1. Introduction and preliminaries
In [1] , Khan introduced and proved fixed point results by the altering distance in metric space. Aliouche [2] proved common fixed point results in symmetric space for weakly compatible mappings under contractive condition of integral type. In [3] , Babu generalized and proved fixed point results using control function. Later Bouhadjera and Godet [4] generalized concept of pair sub compatible maps and proved fixed point results. Also Chaudhari [5] [6] , Chugh & Kumar [7] , Naidu [8] , Sastry et al. [9] generalized and proved some fixed point results. Recently in [10] [11] , Hosseni used contractive rule of integral type by altering distance and generalized common fixed point results. Many authors proved fixed point results with different techniques in different spaces (see [12] - [17] ). In [18] [19] [20] [21] , Wadkar et al. proved fixed point theorems using the concept of soft metric space. In the present paper, we prove two theorems on fixed point under contraction rule of integral type in metric space by altering distance function, first for self map and second for a pair of sub compatible maps. Our results are motivated by V. R. Hosseni, Neda Hosseni.
Definition 1.1: A function is an altering distance functions if is continuous with monotone increasing in all variables and if
The collection of all altering distance is denoted by .
Now let us define a function by for , clearly if and only if .
Examples of are for , (1)
(2)
Definition 1.2: The maps of metric space are called as sub compatible if and only if the sequence in E such that and which satisfies
Example 1.3: Let we define p & q with metric as follows
& (3)
Let us define the sequence in E as , for then
(4)
and (5)
(6)
Thus, we have . (7)
Hence maps p and q are sub-compatible.
On the other hand, we have if and only if , and
Then but , hence p and q are not OWC (Oscillatory weakly commuting).
2. Main Result
Theorem 2.1: Let us consider the mappings of complete metric space be such that for all
(8)
where with , and Lebesgue-integr- able mapping , which is positive, sum able, and for each , then there exist a unique common fixed point in E for U and V.
Proof: Consider arbitrary point of E, for we have
and .
Let (9)
Substituting and in Equation (8), then for all we have
Using Equation (9) for all we get
(10)
As implies that , so we have
(11)
Now by monotone increase of in all variables and using the property that whenever , we get a contradiction i.e. not greater than . Hence we have , for
(12)
Substituting in Equation (8) we have
(13)
By using (12) we consider
(14)
From (10) and (12) we obtain
(15)
From (8) & (11) for all , we have
then
Taking summation in above equation we obtain
,
which implies as . (16)
Now from (13) sequence is convergent and as , We know that is continuous and from Equation (14) we obtain which implies that , i.e. as
(17)
We now show that the sequence is a Cauchy sequence in E. Keeping in mind Equation (15) it is require to show that is a Cauchy sequence. If is not a Cauchy sequence of natural number such that ,
(18)
Hence using (16)
Taking in the inequality above & by result of Equation (15), we arrive at
. (19)
For all
(20)
Also for
. (21)
Making in (18) & (19) respectively by using (15) & (17) we have
and
Therefore, , for (22)
Taking in the above two inequalities and using (15) & (17) we obtain
. (23)
Putting in (8), for all , we obtain
Now in above inequality if we take and by using results of (15), (20) & (21) we get
Then
This is due to monotone increasing fact of in its variable and by using property of that if and only if .
From the above inequality we get a contradiction. So that . This establishes convergent sequence in .
Let as . (24)
Substituting in (8) for all
Taking limit n tends to infinity in the above inequality and using continuity of and and Equations (15), (22) we get
If then monotone increasing and are monotone increasing and if and only if , we obtain
This contradiction, hence we obtain (25)
In similar way we prove that Hence (26)
Hence (25) & (26) shows that z is a common fixed point of U and V.
Theorem 2.2: Let be a complete metric space and p, q, U and V be four mappings from E to itself such that
(27)
for all , where , , for .
i: One of the four mappings p, q, U and V is continuous.
ii: (p, U) & (q, V) are sub compatible.
iii: The pairs and .
iv: Where is Lebesgue-integrable mappings, which is sum able, non negative and such that for each .
Then p, q, U and V have a unique common fixed point in E.
Proof: Consider arbitrary point , we construct the sequence and in E such that
and ,
Let , Substitution and in (27) we have
If then and
(28)
Thus we arrive at a contradiction. Hence , similarly by substituting in (27) we can prove that, , for . Thus , for . Hence the sequence is sequence of positive real numbers, which is decreasing and converges to .
Let . Taking in (27) we have
(29)
In view of (29), to prove sequence is a Cauchy sequence it is sufficient to prove the subsequence of sequence is a Cauchy sequence. If is not a Cauchy sequence there exist & sequence of natural numbers & which are monotone increasing such that .
(30)
Then from (29) we have
(31)
Taking and using (29) we have
(32)
Taking using (29) & (30) in
(33)
We get (34)
Letting and from Equations (29) & (30) in
We get (35)
Putting in (27), for all we obtain
Taking & using (29), (30), (32), (33) & (35) we get
This is contradiction. Hence is a Cauchy sequence and is convergent. Since E is complete there exist such that as we have .
Case I: Assume that U is continuous then , Since (p, U) is sub compatible, we have
Step I: Substituting in (27), we have
It is contradiction if . Hence
Step II: Substituting in (27) and taking limit as n tends to infinity we get .
Step III: We know that then there exist such that . Substituting in (27) we get . Hence and , which gives .
Step IV: Substituting in (27) we have so that . Hence p, q, U & V have a common fixed point z in E.
Case II: Assume that U is continuous then . Similarly we can prove that z is common fixed point of p, q, U & V. When q or V is continuous, then the uniqueness of common fixed point follows easily from (27).
Example: Let with the usual metric . Define such that , , , .
Let ,
then
For all , it follows that the condition (27).
Let be a sequence in E such that & for some z in E. Then z = 0, . Hence is sub compatible. We have common fixed point in E.
3. Conclusion
In this paper, we proved the fixed point theorem for four sub compatible maps under a contractive condition of integral type. These results can be extended to any directions and can also be extended to fixed point theory of non-expansive multi-valued mappings.
Acknowledgements
The authors would like to give their sincere thanks to the editor and the anonymous referees for their valuable comments and useful suggestions in improving the article.
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