Presic Type Fixed Point Theorem for Four Maps in Metric Spaces

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 equationwith 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 thatholds 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), defineLetLet and .
Then and by selection of we haveContinuing in this way, we getNow considerAlsoThus from (10) and (11) we haveNow from considerHence is a Cauchy sequence in . Since is complete there exists such that as . From (2.1.4), there exists such thatNow considerLetting and using (14) so thatSimilarly we haveSince and are jointly -weakly compatible pairs, we haveSimilarly, we haveFrom (20) and (21) we havewhich in turn yields thatFrom (18), (19), and (23), we haveSuppose there exists such that Then from (2.1.2) we haveThis 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 haveThus 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.