#### 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 , economics, and telecommunication (see for more details ).

Istrăţescu  introduced and studied the notion of convex contractions. Recently Miandaragh et al.  proved certain results for generalized convex contractions on complete metric spaces. Salimi et al.  modified the concept of -admissible mappings introduced and studied by Samet et al. , Karapınar and Samet , and Salimi and Karapınar . 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.  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.  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 ). 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 ). 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 , 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  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 ). 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 ). 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,