Abstract and Applied Analysis

Volume 2014 (2014), Article ID 456482, 13 pages

http://dx.doi.org/10.1155/2014/456482

Research Article

Discussions on Recent Results for --Contractive Mappings

N. Hussain,¹ M. A. Kutbi,¹ S. Khaleghizadeh,² and P. Salimi³

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

²Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

³Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

Received 17 December 2013; Accepted 12 January 2014; Published 2 March 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 N. Hussain 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 certain fixed point results for -generalized convex contractions, -weakly Zamfirescu mappings, and -Ćirić strong almost contractions. As an application, we derive some Suzuki type fixed point theorems and certain new fixed point theorems in metric spaces endowed with a graph and a partial order. Moreover, we discuss some illustrative examples to highlight the realized improvements.

1. Introduction

Banach contraction principle states that every contraction mapping defined on a complete metric space has a unique fixed point and that point can be obtained as a limit of repeated iteration of the mapping at any point of . This fundamental fixed point theorem has laid the foundation of metric fixed point theory which is very important due to its applications in different fields such as image processing, physics, computer science [1], economics, and telecommunication (see for more details [2–11]).

Istrăţescu [12] introduced and studied the notion of convex contractions. Recently Miandaragh et al. [13] proved certain results for generalized convex contractions on complete metric spaces. Salimi et al. [14] modified the concept of -admissible mappings introduced and studied by Samet et al. [15], Karapınar and Samet [16], and Salimi and Karapınar [17]. We establish certain fixed point results for -generalized convex contractions, -weakly Zamfirescu mappings, and -Ćirić strong almost contractions. As an application, we shall derive corresponding results in metric spaces endowed with a graph and a partial order.

2. Discussion on --Contractive Mappings

We shall denote by the family of nondecreasing functions such that for each , where is the th iterate of . Clearly, if , then for all .

Samet et al. [15] introduced following concept.

Definition 1. Let be a metric space, let be a self-mapping, and let be a function. One says that is an --contractive mapping if

holds for all , where .

By taking for all and , where , --contractive mapping reduces to Banach contraction mapping.

We suggest the following notion as generalization of --contractive mappings.

Definition 2. Let be a metric space, let be a self-mapping, and let be two functions. One says that is an --contractive mapping if for all with we have

for some .

Example 3. Let be endowed with usual metric and let be defined by , where . Also, let be two functions such that only for some with . Then, is not an --contractive mapping while it is a Banach contraction and --contractive mapping. In fact,

while holds for all where .

Example 4. Let be endowed with usual metric and let be defined by . Also, let be two functions such that only . Then, is not an --contractive mapping while it is a Banach contraction and --contractive mapping. In fact,

while holds for all where .

Similarly, one may develop other examples of self-mappings that are not --contractive mappings while they are Banach contraction and --contractive mappings.

Remark 5. It is worth to notice that there is no Banach contraction mapping which is not --contractive. Indeed, let be a Banach contraction mapping on with contraction constant such that is not an --contractive mapping. Then for all , there exists such that and . But produces a contradiction to the fact that is a Banach contraction mapping.

More recently, Miandaragh et al. [13] introduced the following notions.

Definition 6. Let be a metric space and let be a self-mapping. One says is a generalized convex contraction if there exist with and a function such that

holds for all .

Definition 7. Let be a metric space and let be a self-mapping. One says is a generalized convex contraction of order if there exist with and a function such that

holds for all .

On the basis of the above facts, we suggest the notions of generalized convex contraction and generalized convex contraction of order 2 as follows.

Definition 8. Let be a metric space, let be a self-mapping, and let be two functions. Then is said to be an -generalized convex contraction if

where with .

Definition 9. Let be a metric space, let be a self-mapping, and let be two functions. Then is said to be an -generalized convex contraction of order 2 if

where, and .

Example 10. Let be endowed with usual metric and let be defined by , where . Also, let be two functions such that for some with . Then, is not a generalized convex contraction while it is a convex contraction and -generalized convex contraction. Indeed,

for all with . That is, is not a generalized convex contraction mapping. But if we choose and then,

holds for all . That is, is a convex contraction and -generalized convex contraction mapping.

Example 11. Let be endowed with metric

Let be defined by and let be two functions such that . Then is not a generalized convex contraction of order 2 while it is a convex contraction of order 2 and -generalized convex contraction of order 2 mapping. Indeed, if we choose and then,

holds for all with . That is, is not a generalized convex contraction of order 2. But, if we choose and then,

holds for all with . Moreover, if , then

and so,

holds for all . That is, is a convex contraction of order 2 and -generalized convex contraction of order 2 mapping.

Remark 12. We cannot find a self-mapping and functions such that is a convex contraction mapping (or convex contraction of order 2) which is not a -generalized convex contraction (or -generalized convex contraction of order 2).

3. Fixed Point Results for Modified Convex Contractions

Let be given. A point in a metric space is called an -fixed point of the self-map on whenever . We say that has an approximate fixed point (or has the approximate fixed point property) whenever has an -fixed point for all ; see [18, 19].

Definition 13 (see [14]). Let be a self-mapping on and let be two functions. One says that is an -admissible mapping with respect to if

Note that if we take , then is called -admissible mapping.

We shall need the following result.

Lemma 14 (see [18]). Let be a metric space and let be an asymptotic regular self-map on ; that is, as for all . Then has the approximate fixed point property.

Theorem 15. Let be a complete metric space and let be a modified generalized convex contraction on . If is an -admissible mapping with respect to and for all , then has an approximate fixed point.

Proof. Let for all . Since is an -admissible mapping with respect to , then we deduce that for all . By continuing this process, we get for all and for all . By taking and we have . Let and ; then by (7),

By continuing this process we get

where or . This implies that for all . By applying Lemma 14, has an approximate fixed point.

Let be a self-mapping and let be two functions. We say that has the -property whenever for all with , and there exists such that and . Also for all we have, .

Theorem 16. Let be a complete metric space and let be a modified generalized convex contraction on . Also suppose that is continuous and -admissible mapping with respect to . If there exists an such that , then has a fixed point. Moreover, has a unique fixed point when has -property.

Proof. Define a sequence in by for all . Since is an -admissible mapping with respect to and , we deduce that . By continuing this process, we get for all . Since is a modified generalized convex contraction, so from (7) we get

By taking and we have

where or . Let . Then for with , , and we deduce

Similarly, for and with , , and we get

Now, assume that . Then for with , , and we have

Similarly, for and with , , and we deduce

Hence, for all with we have

Taking limit as in the above inequality we get . That is, is a Cauchy sequence. Since is a complete metric space, then there exists such that as . Since is continuous, then .

Let , where . For prove of uniqueness we consider the following cases.

Case 1. Let . Since is a modified generalized convex contraction, then we have

which is a contradiction.

Case 2. Let . Since has -property, then there exists such that and . Now, since is an -admissible mapping with respect to , then we can deduce and . First we assume that . So by hypothesis we get

By taking and we have

where or . Therefore, . Similarly, we can show that . That is, which is a contradiction. Therefore, has a unique fixed point.

Theorem 17. Let be a metric space and let be a modified generalized convex contraction of order 2 on . If is an -admissible mapping with respect to and for all , then has an approximate fixed point.

Proof. As in proof of Theorem 15 we can conclude that for all and all . Put , , and . From (8) with and we have

which implies that . That is, . Again from (8) with and we get

which implies that . Similarly, and . By continuing this process, we get for all when or . This implies that for all . By Lemma 14   has an approximate fixed point.

Theorem 18. Let be a complete metric space and let be a modified generalized convex contraction of order on . Also suppose that is an -admissible with respect to and continuous mapping. If there exists an such that , then has a fixed point. Moreover, has a unique fixed point when has -property.

Proof. Define a sequence in by for all . Put and and . From (8) with and we have

which implies that . That is, . Again from (8) with and we get

which implies that . Similarly, and . By continuing this process, we get when or . Let . Then for with , , and we deduce

Similarly, for and with , , and we get

Now, assume that . Then for with , , and we have

Similarly, for and with , , and we deduce

Hence, for all with we have

Taking limit as in the above inequality we get . That is, is a Cauchy sequence. Since is a complete metric space, there exists such that as . Now since is a continuous mapping then has a fixed point . If has the -property, then by using a similar method to that in the proof of Theorem 16, we can prove uniqueness of the fixed point of .

4. -Weakly Zamfirescu Mappings

In this section we introduce the notion of -weakly Zamfirescu mapping and establish fixed point results.

Definition 19. Let be a metric space and let be a self-mapping on . Assume there exists with for all , such that

and then is a modified -weakly Zamfirescu mapping.

Theorem 20. Let be a metric space and let be an -weakly Zamfirescu mapping on . If is an -admissible mapping with respect to and for all , then has an approximate fixed point.

Proof. For a given , we define the sequence by . As in proof of Theorem 15 we can conclude that for all and all . Now since is an -weakly Zamfirescu mapping, then

Now if , then

which implies

and so is a nonincreasing sequence and converges to a real number . Assume that . Now since for all and for all , where , thus

for all . This implies , which is a contradiction. Therefore,

for a given . By Lemma 14   has an approximate fixed point.

Theorem 21. Let be a complete metric space and let be an -weakly Zamfirescu mapping on . Also suppose that is an -admissible mapping with respect to and continuous mapping. If there exists an such that , then has a fixed point.

Proof. Let such that . Define a sequence as in Theorem 15. By the similar proof as in proof of Theorem 20 we deduce

for all . As in proof of Theorem 28 [20], we deduce that is a Cauchy sequence. Since is a complete metric space, there exists such that as . Now since is an continuous mapping, so .

Example 22. Let be endowed with usual metric. Define and by

Let and and let be a given function. Then,

That is, is not an -weakly Zamfirescu mapping. Therefore, Theorem 3.3 of [13] can not be applied for this example.

Further, if and , then

That is, is not a weakly Zamfirescu mapping.

But if , then . Therefore,

Put and so

That is, there exists with for all , such that

holds for all with . Then is an -weakly Zamfirescu mapping. Clearly has a fixed point by our result.

5. From -Ćirić Strong Almost Contraction to Suzuki Type Contraction

Definition 23 (see [21]). Let be a metric space and let be a self-mapping on . Then is called a Ćirić strong almost contraction, if there exists a constant such that

for all , where and

Now we generalize the notion of Ćirić strong almost contraction mapping as follows.

Definition 24. Let be a metric space and let be two functions. A mapping is called an -Ćirić strong almost contraction, if there exists a constant such that

for all , where and

Moreover, if we take for all , then we say is a modified -Ćirić strong almost contraction mapping.

Theorem 25. Let be a complete metric space and be a continuous -Ćirić strong almost contraction on . Also suppose that is an -admissible mapping with respect to . If there exists a such that , then has a fixed point.

Proof. Let such that . For a given , we define the sequence by . Now since is an -admissible mapping with respect to , then . By continuing this process we have

for all . Since is an --Ćirić strong almost contraction mapping, so we obtain

where

which implies

Now if , then

which is a contradiction. Hence, for all . Now it is easy to show that is a Cauchy sequence. Since is a complete metric space, so there exists such that as . Continuity of implies that .

Theorem 26. Let be a metric space and let be a self-mapping on . Also, suppose that be two functions. Assume that the following assertions holds true:

(i) is an -admissible mapping with respect to ;(ii) is an --Ćirić strong almost contraction on ;(iii)there exists such that ;(iv)if is a sequence in such that with as , then either

 holds for all .

Then has a fixed point.

Proof. Let be such that . Define a sequence in by for all . Now as in the proof of Theorem 25 we have for all and there exists such that as . Let . From (iv) either

holds for all . Then,

holds for all . Let hold for all . Since is an --Ćirić strong almost contraction, so we get

where

Taking limit as in the above inequality we get

which is a contradiction. Hence, . That is, . By the similar method we can show that if holds for all .

If in Theorem 26 we take for all , then we obtain following corollary.

Corollary 27. Let be a metric space and let be a self-mapping on . Also, suppose that is a function. Assume that the following assertions holds true:

(i) is an -admissible mapping;(ii) is modified -Ćirić strong almost contraction on ;(iii)there exists such that ;(iv)if is a sequence in such that with as , then either

 holds for all .

Then has a fixed point.

If in Theorem 26 we take for all , we obtain following result.

Corollary 28 (Theorem 2.2 of [21]). Let be complete metric space and let be a Ćirić strong almost contraction on . Then has a fixed point.

Example 29. Let . We endow with usual metric. Define , by

Let , and then . On the other hand, for all . Then, . That is, is an -admissible mapping with respect to . If is a sequence in such that with as , then for all . That is,

hold for all . Clearly, . Let . Now, if or , then , which is a contradiction. So, . Therefore,

Therefore is an --Ćirić strong almost contraction. Hence, all conditions of Theorem 26 hold and has a fixed point. Let and ; then

That is, is not a Ćirić strong almost contraction. Hence, Corollary 28 (Theorem 2.2 of [21]) cannot be applied for this example.

As an application of the above results, we obtain the following Suzuki type fixed point theorem [22].

Theorem 30. Let be a complete metric space and let be a self-mapping on . Assume that there exists such that

for all , where

Then has a fixed point.

Proof. Define by

for all , where and . Now, since for all , then for all . That is, conditions (i) and (iii) of Theorem 26 hold true. Let be a sequence with as . Assume that for some . Then, . That is is a fixed point of and we have nothing to prove. Hence we assume for all . Since for all , then from (82) we get

where

which implies

Assume that there exists , such that

Then,

So by (76) we have

which is a contradiction. Hence, either

holds for all . That is, condition (iv) of Theorem 26 holds. Let . So, . Then from (82) we get . Hence, all conditions of Theorem 26 hold and has a fixed point.

Corollary 31 (see [23], Theorem 3.2). Let be a complete metric space and let be a self-mapping on . Define a nonincreasing function by

Assume that there exists such that

for all . Then has a unique fixed point.

6. Fixed Point Results on Metric Spaces Endowed with Graph

Consistent with [1, 24], let be a metric space, and denotes the diagonal of the Cartesian product . Consider a directed graph such that the set of its vertices coincides with , and the set of its edges contains all loops; that is, . We assume has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph (see [24]) by assigning to each edge the distance between its vertices. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that and for . A graph is connected if there is a path between any two vertices. is weakly connected if is connected (see for details [23–25]).

Definition 32 (see [24]). A mapping is called -continuous, if given and sequence :

Definition 33. Let be a metric space endowed with a graph and let be a self-mapping. We say is a graphic convex contraction if

holds for all with , where , .

Definition 34. Let be a metric space endowed with a graph and let be a self-mapping. One says is a graphic convex contraction of order 2 if

holds for all with , where , .

Definition 35. Let be a metric space endowed with a graph and let be a self-mapping on . Assume there exists with for all , such that

holds for all with and

then is a graphic weakly Zamfirescu mapping.

Definition 36. Let be a metric space endowed with a graph . A mapping is called graphic Ćirić strong almost contraction, if there exist a constant such that

holds for all with , where :

Theorem 37. Let be a metric space endowed with a graph and let be a graphic convex contraction on . If for all , then has an approximate fixed point.

Proof. Define by

At first we prove that is an -admissible mapping. Let ; then . Now since is a graphic convex contraction, we have . That is, . Also, clearly, is a modified generalized convex contraction.

Let for all . Then, for all . Hence, all conditions of Theorem 15 hold and has an approximate fixed point.

Similarly, we can deduce the following results.

Theorem 38. Let be a complete metric space endowed with a graph and let be a graphic convex contraction on . Also suppose that is -continuous mapping. If there exists such that , then has a fixed point. Moreover, has a unique fixed point if, for all with , there exists such that and .

Theorem 39. Let be a metric space endowed with a graph and let be a graphic convex contraction of order 2 on . If for all , then has an approximate fixed point.

Theorem 40. Let be a metric space endowed with a graph and let be a graphic convex contraction of order 2 on . Also suppose that is -continuous mapping. If there exists a such that , then has a fixed point. Moreover, has a unique fixed point if, for all with , there exists such that and .

Theorem 41. Let be a metric space endowed with a graph and let be a graphic weakly Zamfirescu mapping on . If for all , then has an approximate fixed point.

Theorem 42. Let be a complete metric space endowed with a graph and let be a graphic weakly Zamfirescu mapping on . Also suppose that is -continuous mapping. If there exists an such that , then has a fixed point.

Theorem 43. Let be a metric space endowed with a graph and let be a self-mapping on . Assume that the following assertions hold true:

(i) is graphic Ćirić strong almost contraction on ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all .

Then has a fixed point.

Let be a partially ordered metric space. Define the graph by

For this graph, the condition “” in Definitions 32–35 translates into “” which means is nondecreasing with respect to this order [6]. Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [2, 25–30] and references therein). From Theorems 37–43 we derive the following new results in partially ordered metric spaces.

Theorem 44. Let be a partially ordered metric space and let be a nondecreasing ordered convex contraction on . If for all , then has an approximate fixed point.

Theorem 45. Let be a complete partially ordered metric space and let be a nondecreasing and ordered convex contraction on . Also suppose that is continuous mapping. If there exists an such that , then has a fixed point. Moreover, has a unique fixed point if, for all with , there exists such that and .

Theorem 46. Let be a partially ordered metric space and let be a nondecreasing, ordered convex contraction of order 2 on . If for all , then has an approximate fixed point.

Theorem 47. Let be a complete partially ordered metric space and let be a nondecreasing and ordered convex contraction of order 2 on . Also suppose that is continuous mapping. If there exists an such that , then has a fixed point. Moreover, has a unique fixed point if, for all with , there exists such that and .

Theorem 48. Let be a partially ordered metric space and let be a nondecreasing, ordered weakly Zamfirescu mapping on . If for all , then has an approximate fixed point.

Theorem 49. Let be a complete partially ordered metric space and let be a nondecreasing and ordered weakly Zamfirescu on . Also suppose that is continuous mapping. If there exists an such that , then has a fixed point.

Theorem 50. Let be a complete partially ordered metric space. Assume that the following assertions hold true:

(i) is nondecreasing and ordered Ćirić strong almost contraction on ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all .

Then has a fixed point.

Conflict of Interests

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

Acknowledgment

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

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