we get the following result.

Corollary 12. *Let be a complete gauge space, and let be a mapping satisfying the following condition.**For all , there exists such that
**
Then, has a unique fixed point. *

Also, by taking in Corollary 12, we get the following result.

Corollary 13. *Let be a complete gauge space, and let be a mapping satisfying the following condition.**For all , there exists such that
**
Then, has a unique fixed point.*

Finally, we give a result involving a condition of integral type. Let denote the set of all functions which satisfy the following conditions.(c1) is Lebesgue integrable on .(c2) For all , we have .

Corollary 14. *Let be a complete gauge space, and let be a mapping satisfying the following conditions.*(i)*For all , for all , for all , and for all .*(ii)*For all and for all , there exist such that
**Then, has a unique fixed point.*

*Proof. *It follows from Theorem 7 that, by taking

*Remark 15. *Corollary 14 is a generalization of the main result of [19].

#### 4. Results for Cyclic Mappings

In [20], Kirk et al. obtained extensions of the Banach contraction principle for cyclic mappings, by considering a cyclical contractive condition as given by the following theorem.

*Definition 16. *Let be nonempty subsets of a nonempty set . Then, is called a cyclic mapping associated with if the following conditions hold:(i),(ii), where .

Theorem 17. *Let be nonempty closed subsets of a metric space , and suppose that is a cyclic mapping associated with satisfying the following:
**
Then, has a unique fixed point.*

Inspired by this result, other fixed-point theorems with cyclical contractive conditions were obtained (see [21–30]). Our aim in this section is to extend Theorem 7 for cyclic mappings.

Theorem 18. *Let be nonempty closed subsets of a complete gauge space , and suppose that is a cyclic mapping associated with satisfying the following conditions.*(i)*For all , for all , for , and for all , ,
*(ii)*For all and for all , there exist and such that, for ,
**Then, has a unique fixed point.*

*Proof. *As in the proof of Theorem 7, one can prove that
Next, in establishing the existence of a fixed point, since , one only needs to prove that . In fact, applying Theorem 7 to the restriction of on the complete gauge space , we obtain that has a fixed point in .

Now, let , and we construct a sequence in such that
Remark that, for all , there exists such that . Then, as in the proof of Theorem 7, we obtain that
We will prove that is a Cauchy sequence in the complete gauge space . Suppose that is not a Cauchy sequence. Then, there exists for which we can find two sequences of positive integers and such that, for all positive integers ,
Using (44) and the triangular inequality, we get
Thus, we have
Letting in the above inequality and using (43), we obtain
Let be a positive integer, and let be the integer such that
Using the triangular inequality, we have
Thus, we have
Letting in the above inequality, using (43) and (47), we get that
Using the same technique, we obtain that
On the other hand, from (52), there exists a positive integer such that
Applying the hypothesis of the theorem by taking and , we get the existence of and such that
that is,
Letting in the above inequality, using (51), (52), the continuity of , and the lower semicontinuity of , we obtain
which implies that , which leads to the contradiction . Finally, we deduce that is a Cauchy sequence in the complete gauge space ; therefore, converges to some . In view of condition (ii) of Definition 16, an infinite number of elements of the sequence lie in each , . As is a closed subset, we deduce that , for . Thus, , and so .

In order to complete the proof, we have to show that uniqueness of the fixed point.

From condition (ii) of Definition 16, any fixed point of lies necessarily in . Suppose that there exist such that , , and . Then, there exists such that , and so, by using (40), we get a contradiction.

*Remark 19. *Theorem 18 is a generalization of the main result of [24].

By taking in Theorem 18, we get the following corollary.

Corollary 20. *Let be nonempty closed subsets of a complete gauge space , and suppose that is a cyclic mapping associated with satisfying the following conditions.*(i)*For all , for all for , and for all ,
*(ii)*For all and for all , there exists such that, for ,
**Then, has a unique fixed point.*

The analogous of Corollaries 10–14 for cyclic mappings can be obtained easily, and so, to avoid repetitions, we omit the details.

#### 5. Application to Ordinary Differential Equation

In this section, we present an example where our obtained results can be applied. Precisely, we study the existence of solution for the following second-order nonlinear initial-value problem: where is a continuous function. This problem is equivalent to the integral equation Denote by the set of continuous functions defined on . For each positive integer , we define the function by This function is a seminorm on . Also, define Then, is a separating family of pseudometrics on , and is a complete gauge space.

We will prove the following result.

Theorem 21. *Suppose that, for all , for each , and for all , one has
**
where is such that the function is bounded on and
**Then, (59) has a unique solution . *

*Proof. *Consider the operator given by
Clearly, is well defined since is a continuous function.

Now, is a solution of (59) if and only if is a fixed point of . Next, we will show that the two conditions of Corollary 9 hold true.*Condition (i).* Let be a positive integer such that , and let such that . This means that
Let us take ; then, we have
Hence,
Thus, the condition is satisfied.*Condition (ii).* Let such that , and let such that . For all , , we have
Then, for all , we have
Hence, for all , for all , and for all such that , we have
where and, obviously, . Finally, applying Corollary 9, we obtain that has a unique fixed point ; that is, is the unique solution of (59).

#### Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for the funding of this research through the Research Group Project no. RGP-VPP-237.

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