Journal of Mathematics
Volume 2016 (2016), Article ID 2121906, 4 pages
http://dx.doi.org/10.1155/2016/2121906

Research Article

Presic Type Fixed Point Theorem for Four Maps in Metric Spaces

K. P. R. Rao,¹ Sk. Sadik,² and S. Manro³

¹Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar 522 510, India
²Department of Mathematics, Sir C.R.R. College of Engineering, Eluru 534 007, India
³Department of Mathematics, Thapar University, Patiala, India

Received 2 July 2016; Accepted 27 October 2016

Academic Editor: Kaleem R. Kazmi

Copyright © 2016 K. P. R. Rao 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 obtained a Presic type fixed point theorem for two pairs of jointly -weakly compatible maps in metric spaces. We also have given an example to illustrate our main theorem.

1. Introduction and Preliminaries

Banach [1], in 1932, proved the following theorem, popularly known as Banach contraction principle.

Theorem 1. Let be a complete metric space and let be a self-mapping on satisfying for all , where . Then has a unique fixed point in .

There are number of generalizations of such principle (see [2–12]). One such generalization is given by Presic [13] in 1965.

Let , where is a positive integer. A point is called a fixed point of if . Consider the -order nonlinear difference equation

with the initial values .

Equation (1) can be studied by means of fixed point theory in view of the fact that is a solution of (1) if and only if is a fixed point of . One of the most important results in this direction is obtained by Presic [13] in the following way.

Theorem 2 (see [13]). Let be a complete metric space, be a positive integer, and . Suppose that

holds for all , where and . Then has a unique fixed point . Moreover, for any arbitrary points in , the sequence defined by , for all converges to .

Later Ciric and Presic [14] generalized the above theorem as follows.

Theorem 3 (see [14]). Let be a complete metric space, be a positive integer, and . Suppose that

holds for all in , where . Then has a fixed point . Moreover, for any arbitrary points in , the sequence defined by , for all converges to . Moreover, if holds for all with , then is the unique fixed point of .

Recently Rao et al. [2, 3] obtained some Presic type theorems for two and three maps in metric spaces. Now we give the following definition of [2, 3].

Definition 4. Let be a nonempty set and and . The pair is said to be -weakly compatible if whenever such that .

Using this definition, Rao et al. [2] proved the following.

Theorem 5 (see [2]). Let be a metric space, be a positive integer, and , be mappings satisfying

(1.4.1) for all in ,(1.4.2) for all in , where ,(1.4.3), for all with .(1.4.4)Suppose that is complete and either or is a -weakly compatible pair.

Then there exists a unique point such that .

Now we give a definition of jointly -weakly compatible pairs.

Definition 6. Let be a nonempty set, be a positive integer, , and . The pairs and are said to be jointly -weakly compatible pairs if and whenever there exists such that and .

In this paper, we obtain a Presic type theorem for four mappings satisfying a slight different contractive condition in metric spaces. We also give an example and two corollaries to our main theorem.

Now we prove our main result.

2. Main Result

Theorem 7. Let be a complete metric space, be a positive integer, and and be mappings satisfying

(2.1.1), ,(2.1.2),  for all and ,(2.1.3) and are jointly -weakly compatible pairs.(2.1.4)Suppose for some whenever there exists a sequence in such that  . Then is the unique point in such that .

Proof. Suppose are arbitrary points in . From (2.1.1), define

Let

Let and .
Then and by selection of we have

Continuing in this way, we get

Now consider

Also

Thus from (10) and (11) we have

Now from consider

Hence is a Cauchy sequence in . Since is complete there exists such that as . From (2.1.4), there exists such that

Now consider

Letting and using (14) so that

Similarly we have

Since and are jointly -weakly compatible pairs, we have

Similarly, we have

From (20) and (21) we have

which in turn yields that

From (18), (19), and (23), we have

Suppose there exists such that

Then from (2.1.2) we have

This implies that .
Thus is the unique point in satisfying (24).

Now we give an example to illustrate our main Theorem 7.

Example 8. Let , , and . Define , , , and for all .
Then, for all , we have

Thus the condition (2.1.2) of Theorem 7 is satisfied. One can easily verify the remaining conditions of Theorem 7. Clearly, 0 is the unique point in such that .

Corollary 9. Let be a metric space and and be mappings satisfying

(2.3.1), ,(2.3.2) for all and ,(2.3.3) is a complete subspace of ,(2.3.4)the pairs or is -weakly compatible.

Then there exists a unique such that .

Corollary 10. Let be a complete metric space and be mappings satisfying

(2.4.1) for all and .

Then there exists a unique such that .

Remark 11. Corollaries 9 and 10 are slight variant of theorems of [2, 3].

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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