Edelstein type fixed point theorems

In this manuscript, we investigate certain conditions that imply the existence of fixed points for almost contraction mappings defined on compact metric spaces. Furthermore we introduce a criteria establishing the uniqueness of fixed points for the mentioned operators. As a result we obtain generalized results by unifying some recent related fixed point theorems on the topic.

In nonlinear functional analysis, fixed point theory is being investigated increasingly by reason of the fact that it has a wide range of applications in fields such as economics (see e.g.[1,2]), computer science (see e.g.[3-7]), and many others. One of the pioneering theorems in this direction is the Banach contraction mapping principle [8] which states that each contraction defined on a complete metric space X has a unique fixed point. Banach’s result is the origin and antecedents results by the fact that he not only proved the existence and uniqueness of a fixed point of a contraction, but also showed how to evaluate this point. After this celebrated result[8], a number of authors have observed various other types of contraction mappings and proved related fixed point theorems (see e.g. such as Kannan [9], Reich [10], Hardy and Rogers [11], Ćirić [12-14], Zamfirescu [15], Arshad et al.[16]). By following this trend Suzuki recently proved the following fixed point theorems:

Theorem 1 (Suzuki [17])

Letbe a compact metric space and letTbe a mapping on X. Assume thatimpliesfor all. ThenThas a unique fixed point.

Theorem 2 (Suzuki [18])

Define a non-increasing functionθfrom [0,1) onto (1/2,1] by

Then for a metric space, the followings are equivalent:

1. Xis complete;

2. Every mappingTonXsatisfying the following has a fixed point: There existssuch thatimpliesfor all.

In the literature Theorem 1 and Theorem 2 attracted considerable attention from many authors (see e.g.[19-23]). Notice that these theorems are inspired by Edelstein’s Theorem [24]:

Theorem 3Letbe a compact metric space and letTbe a mapping onX. Assumefor allwith. ThenThas a unique fixed point.

Motivated by these developments in this area, in this manuscript, we combine well-known results of Suzuki [17], Edelstein [24] and Berinde [25] to complement a multitude of related results from the literature. For the sake of completeness we include the results of Berinde as well:

Theorem 4 (See [25])

Letbe a complete metric space andbe an almost contraction, that is, a mapping for which there exist a constantand somesuch that

for all. Then.

Theorem 5 (See [25])

Letbe a complete metric space andbe an almost contraction, that is, a mapping for which there exist a constantand somesuch that

for all. Then has a unique fixed point.

We start this section by proving the following theorem:

Theorem 6LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a fixed point, that is, .

Proof Set and choose a sequence in X such that . Regarding that X is compact, without loss of generality, assume that and converge to the points z and w in X, respectively.

We claim that θ is equal to zero. To show this, assume to the contrary that . Observe that we have

We can choose in such a way that

for each . As a consequence, we have for each . Due to (3), we get for each . Accordingly, we obtain

which implies that

By taking the definition of θ into account, we conclude that . Notice that the inequality always holds. By applying the condition (3) again, we find

which is equivalent to the inequality . This contradicts with the definition of θ. Hence, we conclude that .

We next assert that T has a fixed point. We use the method of Reductio ad absurdum to show this assertion. Suppose that T has no fixed points. Since the inequality holds for each n, we derive, for every , that

Hence, we find that

for each and

Thus, we get . In other words, and converge to the same point. Due to the triangular inequality and (9), we obtain that

Hence, too converges to z.

Assume that

We use (9), (12) and the triangular inequality, we find that

This is a contradiction. Thus, either

holds for each . Then regarding (3), one of the below holds:

This is equivalent to stating that either

(i) there is an infinite subset I of ℕ so that the inequality (14) holds for all , or,

(ii) there is an infinite subset J of ℕ so that the inequality (15) holds for all .

We first consider the case (14). The inequality

yields that

Thus, we conclude that . For the other case in (15) we get

which implies that

Thus, we reach the conclusion again. This contradicts with the assumption that T has no fixed point. Hence, T has a fixed point. □

Corollary 7LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a unique fixed point, that is, .

Proof The proof of Theorem 6 applies, mutatis mutandis, to show the existence of a fixed point. Let be a fixed point of T.

We shall prove z is the unique fixed point of T. Suppose, to the contrary that, there exists so that and . Then the inequalities and are satisfied. Due to (3), we have

which is a contradiction. Hence, z is the unique fixed point of T. □

Theorem 8LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a fixed point, that is, .

Proof The proof of Theorem 6 applies, mutatis mutandis, to prove Theorem 8. □

Corollary 9LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a unique fixed point, that is, .

Proof The proof of Corollary 7 applies, mutatis mutandis, to prove Corollary 9. □

Theorem 10LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a fixed point, that is, .

Proof As in the proof of Theorem 6, we set and choose a sequence in X such that . Since X is compact, without loss of generality, we assume that and converge to the points z and w in X, respectively.

We aim to show that θ is equal to zero. Let assume the contrary. Recall that

It is possible to choose in such a way that

for each . Consequently, we see that for each . By (21), we derive that

for each . Then it follows that

which implies that

Hence, we find that . As a result, we conclude that when we take the definition of θ into account. Since we always have the inequality , we obtain

by applying (21). But this is equivalent to stating that . This contradicts with the definition of θ. So, we find .

We are ready to show that T has a fixed point. We shall use the method of Reductio ad absurdum again. Suppose that T has no fixed point. Since the inequality is true for each n, the expression

holds for every . In other words, we see that the inequality

is satisfied for each . Therefore, we infer that

So, we also find . Thus, the sequences and converge to the same point. By the triangular inequality, together with the inequality (27), we derive

Hence, also converges to z. Assume that

Regarding (27), (30) and the triangular inequality, we find

This is a contradiction. Thus, we have either

for each . By (21), we conclude that one of the inequalities below

is satisfied. This is equivalent to phrasing that either

(a) there is an infinite subset I of ℕ so that

(b) there is an infinite subset J of ℕ so that

for all holds.

Considering the case (32), we find

which yields that

Thus, we conclude that . For the other case (33), we obtain

which implies

Thus, we reach the same conclusion, that is, . This contradicts with assumption that T has no fixed point. Hence, T has a fixed point, say . □

Corollary 11LetTbe a self mapping on a compact metric space. Assume that

for allwithand. Then, Thas a unique fixed point, that is, .

Proof The proof of Corollary 13 follows, mutatis mutandis, from the proofs of Corollary 7, Corollary 9 and Corollary 11. Therefore T has a fixed point, say .

We need to prove z is the unique fixed point of T. Suppose, to the contrary that, there exists such that and . Then, we have and . By (21), we see that

which in turn implies that . So y is not a fixed point of T. Hence, z is the unique. □

Combining Theorem 6, Theorem 8 and Theorem 10 yields the following:

Theorem 12LetTbe a self mapping on a compact metric space. Assume that

for all. Then, Thas a fixed point, that is, .

Combining Corollary 7, Corollary 9 and Corollary 11 yields the following:

Corollary 13LetTbe a self mapping on a compact metric space. Assume that

for all. Then, Thas a unique fixed point, that is, .

The result below is a corollary of Theorem 6-Theorem 12:

Theorem 14LetTbe a self mapping on a compact metric space. Assume that

for all. Then, Thas a fixed point, that is, .

Proof The proof of Theorem 12 follows from the proofs of the previous theorems verbatim. □

Corollary 15LetTbe a self mapping on a compact metric space. Assume that

for all. Then, Thas a unique fixed point, that is, .

Proof The proof of Corollary 13 follows from the proofs of the previous theorems verbatim. □

Example 16 (cf.[17])

Let and d be the discrete metric

Each self-mapping T on X satisfying (18) has a unique fixed point. It is clear that is complete, but it is not a compact metric space. Let T be a self-mapping on X. If T has a fixed point, it is sufficient to prove that it is unique.

To show that is the unique fixed point of T, we take where . Thus the inequalities and are satisfied. Due to (18), we have

which implies that . So y is not a fixed point of T. Hence, z is the unique fixed point of T. Suppose T has no fixed point. Then, we have

Due to (18), the inequality holds. In other words, we get . Thus the image of T on the domain X consists of only one point which is clearly a unique fixed point. This is a contradiction.

Remark 17 Example 16 can be modified for Theorem 8-Theorem 10 just by replacing the condition (3) with the relevant one. It is clear that proofs are obtained by apply the necessary manipulations in Example 16.

The following theorem is a generalization of [[17], Theorem 5].

Theorem 18LetTbe a self mapping on a metric space. Suppose that there existand aT-invariant complete subsetKofXsuch that

for allwith, andwithand. Then, Thas a unique fixed point, that is, .

Proof Due to Banach [8], there exists a unique fixed point . Consider

for all which implies that

In other words, for all is not a fixed point of T. Hence, u is the unique fixed point of T on X. □

Remark 19 Theorem 18 holds also if we replace one of the conditions below instead of the condition (41):

The author declares that they have no competing interests.

The author express his gratitude to the referees for constructive and useful remarks and suggestions.

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