Abstract and Applied Analysis

Volume 2014 (2014), Article ID 568718, 18 pages

http://dx.doi.org/10.1155/2014/568718

Research Article

Coupled and Tripled Coincidence Point Results with Application to Fredholm Integral Equations

Marwan Amin Kutbi,¹ Nawab Hussain,¹ Jamal Rezaei Roshan,² and Vahid Parvaneh³

¹Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

³Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

Received 25 January 2014; Accepted 29 March 2014; Published 18 May 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 Marwan Amin Kutbi 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.

The aim of this paper is to define weak ---contractive mappings and to establish coupled and tripled coincidence point theorems for such mappings defined on -metric spaces using the concept of rectangular --admissibility. As an application, we derive new coupled and tripled coincidence point results for weak --contractive mappings in partially ordered -metric spaces. Our results are generalizations and extensions of some recent results in the literature. We also present an example as well as an application to nonlinear Fredholm integral equations in order to illustrate the effectiveness of our results.

1. Introduction and Mathematical Preliminaries

The concept of generalized metric space, or a -metric space, was introduced by Mustafa and Sims.

Definition 1 (-metric space [1]). Let be a nonempty set and let be a function satisfying the following properties:

(G1) if and only if ;(G2), for all with ;(G3), for all with ;(G4) (symmetry in all three variables);(G5), for all (rectangle inequality).

Then, the function is called a -metric on and the pair is called a -metric space.

Recently, Aghajani et al. in [2] motivated by the concept of -metric [3] introduced the concept of generalized -metric spaces (-metric spaces) and then they presented some basic properties of -metric spaces.

The following is their definition of -metric spaces.

Definition 2 (see [2]). Let be a nonempty set and let be a given real number. Suppose that a mapping satisfies the following:

(G b1) if ,(G b2) for all with ,(G b3) for all with ,(G b4), where is a permutation of (symmetry),(G b5) for all (rectangle inequality).

Then is called a generalized -metric and the pair is called a generalized -metric space or a -metric space.

Each -metric space is a -metric space with .

Example 3 (see [2]). Let be a -metric space and , where is a real number. Then is a -metric with .

Example 4 (see [4]). Let and . We know that is a -metric space with . Let , it is easy to see that is not a -metric space. Indeed, ( is not true for , , and . However, is a -metric on with .

Definition 5 (see [2]). A -metric is said to be symmetric if , for all .

Proposition 6 (see [2]). Let be a -metric space. Then for each it follows that

(1)if  , then ,(2), (3), (4).

Definition 7 (see [2]). Let be a -metric space. One defines , for all . It is easy to see that defines a -metric on , which one calls the -metric associated with .

Definition 8 (see [2]). Let be a -metric space. A sequence in is said to be

(1)-Cauchy if, for each , there exists a positive integer such that, for all , ;(2)-convergent to a point if, for each , there exists a positive integer such that, for all , .

Proposition 9 (see [2]). Let be a -metric space. Then the following are equivalent:

(1)the sequence is -Cauchy,(2)for any there exists such that for all .

Proposition 10 (see [2]). Let be a -metric space. The following are equivalent.

(1)is -convergent to . (2), as .(3), as .

Definition 11 (see [2]). A -metric space is called -complete if every -Cauchy sequence is -convergent in .

Proposition 12. Let and be two -metric spaces. Then a function is -continuous at a point if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Proposition 13. Let be a -metric space. A mapping is said to be continuous if, for any two -convergent sequences and converging to and , respectively, is -convergent to .

In general, a -metric function for is not jointly continuous in all its variables. The following is an example of a discontinuous -metric.

Example 14 (see [4]). Let and let be defined by

Then it is easy to see that, for all , we have

Thus, is a -metric space with (see [5]).

Let . It is easy to see that is a -metric with which is not a continuous function.

We will need the following simple lemma about the -convergent sequences in the proof of our main results.

Lemma 15 (see [4]). Let be a -metric space with and suppose that ,, and are -convergent to , , and , respectively. Then one has

In particular, if , then we have .

The existence of fixed points, coupled fixed points, and tripled fixed points for contractive type mappings in partially ordered metric spaces has been considered recently by several authors (see [6–28], etc.)

Lakshmikantham and Ćirić [17] introduced the notions of mixed -monotone mapping and coupled coincidence point and proved some coupled coincidence point and common coupled fixed point theorems in partially ordered complete metric spaces.

Definition 16 (see [17]). Let be a partially ordered set and let and be two mappings. has the mixed -monotone property, if is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument; that is, for all , implies for any and for all , implies for any .

Definition 17 (see [7, 17]). An element is called

(1)a coupled fixed point of mapping if and ,(2)a coupled coincidence point of mappings and if and ,(3)a common coupled fixed point of mappings and if and .

Definition 18 (see [17]). Let be a nonempty set. We say that the mappings and are commutative if for all .

Choudhury and Maity [10] have established some coupled fixed point results for mappings with mixed monotone property in partially ordered -metric spaces. They obtained the following results.

Theorem 19 (see [10, Theorem 3.1]). Let be a partially ordered set and let be a -metric on such that is a complete -metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists such that

for all and , where either or .

If there exist such that and , then has a coupled fixed point in ; that is, there exist such that and .

Theorem 20 (see [10, Theorem 3.2]). If, in the above theorem, in place of the continuity of , one assumes the following conditions, namely,

(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all ,

then has a coupled fixed point.

Definition 21 (see [29]). Let be a partially ordered set and let be a -metric on . One says that is regular if the following conditions hold.

(i)If is a nondecreasing sequence with , then for all .(ii)If is a nonincreasing sequence with , then for all .

Definition 22 (see [10]). Let be a generalized -metric space. Mappings and are called compatible if

hold whenever and are sequences in such that

On the other hand, Berinde and Borcut [25] introduced the concept of tripled fixed point and obtained some tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. For a survey of tripled fixed point theorems and related topics we refer the reader to [25–28, 30].

Definition 23 (see [25, 26]). Let be a partially ordered set, , and .

(1)An element is called a tripled fixed point of if , , and .(2)An element is called a tripled coincidence point of the mappings and if , , and .(3)An element is called a tripled common fixed point of and if , , and .(4)One says that has the mixed -monotone property if is -nondecreasing in , -nonincreasing in , and -nondecreasing in ; that is, if, for any ,

Definition 24 (see [28]). Let be a nonempty set. One says that the mappings and commute if , for all .

In [26], Borcut obtained the following.

Theorem 25 (see [26, Corollary 1]). Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let and be such that has the -mixed monotone property. Assume that there exists such that

for all with , , and . Suppose and is continuous and commutes with and also suppose either

(a) is continuous, or(b) has the following properties:

(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .

If there exist such that , , and , then and have a tripled coincidence point.

Definition 26. Let be a generalized -metric space. Mappings and are called compatible if

hold whenever ,, and are sequences in such that

Let satisfies the following:

(i) is continuous and nondecreasing,(ii) if and only if .

That is, is an altering distance function.

In this paper, we obtain some coupled and tripled coincidence point theorems for nonlinear weakly contractive mappings which are --admissible with respect to another function in partially ordered -metric spaces. These results generalize and modify several comparable results in the literature.

2. Main Results

Samet et al. [31] defined the notion of -admissible mapping as follows.

Definition 27. Let be a self-mapping on and let be a function. One says that is an -admissible mapping if

Definition 28 (see [32]). Let be a -metric space, let be a self-mapping on , and let be a function. One says that is an --admissible mapping if

Following the recent work in [33–35] we present the following definition in the setting of -metric spaces.

Definition 29. Let be a -metric space and let and . One says that is a rectangular --admissible mapping with respect to if

(R1), ,(R2)implies .

Lemma 30. Let be a rectangular --admissible mapping with respect to such that . Assume that there exists such that . Define sequence by . Then

Now, we prove the following coincidence point result.

Theorem 31. Let be a generalized -metric space and let satisfy the following condition:

for all , where are two altering distance mappings, , and is a rectangular --admissible mapping with respect to .

Then, maps and have a coincidence point if

(i),(ii)there exists such that ,(iii) and are continuous and compatible and is complete,(iii′)one of or is complete and assume that whenever in is a sequence such that for all and as , we have for all .

Proof. Let be such that . According to (i) one can define the sequence as for all .

As and since is an --admissible mapping with respect to , then . Continuing this process, we get for all .

If , then is a coincidence point of and .

Now, assume that for all ; that is,

for all . Let . Then, from (14) we obtain that

We prove that for each . If for some , then from (16) we have which implies that , a contradiction to (15).

Hence, we have for each . Thus, the sequence is nonincreasing and so there exists such that .

Suppose that . Then from (16), taking the limit as implies that

a contradiction. Hence,

Since for every , so by property (G b3) we obtain

Hence,

Also, by part (3) of Proposition 6 we have

Now, we prove that is a -Cauchy sequence. Assume on contrary that is not a -Cauchy sequence. Then there exists for which we can find subsequences and of such that is the smallest index for which and

This means that

Since is a rectangular --admissible mapping with respect to , then from Lemma 30  . Now, from (14) we have

Using (G b5) we obtain that

Taking the upper limit as and using (20) and (23) we obtain that

Using () we obtain that

Taking the upper limit as and using (20) and (23) we obtain that

Taking the upper limit as in (24) and using (23) and (26) we obtain that

which implies that

so , a contradiction to (28). It follows that is a -Cauchy sequence in .

Suppose first that (iii) holds. Then there exists

Further, since and are continuous and compatible, we get that

We will show that . Indeed, we have

and it follows that . It means that and have a coincidence point.

In the case (iii′), if we assume that is -complete, then

for some . Also, from (iii′) we have . Applying (14) with and , we have

It follows that when ; that is, . Uniqueness of the limit yields that . Hence, and have a coincidence point .

Theorem 32. Let be an ordered generalized -metric space and let satisfy the following condition:

for all , with , where are two altering distance functions.

Then, maps and have a coincidence point if

(i) is -nondecreasing with respect to and ;(ii)there exists such that ;(iii) and are continuous and compatible and is complete, or(iii′) is regular and one of or is complete.

Proof. Define by

First, we prove that is a rectangular --admissible mapping with respect to . Assume that . Therefore, we have . Since is -nondecreasing with respect to , we get ; that is, . Also, let and , and then and . Consequently, we deduce that ; that is, . Thus, is a rectangular --admissible mapping with respect to . Since

for all , with , then

Moreover, from (ii) there exists such that ; that is, . Hence, all the conditions of Theorem 31 are satisfied and therefore and have a coincidence point.

If for all in Theorem 31, then we obtain the following coincidence point result.

Theorem 33. Let be a generalized -metric space and let satisfy the following condition:

for all , where are two altering distance functions.

Then, maps and have a coincidence point if

(i),(ii) and are continuous and compatible and is complete, or(ii′) one of or is complete.

3. Coupled Fixed Point Results

We will use the following simple lemma in proving our next results. A similar case in the context of -metric spaces can be found in [24].

Lemma 34. Let be a generalized -metric space (with the parameter ) and let and . Suppose that is given by

and is defined by

(a)If a mapping is given by

then is a generalized -metric space (with the same parameter ). The space is -complete if and only if is -complete.(b)If is continuous from to , then is continuous in .(c)If and are compatible, then and are compatible.(d)The mapping is --admissible with respect to ; that is,

if and only if the mapping is --admissible with respect to ; that is,

where is a function.(e)The statement (d) holds if we replace the --admissibility by rectangular --admissibility.

Let be a generalized -metric space, , and . In the rest of this paper unless otherwise stated, for all , let

Now, we have the following coupled coincidence point result.

Theorem 35. Let be a generalized -metric space with the parameter and let and . Assume that

for all , where are altering distance functions, , and is a rectangular --admissible mapping with respect to .

Assume also that

(1);(2)there exist such that

Also, suppose that either

(a) and are continuous, the pair is compatible, and is -complete, or(b) is -complete and assume that whenever and in are sequences such that

for all and , as , we have

for all .

Then, and have a coupled coincidence point in .

Proof. Let be the generalized -metric on defined in Lemma 34. Also, define the mappings by and , as in Lemma 34. Then, is a generalized -metric space (with the same parameter as ), such that . Moreover, the contractive condition (47) implies that

holds for all . Also, one can show that all conditions of Theorem 31 are satisfied for and and we have proved in Theorem 31 that, under these conditions, it follows that and have a coincidence point which is obviously a coupled coincidence point of and .

In the following theorem, we give a sufficient condition for the uniqueness of the common coupled fixed point (see also [23]).

Theorem 36. In addition to the hypotheses of Theorem 35, suppose that and are commutative and that, for all and , there exists , such that and . Then, and have a unique common coupled fixed point of the form .

Proof. We will use the notations as in the proof of Theorem 35. It was proved in this theorem that the set of coupled coincidence points of and ; that is, the set of coincidence points of and in is nonempty. We will show that if and are coincidence points of and , that is,

then .

Choose an element such that and . Let and choose so that . Then, we can inductively define a sequence such that .

As , ,  and is rectangular --admissible with respect to , then ; that is, yields that

Therefore, by the mathematical induction, we obtain that , for all .

Applying (47), one obtains that

From the properties of , we deduce that the sequence is nonincreasing. Hence, if we proceed as in Theorem 31, we can show that

that is, is -convergent to .

Similarly, we can show that is -convergent to . Since the limit is unique, it follows that .

The compatibility of and yields that and are compatible, and hence and are weak compatible. Since , we have . Let . Then, . Thus, is another coincidence point of and . Then, . Therefore, is a coupled common fixed point of and .

To prove the uniqueness, assume that is another common fixed point of and . Then, and also . Thus, . Hence, the coupled common fixed point is unique. Also, if is a common coupled fixed point of and , then is also a common coupled fixed point of and . Uniqueness of the common coupled fixed point yields that .

Let be given by

and then is a generalized -metric space (with the same parameter ).

Let be a generalized -metric space, , and . For all , let

Remark 37. The result of Theorems 35 and 36 holds, if we replace , , and by , , and , respectively.

4. Coupled Fixed Point Results in Partially Ordered Generalized -Metric Spaces

We will use the following simple lemma in proving our results.

Lemma 38. Let be an ordered generalized -metric space (with the parameter ) and let and . Let be given by

and is defined by

(a) If a relation is defined on by

then and are ordered generalized -metric spaces (with the same parameter ).

(b) If the mapping has the -mixed monotone property, then the mapping is -nondecreasing with respect to ; that is,

Theorem 39. Let be a partially ordered generalized -metric space with the parameter and let and . Assume that

for all with and , where are altering distance functions.

Assume also that

(1) has the mixed -monotone property and ;(2)there exist such that and .

Also, suppose that either

(a) and are continuous, the pair is compatible, and is -complete, or(b) is regular and is -complete.

Then, and have a coupled coincidence point in .

Proof. By Lemma 38, is an ordered generalized -metric space (with the same parameter ).

Define by

First, we prove that is a rectangular --admissible mapping with respect to . Hence, we assume that , where ,, and . Therefore, we have . Since has the mixed -monotone property, then from Lemma 38, the mapping is -nondecreasing with respect to ; that is,

that is, . Also, let and ; then and . Consequently, we deduce that ; that is, . Thus, is a rectangular --admissible mapping with respect to .

From (62) and the definition of and ,

for all with . Moreover, from (2) there exists such that

Hence, all the conditions of Theorem 32 are satisfied and so and have a coincidence point which is a coupled coincidence point of and .

In the following theorem, we give a sufficient condition for the uniqueness of the common coupled fixed point (see also [25, 28, 30]).

Theorem 40. In addition to the hypotheses of Theorem 39, suppose that, for all and , there exists , such that is comparable with and . Then, and have a unique common coupled fixed point of the form .

Proof. It was proved in Theorem 39 that the set of coupled coincidence points of and , that is, the set of coincidence points of and in , is nonempty. We will show that if and are coincidence points of and , that is,

then . There exists , such that is comparable with and . Without any loss of generality, we may assume that and . According to the definition of in the above theorem, and .

Now, following the proof of Theorem 36, one can obtain that . The remainder part of proof is analogous to the proof of Theorem 36 and so we omit it.

Remark 41. In Theorem 39, we can replace the contractive condition (62) by the following:

Remark 42. Theorem 39 provides conclusions of Theorems 3.1 and 3.2 of [4] for more general pair of compatible maps.

In Theorem 39, if we take for all , we obtain the following result.

Corollary 43. Let be a partially ordered -complete generalized -metric space with the parameter and let . Assume that

for all with and , where is an altering distance function.

Assume also that

(1) has the mixed -monotone property and ;(2)there exist such that and .

Also, suppose that either

(a) is continuous, or(b) is regular.

Then, and have a coupled coincidence point in .

In addition, suppose that, for all and , there exists , such that is comparable with and . Then,andhave a unique coupled fixed point of the form .

In Corollary 43, if we take for all , where , we obtain the following extension of the results by Choudhury and Maity (Theorems 19 and 20).

Corollary 44. Let be a partially ordered -complete generalized -metric space with the parameter and let . Assume that

for all with and , or and , where .

Assume also that

(1) has the mixed monotone property;(2)there exist such that and .

Also, suppose that either

(a) is continuous, or(b) is regular.

Then, has a coupled fixed point in .

In addition, suppose that, for all and , there exists , such that is comparable with and . Then, has a unique coupled fixed point of the form .

Now, we present an example to illustrate Theorem 39 and Remark 41.

Example 45. Let be endowed with the usual ordering and let -metric on be given by , where . Define as

We define by

where . Also, has mixed monotone property and satisfies the condition (68). Indeed, for all with and , we have

Similarly,

So,

Finally, there are obviously such that and . Thus, we conclude that the mapping has a coupled fixed point (which is ).

Consider now the same example, but with the -metric

on . The respective contractive condition

does not hold for all such that and . Indeed, for , it reduces to

We conclude that, using a -metric instead of the standard one, one has more possibilities for choosing a control function in order to get a coupled fixed point result.

Remark 46. Theorems 3.5, 3.6, and 3.7 of [4] are special cases of Theorem 39.

5. Tripled Coincidence Point Results

In this section we prove some tripled coincidence and tripled common fixed point results.

Lemma 47. Let be an ordered generalized -metric space (with the parameter ) and let and .

(a) If a relation is defined on by

and a mapping is given by

for all , , and , then is an ordered generalized -metric space (with the same parameter ). The space is -complete if and only if is -complete.

(b) If the mapping has the -mixed monotone property, then the mapping given by

is -nondecreasing with respect to ; that is,

where is defined by

(c) If is continuous from to , then is continuous in .

(d) If and are compatible, then and are compatible.

(e) The mapping is --admissible with respect to ; that is,

if and only if the mapping is --admissible with respect to ; that is,

where is a function.

Let be an ordered generalized -metric space, , and . For all , let

Theorem 48. Let be a generalized -metric space with the parameter and let and . Assume that

for all where are altering distance functions and is a mapping such that is a rectangular --admissible mapping with respect to .

Assume also that

(1);(2)there exist such that

Also, suppose that either

(a) and are continuous, the pair is compatible, and is -complete, or(b) is -complete and assume that whenever ,, in are sequences such that

for all and , , and as , we have

for all .

Then, and have a tripled coincidence point in .

Theorem 49. In addition to the hypotheses of Theorem 48, suppose that and are commutative and that, for all and , there exists , such that

Then, and have a unique common tripled fixed point of the form .

Let

Remark 50. In Theorem 48, we can replace the contractive condition (87) by the following:

The following tripled fixed point results in ordered metric spaces can be obtained.

Theorem 51. Let be a partially ordered generalized -metric space with the parameter and let and . Assume that

for all with , , and , where are altering distance functions.

Assume also that

(1) has the mixed -monotone property and ;(2)there exist such that , , and .

Also, suppose that either

(a) and are continuous, the pair is compatible, and is -complete, or(b) is regular and is -complete.

Then, and have a tripled coincidence point in .

Theorem 52. In addition to the hypotheses of Theorem 48, suppose that and are commutative and that, for all and , there exists , such that is comparable with and . Then, and have a unique common tripled fixed point of the form .

Remark 53. In Theorem 51, we can replace the contractive condition (94) by the following:

Remark 54. Theorem 51 extends Theorem 2.1 of [36] to a compatible pair.

Remark 55. Corollaries 2.2, 2.3, 2.6, 2.7,  and 2.8 of [36] are special cases of Theorem 51.

Remark 56. Theorem 25 is a special case of Theorems 51.

6. Application to Integral Equations

As an application of the (coupled) fixed point theorems established in Section 4, we study the existence and uniqueness of a solution to a Fredholm nonlinear integral equation.

In order to compare our results to the ones in [37, 38] we will consider the same integral equation; that is,

where .

Let denote the set of all functions satisfying the following.

is nondecreasing and , for all .There exists such that , for all .is nonempty, as with is an element of .

Like in [38], we assume that the functions , , , and fulfill the following conditions.

Assumption 57. Consider the following:

(i) and , for all ;(ii)there exist two positive numbers and and such that, for all with , the following Lipschitzian type conditions hold:

(iii)

Definition 58 (see [38]). A pair with is called a coupled lower-upper solution of (96) if, for all ,

Theorem 59. Consider the integral equation (96) with

Suppose that there exists a coupled lower-upper solution of (96) with and that Assumption 57 is satisfied. Then the integral equation (96) has a unique solution in .

Proof. Consider on the natural partial order relation; that is, for ,

It is well known that is a complete metric space with respect to the sup metric:

Now for we define

Define

It is easy to see that is a complete -metric space with (see Example 3).

Now define on the following partial order:

for all

Obviously, for any , the element is comparable with and .

Define now the mapping by

It is not difficult to prove, like in [38], that has the mixed monotone property. Now for with and , we have

Let us first evaluate the expression in the right hand side. According to the computations done by Berinde in [37],

Since the function is nondecreasing and and , we have

Hence, by (108), in view of the fact that , we obtain that

as all quantities in the right hand side of (108) are nonnegative.

Now from (108) we have

So, we have

Similarly, one can obtain that

Taking the supremum with respect to and using (98) we get

Now, since is nondecreasing, we have

and so

by the definition of . Finally, from (114) we get that

Similarly, we can obtain that

which is just the contractive condition (69) in Corollary 43.

Now, let be a coupled upper-lower solution of (96). Then we have

for all , which show that all hypotheses of Corollary 43 are satisfied.

This proves that has a coupled fixed point in .

Since , by Corollary 43 it follows that ; that is,

and therefore is the solution of the integral equation (96).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publications of this paper.

Authors’ Contribution

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

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and second authors express their thanks to DSR, KAU for financial support.

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