Some Fixed Point Theorems under Weak Semicompatibility

Abstract

The aim of the present paper is to prove some fixed point theorems by using the recent notion “weak semicompatibility.” The new notion is proper generalization of semicompatibility and can be applicable on commuting and compatible maps. We used compatible and absorbing mappings to prove theorems which also include (E.A.) property.

1. Introduction and New Definitions

In 1995, Cho et al. [1] introduced the concept of semicompatibility and obtained the first result that established a situation in which a collection of mappings has a fixed point. They defined a pair of self-maps to be semicompatible if(a) and(b). For some , implying holds.

Singh and Jain [2] observe that (b) implies (a). Hence they defined the semicompatibility by condition (b) only.

Let be a metric space and let and be two maps from into itself. and are commuting maps if for all in .

To generalize the notion of commuting maps, Sessa [3] introduced the concept of weakly commuting maps. He defines and as being weakly commuting if for all . Obviously, commuting maps are weakly commuting but the converse is not true.

In 1986, Jungck [4] gave more generalized commuting and weakly commuting maps called compatible maps. and are called compatible if Whenever is a sequence in such that for some , clearly, weakly commuting maps are compatible, but the implication is not reversible (see [4]).

Afterwards, Jungck et al. [5] made another generalization of weakly commuting maps by introducing the concept of compatible maps of type (). Previous and are said to be compatible of type () if in place of (1) we have the two following conditions: It is clear to see that weakly commuting maps are compatible of type (); from [5] it follows that the implication is not reversible.

Two self-maps and of metric space are called -compatible ([6] cited from [7]) if , whenever is a sequence in such that for some in . Similarly, two self-maps and of metric space are called -compatible ([6] cited from [7]) if , whenever is a sequence in such that for some in .

Two self-mappings and of a metric space are called weakly commuting [8] at a point in if for some .

The two self-maps and of a metric space are called weakly commuting of type [9] if there exists some positive real number such that for all in .

The two self-maps and of a metric space are called weakly commuting of type [9] if there exists some positive real number such that for all in . It may be noted that compatible mappings and can be weakly commuting of types and .

Let and be two self-maps of metric space ; then will be called -absorbing [10] if there exists a real number such that for all . Similarly, let and be two self-maps of metric space ; then will be called -absorbing [10] if there exists a real number such that for all .

Let and be two self-mappings of metric space . The maps and satisfy the E.A. property [11] if there exists a sequence in such that for some .

Two self-maps and of metric space are said to be -compatible of type [12] if , whenever is a sequence in such that for some in . Similarly, two self-maps and of metric space are said to be -compatible of type [12] if , whenever is a sequence in such that for some in .

Pant et al. [13] introduced a notion of weak reciprocal continuity as follows.

Definition 1. Two self-mappings and of metric space will be called weakly reciprocally continuous if or , whenever is a sequence in such that for some in .
Further Saluja et al. [14] introduced a notion of weak semicompatibility as follows.

Definition 2. Two self-mappings and of a metric space will be called weak semicompatible mappings if or , whenever is a sequence in such that for some in .

Example 3 (see [14]). Let and be the usual metric on . Define by Taking , since , we have Therefore and are weak semicompatible.
Also here maps and have no common fixed point.

Now we give some more definitions to improve this result.

Definition 4 (see [15]). Let be a set and and be self-maps of . A point in is called coincidence point of and if and only if . One will call a point of coincidence of and .

Definition 5 (see [16]). Two self-maps and of a set are occasionally weakly compatible (owc) if and only if there is a point in which is coincidence point of and at which and commute.

Lemma 6 (see [15]). Let be a set and let and be owc self-maps on . If and have a unique point of coincidence, , then is the unique common fixed point of and .

Definition 7 (see [17]). Two self-mappings and on a metric space are called conditionally commuting if they commute on nonempty subset of the set of coincidence points whenever the set of their coincidence points is nonempty.
From the definition itself, it is clear that if two maps are weakly compatible or owc, then they are necessarily conditionally commuting; however, the conditionally commuting mappings are not necessarily weakly commuting or owc.

Definition 8. Let be a set. A symmetric on is a mapping such that

Theorem 9. Let be a set with a symmetric . Suppose that and are owc self-maps of satisfying where , , such that . Then and have a unique common fixed point.

Proof. Since the maps are owc, there exists a point such that and . Substituting (a) with gives
Thus, , which implies that is a fixed point of . But, since , is also a fixed point of .
Suppose that and are common fixed point of and . Substituting (1) with implies that .

Lemma 10 (see [18]). If and are compatible of type (A), then they are owc, but the converse is not true in general.
Now we give the following lemma with the fact of [18].

Lemma 11. If and are either -compatible or -compatible, then they are owc, but the converse is not true in general. We give the following examples to prove this.

Example 12. Let with the usual metric. Define by We have and ; that is, and are owc. Now consider for ; we have and . But and ; therefore ; hence and are not -compatible.

Example 13. Let with the usual metric. Define by We have and ; that is, and are owc. Now consider for ; we have and . But and ; therefore ; hence and are not -compatible.

Lemma 14. If and are -compatible of type () or -compatible of type (), then they are owc, but the converse is not true in general. Here we prove the converse condition.

Proof. If with the usual metric, define and by We have , , and also , but . This implies that and are owc.
If we take sequence This implies that and are not -compatible of type ().

2. Main Results

Theorem 15. Let and be weak semicompatible, weakly commuting type of , and self-mappings of a complete metric space such that (a); (b);   and such that ;(c) and are either -compatible of type () or -compatible of type ().If and are conditional commuting, then and have a common fixed point in .

Proof. Let be any point in . Since , there exist such that . Similarly, we can have a sequence Now by (b), we have Since , Similarly, we can obtain . Therefore by (15) we have . By the same way, we can have Now we will show that is a Cauchy sequence. For any integer , we get Since , then taking limit we have . Therefore is a Cauchy sequence. Since is complete, there exists a point in such that as ; moreover for .

Case 1 ( and are -compatible of type ()). Since and are weak semicompatible, this yields either or . First we take .
Since and are -compatible of type (), this yields . Now by (b), we have Now limiting yields Since , this yields .
Now the conditional commutativity of and implies or . Now by (b), we have Since , this implies or . Hence is common fixed point of and .
Now we take .
Since and are -compatible of type (), this yields .
Also and are -weakly commuting type of and this implies .
Now limiting yields . Again by (b), we have
Now limiting yields
Since implies , now by (b), we get
Now limiting with yields , and hence and is common fixed point of and .

Case 2 (and are -compatible of type ()). Since and are weak semicompatible, this yields either or .
First we take ; since and are -compatible of type (), this yields . Also and are -weakly commuting type of and this implies . Now limiting , we get . Now by (b), we get Now limiting , we get , Since , this yields .
Now the conditional commutativity of and implies or . Now by (b), we get Since , this implies or . Hence is common fixed point of and .
Now we take .
Since and are -compatible of type (), this yields . Now by (b) . Limiting yields , since yields .
Now the conditional commutativity of and implies or . Now by (b), we get Since implies or . Hence is common fixed point of and .

Example 16. Let and and be the usual metric on . Define as follows: When we take constant sequence , then , , and , but if we take sequence , , and also Therefore maps and are weak semicompatible.

It can easily be shown that maps and satisfy conditional commutativity at . Moreover if we take sequence , .

Also, and . Hence maps and are -compatible of type (). The mappings and are -weakly commuting of type ; that is,

Moreover the pair satisfies equally the condition of (b) if . Again the pair satisfies equally the condition of (b) if and for , , and .

Now finally if and for , the left hand side of given inequality yields and right hand side gives Therefore the pair is satisfied greater than the condition of (b) when , , and , and is a common fixed points of and .

Corollary 17. Let be a set and let be the symmetric on . Let maps and satisfy all the conditions of Theorem 15. Since either and are -compatible of type (E) or and are -compatible of type (), then by Lemma 14 the pair will be owc and therefore the conclusion of Theorem 15 follows from Theorem 9.

Theorem 18. Let and be weak semicompatible and self-mappings of a metric space such that(a); (b);   and such that ;(c) and satisfying E.A. property.
If is -absorbing or is -absorbing, then and have a common fixed point in .

Proof. Since and satisfy E.A. property, then there exists a sequence in such that for some in .

Case 1 (suppose that is -absorbing). Since and are weak semicompatible, this yields either or .
First we take .
Since , for each sequence, there exists a sequence in such that . Therefore, and . By using (b), . Limiting with yields Since is -absorbing, this yields . Limiting yields . By (32) it is easy to see that . Now by (b), . Limiting yields . Since , it gives . Since , therefore .
Since , then there exists a point in such that . Now by (b), . Limiting yields Since , this yields . Since is -absorbing, this yields implying or . Again by (b), we have Since , this yields or . Hence is common fixed point of and .
Now we take .
Since is -absorbing, this yields . Limiting yields , which implies . Now by (b), . Limiting yields . Since , this yields . Hence or is common fixed point of and .

Case 2 (suppose is -absorbing). By virtue of (32) it is easy to see that .
Since and are weak semicompatible mappings, this yields & .
First we take .
Since is -absorbing, this yields . Limiting yields . This implies . Now by (b), . Limiting yields . Since , this yields or .
Since , then there exists a point in such that . Now by (b), . Limiting yields Since , this yields . Since is -absorbing, this yields implying . Again by (b), we have
Since , this yields or . Hence is common fixed point of and .
Now we take .
Since is -absorbing, this yields and . Limiting yields . By virtue of (32)   .
Now by (b), . Limiting yields . Since , this yields .
Now by (b), . Limiting yields . Since , this yields . Hence or is common fixed point of and .

Example 19. Let and and be the usual metric on . Define as follows: When we take constant sequence , then ,  , and . But if we take sequence , , and also

Therefore maps and are weak semicompatible.

It can be easily seen that maps and satisfy E.A. property. Moreover map is -absorbing at . Map is -absorbing at for and map is -absorbing at for . Moreover the pair satisfies equally the condition of (b) if and the pair satisfies equally the condition of (b) for when , and . Now finally if and for , the left hand side of the given inequality yields and right hand side gives

Therefore the pair is satisfied greater than or is equal to the condition of (b) when , , and , and is a common fixed points of and .

Theorem 20. Let and be weak semicompatible and self-mappings of a metric space such that (a); (b);   and such that ;(c) and satisfying E.A. property.
If and are -compatible or and are -compatible, then and have a common fixed point in .

Proof. Since and satisfy E.A. property, then there exists a sequence in such that for some in .

Case 1 (suppose and are -compatible). Since and are weak semicompatible, this yields either or .
First we take .
Since , for each sequence, there exists a sequence in such that . Therefore, , and . By using (b), . Limiting with yields , so
Since and are -compatible, this yields or . By virtue of (40)   . Therefore, . Now by (b), . Limiting yields . Since , this yields .
Since -compatibility of and implies commutativity at coincidence point, this gives or . Now by (b), we get Since , this implies or . Hence is common fixed point of and .
Now we take .
Since and are -compatible, this yields or . Therefore, . Now by (b), . Limiting yields . Since , this implies .
Since -compatibility of and implies commutativity at coincidence point, this gives or . Now by (b), we get Since , this implies or . Hence is common fixed point of and .

Case 2 (suppose and are -compatible). Since and are weak semicompatible, this yields either or .
First we take .
Since and are -compatible, this yields or . Therefore . Now by (b), . Limiting yields . Since , this implies .
Since -compatibility of and implies commutativity at coincidence point, this gives or . Now by (b), we have Since implies or . Hence is common fixed point of and .
Now we take .
Since and are -compatible, this yields
Since and satisfy E.A. property, this yields with (44)   . Now by (b), . Limiting yields . Since , this implies .
Since -compatibility of and implies commutativity at coincidence point, this gives or . Now by (b), we have
Since , this implies or . Hence is common fixed point of and .

Example 21. Let and and be the usual metric on . Define as follows: if we take constant sequence , , , and , but when we take sequence , then and or . Moreover and .

Therefore and are weak semicompatible. It can be easily shown that mappings and satisfy E.A. property. When we take sequence , it is easy to show that maps and are -compatible. When we take sequence , then and ; therefore .

Now and . Thus . Therefore maps and are -compatible. Again if we take sequence , then and ; thus . Now and . Thus . Therefore maps and are -compatible.

Moreover the pair is satisfying equally the condition of (b) if and if we take and , for , left hand side gives and right hand side gives Therefore the pair is satisfied greater than or is equal to the condition of (b) when , , and .

Now finally if and for . The left hand side of given inequality yields and right hand side gives Therefore the pair is satisfied greater than the condition of (b) when , , and , and is a common fixed points of and .

Corollary 22. Let be a set and let be the symmetric on . Let maps and satisfy all the conditions of Theorem 20. Since and are either -compatible or -compatible, then, by Lemma 11, pair will be owc and therefore the conclusion of Theorem 20 follows from Theorem 9.

Conflict of Interests

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