Research Article | Open Access

# Nonlinear Dynamics, Fixed Points and Coupled Fixed Points in Generalized Gauge Spaces with Applications to a System of Integral Equations

**Academic Editor:**Gabriele Bonanno

#### Abstract

We will discuss discrete dynamics generated by single-valued and multivalued operators in spaces endowed with a generalized metric structure. More precisely, the behavior of the sequence of successive approximations in complete generalized gauge spaces is discussed. In the same setting, the case of multivalued operators is also considered. The coupled fixed points for mappings and are discussed and an application to a system of nonlinear integral equations is given.

#### 1. Introduction

There are several generalizations of the Banach contraction principle. One of the most interesting ones was realized by Perov [1] (see also [2]), by replacing the context of a metric space with that of a space endowed with a vector-valued metric. Some contributions to fixed point theory in complete metric spaces in the sense of Perov can be found in [3–8] (for the case of single-valued operators) and, respectively, in [9–13] (for the case of multivalued operators).

Let be a nonempty set. A mapping is called a vector-valued metric on if the following properties are satisfied:(a) for all ; if , then (where );(b) for all ;(c) for all .

Notice that, if , , and , then, by definition Moreover, we will writeNotice that, through this paper, we will make an identification between row and column vectors in .

A nonempty set endowed with a vector-valued metric is called a generalized metric space in the sense of Perov (in short a generalized metric space) and it will be denoted by . The notions of convergent sequence, Cauchy sequence, completeness, open and closed subset, open and closed ball, and so forth are similar to those for usual metric spaces.

Notice that the generalized metric space in the sense of Perov is a particular case of the cone metric spaces (or -metric space) (see [14, 15]).

On the other hand, in 1959, Marinescu [16] extended the Banach Contraction Principle to locally convex spaces and Colojoară [17] and Gheorghiu [18] did the same for the case of gauge spaces, while Knill [19] considered the framework of an uniform space. In 1971, Cain Jr. and Nashed [20] extended the notion of contraction to Hausdorff locally convex linear spaces. They showed that, on sequentially complete subset, the Banach contraction principle is still valid. In 2000, Frigon [21] introduced the notion of generalized contraction in gauge spaces and proved that every generalized contraction on a complete gauge space (sequentially complete gauge space) has a unique fixed point. For a nice survey on the same subject see also Frigon [22].

*Definition 1. *Let be any set. A mapping is called a pseudometric (or a gauge) in whenever(1), for all ;(2); then ;(3), for all ;(4), for every triple of point.

*Definition 2. *A family of pseudometrics on (or a gauge structure on ), where is a directed set, is said to be separating if for each pair of points , with , there is a such that .

A pair of a nonempty set and a separating gauge structure on is called a gauge space.

It is well known (see [23, pages 198–204]) that any family of pseudometrics on a set induces on a uniform structure and, conversely, any uniform structure on is induced by a family of pseudometrics on . In addition, we have that is separating (or Hausdorff) if and only if is separating. Thus, we may identify gauge spaces and Hausdorff uniform spaces.

If map takes the values in (i.e., and satisfies the axioms of Definition 1), then it is called a vector-valued gauge (or a generalized gauge) on . In this case, the pair (where is a family of separating vector-valued gauges on ) is called a generalized gauge space; see [24]. The properties of the generalized gauge spaces (i.e., the notions of convergent sequences, Cauchy sequences and completeness, open and closed sets, etc.) are similar to those for gauge spaces.

An important concept in the study of different kinds of systems of operatorial equations is that of coupled fixed point. The concept of coupled fixed point for nonlinear operators was considered first by Opoitsev (see [25–27]) and then, in 1987, by Guo and Lakshmikantham (see [28]) in connection with coupled quasisolutions of an initial value problem for ordinary differential equations. A new research direction for the theory of coupled fixed points in ordered metric space was initiated by Bhaskar and Lakshmikantham in [29] and by Lakshmikantham and Ćirić in [30] using contraction type conditions on the operator. For other results on coupled fixed point theory see [29–33] and so forth.

In this paper, we will present fixed point and coupled fixed point theorems for single-valued and multivalued operators in spaces endowed with some generalized metrics. More precisely, the case of complete generalized gauge spaces is discussed. The dynamics of the sequence of successive approximations in each case is considered. An application to a nonlinear system of mixed integral equations is given.

#### 2. Preliminaries

Let us recall first some important preliminary concepts and results.

We denote by the set of all matrices with positive elements, by the identity matrix, and by the zero matrix.

*Definition 3. *A square matrix of real numbers is said to be convergent to zero if * as *; see, for example [34].

A classical result in matrix analysis is the following theorem (see [34, 35]).

Theorem 4. *Let . The following assertions are equivalent:*(i)* is convergent towards zero;*(ii)*its spectral radius is strictly less than 1; that is, , for every with ;*(iii)*the matrix is nonsingular and*(iv)*the matrix is nonsingular and has nonnegative elements;*(v)* and as , for each ;*(vi)*the matrices and converge to , for each , where .*

If is a nonempty set and is an operator, then We recall now Perov’s fixed point theorem (see [1]; see also [2]).

Theorem 5 (Perov). *Let be a complete generalized metric space and the operator with the property that there exists a matrix convergent towards zero such that **Then,*(1)*;*(2)*the sequence of successive approximations , is convergent in to , for all ;*(3)*one has the following estimation:*

Notice that in Precup [8] as well as in [4, 6] are pointed out the advantages of working with vector-valued norm with respect to the usual scalar norms.

There is a vast literature concerning this approach; see also, for example, [3, 5, 7, 8, 36].

Let be a (generalized) gauge space. Then, a sequence of elements in is said to be Cauchy if, for every and , there is an with for all and . The sequence is called convergent if there exists an such that, for every and , there is an with for all . We write as .

*Definition 6. *A (generalized) gauge space is called sequentially complete if any Cauchy sequence is convergent. A subset of is said to be sequentially closed if it contains the limit of any convergent sequence of its elements.

If is a (generalized) gauge space, then is continuous with respect to if, for any sequence which converges (with respect to ) to , we have that the sequence converges (with respect to ) to .

For further details see Dugundji [23] and Granas and Dugundji [37].

We will focus our attention on the following system of operatorial equations:where are two given single-valued operators.

By definition, a solution of the above system is called a coupled fixed point for the operators and Notice that if is an operator and we definethen we get the classical concept of coupled fixed point for operator introduced by Opoitsev and then intensively studied in several papers by many authors.

The case of the coupled fixed point problem in the multivalued setting is defined as follows: find solution of the following system of operatorial inclusions:where are two given multivalued operators.

The concept of coupled fixed point for a multivalued operator is accordingly defined.

#### 3. Fixed Point Theorems in Gauge Spaces

Let us consider first the single-valued case. We will point out first the framework of our study.

If and are two separating (generalized) gauge structures on a set (where and are directed sets), then for (with for every ) and we will denote by the closure of in , whereIn this case, the set is sequentially closed in . If there is just one separating (generalized) gauge structure on , then it is well known that is sequentially closed in .

We can prove the following local fixed point theorems.

Theorem 7. *Let be a nonempty set endowed with two separating generalized gauge structures , (where and are directed sets), (with for every ), and a continuous operator with respect to . One supposes that the following hold.*(i)* is a sequentially complete generalized gauge space.*(ii)*There exist a function and (with for every ) such that*(iii)*There exists a function and with such that, for every , the following implication holds: *(iv)*, for each and every .*(v)*If (with for every ) is given by the expression , then , for each .**Then, has a unique fixed point and the sequence of successive approximations of converges to , for any .*

*Proof. *Notice first that the set is invariant with respect to ; that is, . Indeed, let . Then, there exists a sequence in which converges (with respect to ) to . Since is continuous with respect to , we get that the sequence converges (with respect to ) to . So, we should show now that , for every . Then, using the assumption (v), for each , we haveNow, in a classical manner (see, e.g., Theorem in Novac and Precup [24]), we get that, for any , the sequence is Cauchy in . By assumption (ii), the sequence is also Cauchy in . Notice now that, since is a sequentially complete generalized gauge space, we have that is a sequentially complete generalized gauge space too. Thus, the sequence is convergent (with respect to ) to a certain element . By the continuity of with respect to , we get that . The uniqueness follows from assumptions (iii) and (iv). Indeed, if and are two distinct fixed points of , then, for each , we have Then, by (iv), we get that , for each . Since the family is separating, we obtain that .

In particular, from the above proof, we can obtain the following result.

Theorem 8. *Let be a nonempty set endowed with two separating generalized gauge structures , (where and are directed sets), (with for every ), and a continuous operator with respect to . One supposes that the following hold: *(i)* is a sequentially complete generalized gauge space.*(ii)*There exist a function and (with for every ) such that *(iii)*There exists a function and such that, for every , the following implication holds: *(iv)*, for each .*(v)*If (with for every ) is given by the expression , then , for each .*(vi)*There exists (with ), such that , for every .**Then, has a unique fixed point and the sequence of successive approximations of converges to .*

Notice that if, in particular, the operator in the previous results is a self-one on the whole space (i.e., ), then we obtain some global fixed point theorems on a set with two separating gauge structures.

Let us also remark the following particular cases with respect to the function :

if , then if the matrix converges to zero for each , then assumptions (iv) in Theorems 7 and 8 takes place;

if for some with , then assumptions (iv) in Theorems 7 and 8 take place if we suppose that converges to zero, for each .

*Remark 9. *The above results extend (to the case of nonself-operators on a set endowed with two separating gauge structures) some fixed point theorems given in [3, 18, 22, 24] and so forth.

We illustrate the above remarks by the following consequence of the previous results.

Theorem 10. *Let be a nonempty set endowed with two separating generalized gauge structures , (where and are directed sets) and a continuous operator with respect to . One supposes that the following hold: *(i)* is a sequentially complete generalized gauge space.*(ii)*There exist a function and (with for every ) such that *(iii)*There exists a function and such that, for every , the following implication holds: *(iv)*, for each .*(v)*There exists and (with ), such that , for every .**Then, has a unique fixed point and the sequence of successive approximations of converges to .*

A similar result, given for the cartesian product of two gauge spaces will be useful in applications.

Theorem 11. *Let and be two nonempty sets endowed (resp.) with the separating generalized gauge structures and, respectively, (where and are directed sets) and denote **Let be a continuous operator with respect to the product gauge structures . We suppose that the following hold:*(i)* and are sequentially complete generalized gauge spaces.*(ii)*There exists a function and such that, for every and every , the following condition holds: *(iii)*, for each and .*(iv)*There exists and (with ), such that **Then, there exists a unique such that and the sequence converges to , where , and**for all .**Moreover, for every and , we have the following estimation: *

Let us discuss now the multivalued case. Recall first that Frigon in [38] introduced a concept of multivalued admissible contraction and proved some interesting fixed point and continuation theorems on complete gauge spaces. We will present here some extensions of those results.

Let (with ) be a gauge space and the set of all nonempty subsets of . For , we denote the diameter of the set (for ) by We will also use the following symbols: The gap functional between two sets and from is given by while the (generalized) Hausdorff-Pompeiu functional is defined by The diameter (generalized) functional between two sets and from is given by

If is a multivalued operator, then is called fixed point for if and only if . The set is called the fixed point set of . The symbol is the graph of the operator .

Our first result in this direction is as follows.

Theorem 12. *Let be a nonempty set endowed with two separating gauge structures , (where and are directed sets), , and a multivalued operator having closed graph with respect to . One supposes that the following hold:*(i)* is a sequentially complete gauge space.*(ii)*There exist a function and such that *(iii)*There exists such that, for every , the following implication holds: *(iv)*For every and for every there exists such that *(v)*, for each .**Then, has at least one fixed point and there exists a sequence of successive approximations of starting from which converges to an element .*

*Proof. *By (v), we get that there exists such that , for each . Obviously, . On the other hand, by (iv), there exists such that for every we have ThusChoosing we get thatMoreover, , sinceBy this procedure, we can obtain a sequence having the following properties:(a), for each ;(b), for every ;(c), for each .

By a standard procedure, we obtain that is Cauchy in . Thus, by (ii) sequence is Cauchy in too. Notice now that (i) implies that is a sequentially complete gauge space. Thus, there exists such that as (where the convergence is with respect to ). By (b) and the closed graph hypothesis on , we obtain that . The proof is now complete.

*Remark 13. *The above results extend (to the case of nonself-multivalued operators on a set endowed with two separating gauge structures) some fixed point theorems given by Frigon; see [21, 22].

Notice that if, in particular, , then we obtain a global fixed point theorem on a set with two separating gauge structures.

If, in the previous theorem, we replace (v) with a stronger condition, then we obtain the following result.

Theorem 14. *Let be a nonempty set endowed with two separating gauge structures , (where and are directed sets), , and a multivalued operator having closed graph with respect to . One supposes that the following hold:*(i)* is a sequentially complete gauge space.*(ii)*There exist a function and such that *(iii)*There exists such that, for every , the following implication holds: *(iv)*For every and for every there exists such that *(v)*, for each .**Then, has at least one fixed point and there exists a sequence of successive approximations of starting from any which converges to an element .*

*Proof. *Notice that (v) implies that Indeed, if , then there exists a sequence such that as . Then, for any and for any , we haveThus and so . Now, the rest of the proof follows in a standard manner; see, for example, the proof of Theorem 7 and the proof of Theorem in Frigon [22].

*Remark 15. *It is an open question to prove similar local fixed point results for multivalued operators on a set endowed with two separating generalized gauge structures.

#### 4. Existence, Uniqueness, and Stability for the Coupled Fixed Point Problem in Generalized Gauge Spaces

We consider through this section the coupled fixed point problem in gauge spaces.

Let , be nonempty sets endowed (resp.) with a separating gauge structure and, respectively, (where and are directed sets).

We consider again the following system of operatorial equations:where and are two given single-valued operators. Using the above fixed point results, we can prove the following theorems.

Theorem 16. *Let and be two complete generalized gauge spaces, and let and be two operators. Suppose there exists such that, for each and each , one has**for all (where ).**Suppose that has the property*(i)*, for each and ,* *and the following assumption takes also place:*(ii)*there exists and (with ), such that* *for every and .**Then,*(i)*there exists a unique element such that*(ii)*the sequence converges to as , where* *for all ;*(iii)*we have the following estimation:*

*Proof. *(i)-(ii) Let us define again by Denote and consider ,Then we haveIf we denote , , we get that Applying Theorem 11, we get that there exists a unique element such that and is equivalent with Moreover, for each , we have that as , wherewhereThus, for all , we have that

As an application of the above results, we establish an existence and uniqueness theorem for a system of nonlinear integral equations on the real axis.

Theorem 17. *Consider the following system:**for , where**Suppose that the following assumptions take place:*(i)*the functions and are continuous;*(ii)*there exists and such that * *for every and ;*(iii)*for every the matrix* *(where can be arbitrary chosen) converges to zero;*(iv)*the functions are continuous.**Then, there exists a unique solution of system (55).*

*Proof. *Let endowed with the family of gauges and endowed with the family of gauges where .

We consider the operators and given by whereThen we haveThusSince , we obtain thatFor the second operator, for , we successively have Thus, taking the maximum over , we get