1. Introduction

The fixed point theorem most frequently cited in literature is Banach contraction principle, which asserts that if is a complete metric space and is a contractive mapping ( is contractive if there exist such that for all , ) then has a unique fixed point.

In 1968, Kannan ^[3] established a fixed point for mapping satisfying:

for and for all

Kannan’s paper ^[3] dealing with the generalization of Banach fixed point theorem was followed by a spate of papers containing a variety of contractive definitions in metric spaces.

Rhodes ^[6] considered 250 types of contractive definitions and analyzed the interrelation among them.

Jungck and Rhoades ^[8] introduced the concept of weakly compatible maps for extending some well known fixed point theorems to the setting of set valued non continuous functions. Here the fixed point theorems were proved for set valued functions without appealing to the continuity.

In 2008 Azam and Arshad ^[1] extended Kannan’s theorem for the generalized metric space introduced by Branciari ^[2] by replacing triangular inequality by rectangular one in the context of fixed point theorem.

In 2010 Moradi and Beiranvand ^[7] and Moardi and Omid ^[5] introduced new classes of contractive functions as following and established the Banach contraction principle.

Let be a metric space. A mapping is said to be sequentially convergent if we have, for every sequence, if is convergent then is also convergent. is said to be subsequentially convergent if we have, for every sequence , if is convergent then has a convergent subsequence.

In 2011, Moradi and Alimohammadi ^[4] extended the Kannan’s theorem and the theorem due to Azam and Arshad ^[1] as following:

Let be a complete metric space and be mappings such that is continuous, one- to- one and subsequentially convergent. If and for all

then has a unique fixed point. Also if is sequentially convergent then for every the sequence of iterates converges to this fixed point.

In the present paper, sufficient conditions were obtained for the existence of the unique fixed point of Kannan’s type mapping on complete metric spaces depending on another function. Of course, a variation of this aspect has been discussed by Patel and Deheri ^[9] in 2013 where common fixed point theorems were proved in the light of another function. Of course, the underlying spaces were Banach spaces.

Indeed, It has been deemed proper to provide some generalizations and variations of the main results presented in Moradi and Alimohammadi ^[5].

2. Main Results

The main result of the paper is contained in

Let be a complete metric space and be mappings such that is continuous, one- to- one and subsequentially convergent. If and for all

(1)

then has a unique fixed point. Also if is sequentially convergent then for every the sequence of iterates converges to this fixed point.

Proof Let be an arbitrary point in . We define the iterative sequence by (equivalently, ), .Using equation (1) one gets

(2)

leading to

(3)

Using induction and equation (3), one finds that

(4)

By (4), for every such that one obtains

(5)

Letting in equation (5) one concludes that is a Cauchy sequence, and since is a complete metric space, there exists such that

(6)

Since is a subsequentially convergent, has a convergent subsequence. So there exists and such that . Since is continuous and , .

By equation (6) one gets that , which results in

(7)

Letting in equation (7) one gets

This implies .

Since is one-to-one and has a fixed point.

For the uniqueness of the fixed point let us assume that there are and in, such that and .

Then by equation (1) one get

implies

Since is one-to-one and one finds

Also, if is sequentially convergent, by replacing with we conclude that and this shows that converges to the fixed point of .

To understand the importance of this result one can cast a glance at the following:

Let endowed with the Euclidean metric. Define by and for all . Obviously the condition in Kannan’s theorem is not true for every . So we cannot use Kannan’s theorem. By defining by and for all one have, for ,

Therefore, by theorem 2.1 has a unique fixed point

Now, a little variation in the inequality of above result leads to:

Let be a complete metric space and be mappings such that is continuous, one- to- one and subsequentially convergent. If and for all

(8)

then has a unique fixed point. Also if is sequentially convergent then for every the sequence of iterates converges to this fixed point.

Proof Let be an arbitrary point in . We define the iterative sequence by (equivalently, ), .Using equation (8) one gets

(9)

leading to

(10)

Using induction and equation (10), one finds that

(11)

By equation (11), for every such that one obtains

(12)

Letting in equation (12) one concludes that is a Cauchy sequence, and since is a complete metric space, there exists such that

(13)

Since is a subsequentially convergent, has a convergent subsequence. So there exists and such that . Since is continuous and , .

By equation (13) one gets that , which leads to

hence,

(14)

Letting in equation (14) one gets

Since is one-to-one and has a fixed point.

Uniqueness

For the uniqueness of the fixed point let us assume that there are and in, such that and .

Then by equation (8) one obtains

which implies that

Since is one-to-one and one finds

Also if is sequentially convergent, by replacing with we conclude that and this shows that converges to the fixed point of .

The following result marks the end of the discussion:

Let be a complete metric space and be mappings such that is continuous, one- to- one and subsequentially convergent. If and for all

(15)

then has a unique fixed point. Also if is sequentially convergent then for every the sequence of iterates converges to this fixed point.

Proof Let be an arbitrary point in. We define the iterative sequence by (equivalently, ) .Using equation (15) one gets

(16)

leading to

(17)

Using induction and equation (17), one finds that

(18)

By equation (18), for every such that one obtains

(19)

Letting in equation (19) one concludes that is a Cauchy sequence, and since is a complete metric space, there exists such that

(20)

Since is a subsequentially convergent, has a convergent subsequence. So there exists and such that . Since is continuous and , .

By equation (20) one gets that , which results in

(21)

Letting in equation (21) one gets

This implies.

Since is one-to-one and has a fixed point.

Uniqueness

For the uniqueness of the fixed point let us assume that there are and in, such that and .

Then by equation (15) one obtains

which implies that

Since is one-to-one and one finds

Also, if is sequentially convergent, by replacing with we conclude that and this shows that converges to the fixed point of .

3. Conclusion

As can be seen the results presented here not only are far more generalized version, but also improve some of the well known classical results, addressing convergence intervals.

Acknowledgement

The corresponding author acknowledges the funding agency CSIR.

References

[1]	Azam, A. and Arshad, M., “Kannan fixed point theorem on generalized metric spaces,” The J. Nonlinear Sci. Appl., 1(1), 45-48, Jul.2008.
	In article
[2]	Branciari, A., “A fixed point theorem of Banach- Caccippoli type on a class of generalized metric spaces,” Publ. Math. Debrecen, 57 (1-2), 31-37, 2000.
	In article
[3]	Kannan, R.., “Some results on fixed points,” Bull. Calcutta Math. Soc., 60, 71-76, 1968.
	In article
[4]	Moradi, S. and Alimohammadi, D., “New extensions of Kannan fixed-point theorem on complete metric and generalized metric spaces,” Int. Journal of math. Analysis, 5(47), 2313-2320, 2011.
	In article
[5]	Moradi, S. and Omid, M., “A fixed point theorem for integral type inequality depending on another function,” Int. J. Math. Anal., 4, 1491-1499, 2010.
	In article
[6]	Rhoades, B. E., “A Comparison of Various Definitions of Contractive Mappings,” Amer. Math. Soc., 226, 257-290,1977.
	In article		View Article
[7]	Moradi, S. and Beiranvand, A., “Fixed Point of TF-contractive Single-valued Mappings,” Iranian Journal of Mathematical sciences and informatics, 5, 25-32, 2010.
	In article
[8]	Jungck, G. and Rhoades, B. E., “Fixed points for set valued functions without continuity”, Indian J. pure appl. Math., 29(3), 227-238, March 1998.
	In article
[9]	Patel, K. and Deheri, G., “Extension of some common fixed point theorems”, International Journal of Applied Physics and Mathematics, 3(5), 329-335, Sept.2013.
	In article		View Article