Abstract

In this paper we study and generalize some common fixed point theorems in compact and Hausdorff spaces for a pair of commuting mappings with new contraction conditions. The results presented in this paper include the generalization of some fixed point theorems of Fisher, Jungck, Mukherjee, Pachpatte and Sahu and Sharma.

1. Introduction

Fixed point theorey is a fascinating topic for research in modern analysis and topology. The study and research in fixed point theory began with the pioneering work of Banach ², who in 1922 presented his remarkable contraction mapping theorem popularly known as Banach contraction mapping principle. It has widespread applications in both pure and applied mathematics. In 1961, Edelstein ⁸ for the first time introduced the concept of contractive mapping defined on compact metric spaces. According to Edelstein "if is a continuous mapping of a compact metric space into itself satisfying for all then has unique fixed point in ". In 1976, Jungck ¹⁶ generalized contraction mapping theorem for a pair of commuting mappings. Later on several mathematicians have been generalized and improved The Banach contraction mapping theorem for fixed points in several different ways viz, Bondar ³, Browder ⁴, Chatterjee ⁵, Ciric ⁶ are a few to name. For more results in this direction, we refer to ^{12, 13, 19, 20, 22, 23} and references therein. In this paper we want to establish fixed point results in complete, compact and Hausdorff spaces for a pair of commuting mappings. The obtained results are generalizations of some fixed point theorems of Fisher ⁹, Jungck ¹⁶, Mukherjee ¹⁷, Pachpatte ¹⁸ and Sahu and Sharma ²¹.

The following fixed point theorems were proved in ^{9, 16, 17, 18} and ²¹.

Theorem 1.1. ⁹ Let T be a mapping of the complete metric space X into itself satisfying the inequality

and then has a fixed point in

Theorem 1.2. ¹⁶ Let f be a continuous mapping of the complete metric space into itself. Then f has a fixed point in X if and only if there exists and a mapping which commutes with f and satisfies and

then f and g have a common fixed point in X.

Theorem 1.3. ¹⁷ Let f and g be mappings of a complete metric space X into itself with f continuous. Let f and g commute with each other and Also, let g satisfiy the following conditions:

with for all i and then f and g have a unique common fixed point in X.

Theorem 1.4. ¹⁸ Let T be a mapping of the complete metric space X into itself satisfying the inequality

and such that then T has a unique fixed point.

Theorem 1.5. ²¹ Let T be a mapping of the complete metric space X into itself satisfy the conditions:

and such that then T has a unique fixed point in X.

2. Preliminaries

Definition 2.1. Let be a metric space. A mapping is said to be sequentially convergent if for every sequence of if is convergent then has a convergent subsequence.

Definition 2.2. Let be a continuous self-map of into itself. Then is said to be contractive if

Definition 2.3. If X is a non empty set and is a mapping satisfy the conditions:

(i) and if and only if

(ii)

(iii)

Then is called a metric on and the pair is called a metric space.

Definition 2.4. A point is said to be a fixed point of a self-map if

Definition 2.5. (i) A sequence in a metric space is said to converge to a point if for every there exists such that denoted by

(ii) is called Cauchy sequence if for some there exists such that for we have

(iii) A metric space is said to be complete if and only if every Cauchy sequence in converges to a point of

3. Main Results

In this section we prove our main results on the common fixed points of commuting self-mappings on complete, compact and Hausdorff spaces. We start with the following result.

Theorem 3.1. Let be a complete metric space and let be commuting self maps of into itself with continuous such that and satisfy the conditions

(3.1)

where such that

(3.2)

then f and g have a unique common fixed point in X.

Proof Let be an arbitrary point in then there exists such that we construct sequence in such that since with Now by using (3.1), we get

or

or

or

where On continous repeation of the above process, we get that

(3.3)

on taking limit as we get

Since X is complete, there exists such that

Since f is continuous and f, g commute, we have

also

Now by (3.1) we have

Now, since f is continuous, we have

which implies

On taking lim as and use continuity, we obtain

Since f and g commute, we have

This implies that

We now prove that is a common fixed point of f and g. By (3.1) we have

or

and the inequality is possible only if

Hence, we have

Uniqueness: To see that f and g have only one common fixed point. Suppose and By (3.1) we get

or

which is possible only if and thus we have which shows the uniqueness of fixed point of mappings f and

Corollary 3.2. Let be a complete metric space and be a self map of into itself such that satisfy the condition

where such that

then g has a unique fixed point in X.

Proof Put (Identity mapping) in Theorem (3.1) we get the required result.

Remark: If we put then we get theorem (1.3) in ¹⁷

Remark: If we put then we get theorem (1.1) in ¹⁶

Remark: If and we get Theorem (1.3) in ²¹

Example: Let is a non-empty set and is a metric on given by

If and on are given by Then and commute with each other such that and

with 0 as the only common fixed point of and .

Now we give the proof of the above Theorem(3.2) in the context of complete compact metric space.

Theorem 3.3. Let f and g be two mappings of a compact metric space with f is continuous. Let f and g commute with each other such that and g satisfy the following condition:

(3.4)

and with

(3.5)

then f and g have a unique common fixed point in X.

Proof Let be an arbitrary point of X. Since there exists such that Continuing this process we get a sequence in X such that

Now from (3.4) we get

or

or

and Hence the sequence is a monotonic decreasing sequence of non-negative real number and so converges to a limit such that

Since X is compact, using sequential compactness of X, there exists a subsequence of such that for any and we get

Now, we use the continuity of g and d to obtain

Since we have Similarly,

(3.6)

Since the sequence is a subsequence of the sequence we get

Next we claim Suppose then By (3.6) we obtain

which is contradiction, hence d = 0. Hence,

(3.7)

Since for each From (3.7), we get

(3.8)

Since the mapping defined by is continuous, there exists such that

By (3.8) and so Now, since f and g commute, we have

Thus

(3.9)

and so v is a common fixed point of f and g.

Uniqueness: Next we claim that v is the unique common fixed point of f and g. Suppose on the contrary that there exists another point such that with Using condition (3.4) we get

which implies that or and this gives the uniqueness of the fixed point.

Corollary 3.4. Let be a compact metric space and let be self mappings of satisfying the condition

where with then f and g have unique common fixed point in X.

Proof Putting and in Theorem (3.3) we get the required result.

Corollary 3.5. Let be a complete compact metric space and be a self mapping satisfying

for all and such that then g has a unique fixed point in

Proof Putting and (Identity mapping) in Theorem (3.8), we get the required result.

Corollary 3.6. Let be a compact metric space and be a self map of into itself satisfying the inequality

for all such that then g has a unique fixed point in X.

Proof Putting in Theorem (3.8) we get the required result

Remark: Corollary (3.4) is the result of Sahu and Sharma in ²¹

Remark: Corollary (3.5) is the result of Pachpatte ¹⁸

Remark: Corollary (3.6) is the result of Fisher ⁹

Now we give an example to support the validity of the above theorem (3.3).

Example: Let be any non-empty set and let d be the metric with ordinary distance. Let the functions f and g on X be defined by

Then it is clear that with f and g commute, continuous and is a compact metric space. Now it is easy to show that the above example satisfy all the conditions of Theorem (3.3) with 9 as the only common fixed point in X.

Theorem 3.7. Let f and g be two continuous mappings of a hausdorff space X into itself and let f, g commute with each other such that Let be a continuous function such that for each pair of with

(3.10)

and such that If for some the sequence in X has a convergent subsequence. Then f, g have a common fixed point.

Proof Since So, for we choose such that with the sequence defined by

Now by (3.10) we have

or

(3.11)

Let Then (3.11) gives us

Similarly,

Since

Repeating the above process, we get

This shows that the sequence is bounded which converges along with all its subsequences to some positive real number If has a convergent subsequence of which converges to the real number Then,

Next we show that z is a fixed point of f and g. First we show that z is a fixed point of g. Suppose, then by (3.5) we have

or

which is contradiction because Hence, Thus, is a fixed point of g. Since f and g commute and are continuous, we have

By the uniqueness of limit we have

Uniqueness: Now we claim that z is the unique common fixed point of f and g. Suppose for contradiction that w is another fixed point of f and g such that then by (3.10) we obtain

This implies that which is contradiction because

Hence, Hence z is the unique common fixed point of f and g.

Finally, we provide example to check the validity of Theorem (3.7).

Corollary 3.8. Let be a continuous mappings of a hausdorff space X into itself and let be a continuous function such that for each pair of with

and such that

If for some the sequence in has a convergent subsequence. Then f has a unique fixed point in X.

Proof Put (Identity mapping) in Theorem (3.7) we get the required result.

Corollary 3.9. Let f and g be two continuous mappings of a hausdorff space X into itself and let f, g commute with each other such that Let be a continuous function such that for each pair of with

and such that If for some the sequence in has a convergent subsequence. Then f, g have a common fixed point.

Proof Put in Theorem (3.7) we get Corollary (3.9).

Remark: Corollary (3.8) is the result of ¹⁷

Remark: Corollary (3.9) is the result of ⁵

Example: Let We define and by

Now, it is clear that the conditions of Theorem (3.7) are satisfied and that 5 is the common fixed point of f and g.

References