Some Results on N-Tupled Coincidence and Fixed Points of Graphs on Metric Spaces and an Application to Integral Equations

Abstract

In this work, we establish some N-tupled common coincidence and N-tupled common fixed points for the mappings satisfying a type contractive condition in a complete metric space endowed with a directed graph (for short digraph). Also, we apply our theoretical results to study the existence and uniqueness of solutions for systems of integral equations.

1. Introduction

The theory of fixed point has become very important and useful to solve many mathematical problems in different subjects of sciences such as dynamical programming [1], optimization theory [2], signal processing [3], and iterative process [4], and other times it is used to prove the existence of solution of differential and integral equations [5, 6].

Coupled fixed point was started by Guo and Lakshmikantham [7] in 1987 which is additionally trailed by Bhasker and Lakshmikantham [8] wherein authors gave the notion of mixed monotone property for a generalized contraction mapping (where is an ordered metric space) and utilized the same to introduce some results on existence of coupled fixed points which are seen as coupled definition of specific results of Nieto and Lopez [9]. In 2009, Lakshmikantham and Ciric [10] extended these results for nonlinear contraction mappings by giving the notions of mixed g-monotone property and coupled coincidence point. Recently, the idea of coupled fixed point is reached out to higher dimensions by numerous authors. In fact, in 2010, Samet and Vetro [11] defined the notion of fixed point of n-tupled fixed point, and , as natural extension of coupled fixed point and proved some theorems for n-tupled fixed point in complete metric spaces. In 2013, Imdad et al. [12] extended the notion of n-tupled fixed point for a pair of mappings F and g by defining n-tupled coincidence point and also introduced the general concept of mixed g-monotone property on for merely even n. Also, they employed the same to prove an even-tupled coincidence theorem for nonlinear contractions satisfying mixed g-monotone property. In fact, this result is true for only even n but not for odd ones. Recently, Kir et al. [13] considered the existence of a coupled fixed point of contractive mappings endowed with a digraph and applied the obtained results to the system of two integral equations. There already exists an extensive literature on this topic. Keeping in view the relevance of this paper, we refer to [14–21].

In this paper, we will obtain more general results of Kir et al. [13], which are common N-tupled coincidence points and common N-tupled fixed points of operators satisfying contractive condition in metric spaces endowed with a digraph.

Our paper is organized as follows. In Section 2, some basic definitions are introduced. Section 3 contains some results on N-tupled coincidence points. In Section 4, the existence of a common N-tupled fixed point of operators satisfying contractive conditions in metric spaces endowed with a digraph is considered. Section 5 gives an application to integral equations to support availability of the obtained results.

2. Preliminaries

Through all of this paper let be a complete metric space. We recall the following definition which is given in [22].

Definition 1. An element is said to be an N-tupled fixed point of the function if Therefore, an element is said to be a common N-tupled coincidence point of the functions and if Also, an element is said to be a common N-tupled fixed point of the functions and if is N-tupled fixed point for both and

The following definition of commutative mappings in is the generalization of the definition of Lakshmikntham and Ciric [10].

Definition 2. The two mappings and are said to be commutative if for all

Inspired by the definition of compatible mappings in of Choudhury and Kundu [23], we introduce the following definition.

Definition 3. The mappings and are called compatible if where are sequences in such that where

Definition 4 (see [24]). Let be the set of all functions such that, for any , all following conditions are satisfied:
is lower semi-continuous and (strictly) increasing;
for all ;
for all ;
one can note that if and only if for ;
for denote all functions which satisfy the following:(4A) if ;(4B) if for a sequence .

Now, let be a metric space. Define to be a diagonal of and is digraph which does not have parallel edges, where is the set of vertices of and is the set of edges of

Through all of this paper, let coincide with and Chifu and Petrusel [25] considered the concept of continuous mappings which is extended in by the following definition.

Definition 5. Let and be two mappings. Then (I) is said to be continuous if for all and any positive sequence as and We have (II) (X, d, G) has property if any sequence with as and , then

Consider the set of all N-tupled coincidence points of the mappings and , i.e.,Therefore, is defined asIn 2015, Suantai et al. [26] considered the definition of edge preserving which is extended by the following definition.

Definition 6. The mappings and are said to be edge preserving if implies that

Definition 7 (see [26]). satisfies the transitivity property in if and only if , where

3. An N-Tupled Coincidence Point Theorem

Firstly, we give the following definition which will be used in our main results.

Definition 8. Let be a complete metric space endowed with digraph The mappings and are called contractive if
and are edge preserving,
there exist and such that, for all and , then

We study the existence of N-tupled coincidence point of the mappings and under the following hypotheses:

is continuous and is closed;

, and is compatible;

is continuous;

the tripled has a property

Now, we are ready to introduce our tupled coincidence point theorem.

Theorem 9. Let and be contractive mappings, where is a complete metric space and is the digraph endowed by . If the hypotheses , and hold, then if and only if

Proof. Consider Since , then there exist
such that Also, we have that By doing this process, then we get the sequences , such thatAlso, we have thatNow, suppose that Let . Thus, we have that Hence, we get that Hence, Then, we obtain
Let , then we have that By using (15), we get that Since and are edge preserving, we have that By doing this process, we have Let We get that for all , then we have that By using (26) and because is monotonic function, we obtain is a positive decreasing sequence. Then, as for some Consider . In (26), taking the limit as and from the properties of , we have that We obtain contradiction. Thus and then we get Now, we will show that are Cauchy sequences. Let be not Cauchy sequences. Then there exist , positive integers and we can find such that Therefore, we can find is the smallest integer for which (29) holds, and we have that Using (29) and (30), we get From (27) and (31), we get that Therefore, we obtain that Using the properties of , we get thatAgain from the properties of which is lower semi-continuous, if we take the limit as , we have thatThat is a contradiction. Hence, are Cauchy sequences. Then, there exist such that Since is compatible, we get Since is continuous, we get, for all , Taking the limit as and using (H1) and (H2), we obtain that Thus, we have that Hence, .

Theorem 10. Let and be contractive mappings, where is a complete metric space and is the digraph endowed by . If the hypotheses , and hold, then if and only if

Proof. Let such that Following the same argument as in the proof of Theorem 9, we get sequences such that Thus, we have that Thus, we obtain that which implies that Again using the monotonicity property of , we get By taking the limit as , we obtain that Hence,

4. The Existence and Uniqueness of N-Tupled Fixed Points

Let the set of all common N-tupled fixed points of the mappings and denoted by and defined as

Now, we consider the following fixed point property:

Whenever and are two mappings, then have a unique common N-tupled fixed point.

Theorem 11. Let and be commutative contractive mappings, where is a complete metric space and is the digraph endowed by . If the hypotheses , and hold and for any two elements there exists such that Then if and only if have the fixed point property (FP).

Proof. By doing the same argument as in the proof of Theorem 9, there exists such that Assume there exists another element such that We will show that We can find such that By doing the same process as in (17) and (18), we get the sequences ,, such that and Therefore, we obtain that and we obtain that for all . Then we have that therefore, we get that Similarly, one can get that Using the properties of , we get that We obtain that where The sequence is monotonic decreasing sequence of positive real numbers; then such that Let Take the limit as for (59), This is a contradiction; hence . Therefore, which gives that and according to , then we get that Since for all , then by taking the limit as , we have that Thus, , Hence the uniqueness of N-tupled common coincidence point for and is obtained. Let be the unique N-tupled common coincidence point for and , then it is clear that Since are commutative, we have Hence, is another N-tupled common coincidence point for and . Then, we obtain that To prove the uniqueness of N-tupled fixed point, let be another fixed point. Then there exists such that and we obtain that for all Then we have that Then we have that Therefore, by putting we obtain By doing the same steps above, we can show that there exist such that as If , which is a contraction. Hence it must be . Then we have that Similarly, we can prove Hence, we get

Theorem 12. Let and be commutative contractive mappings, where is a complete metric space and is the digraph endowed by . If the hypotheses , and hold and for any two elements there exist such that , then if and only if have the fixed point property (FP).

5. An application

Now, we apply the N-tupled fixed point theorem to study the existence and uniqueness of the solution of the system of nonlinear integral equations. This system can be written aswhere Define the space is continuous , where, for all , Therefore, and is a complete metric space. Let be a digraph which coincides with and can be obtained by the the partial order relation Thus is endowed with a digraph It is clear that the transitivity property is satisfied on and has property To prove the existence of unique solution of (80), we consider

and are continuous,

there exists a such that

, there exists a continuous mapping such that

there exists such thatfor all

Theorem 13. If conditions (C1), (C2), (C3), and (C4) are satisfied, then system (80) has a unique solution in

Proof. Define , where Define as System (80) can be written as Let such that Then we get that , andHence, are N-G-edge preserving. By using conditions (C2) and (C3), and for all , we get that Take and We get Thus and are -contractive. Therefore by using (C4) there exists such that Hence Hence there exists a unique N-tupled common fixed point of and is the unique solution of system (80).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number G.R.P-53-39.