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Meir-Keeler ɑ-contractive fixed and common fixed point theorems
Fixed Point Theory and Applicationsvolume 2013, Article number: 19 (2013)
The Erratum to this article has been published in Fixed Point Theory and Applications 2013 2013:110
Generalized Meir-Keeler α-contractive functions and pairs are introduced and their fixed and common fixed point theorems are obtained. Also, the so-called generalized Meir-Keeler α-f-contractive maps commuting with f are introduced and their coincidence and common fixed point theorems are investigated. New sufficient conditions different from those in (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012) are used. An application to the coupled fixed point is established as well. An example is given to show that the α-Meir-Keeler generalization is real.
AMS Subject Classification: 47H10, 54H25.
Fixed point theory is of wide and endless applications in many fields of engineering and science. Its core, the Banach contraction principle, has attracted many researchers who tried to generalize it in different aspects. Some dealt with the contractive condition itself, of worth mentioning Meir-Keeler contractive type [1–4], some extended it to more generalized metric-type spaces [5–11] and others applied to common , coupled and tripled versions (see [13, 14] and the references therein). In 1969 Meir and Keeler  established a fixed point theorem in a metric space for mappings satisfying the following condition, called the Meir-Keeler type contractive condition:
In 1978 Maiti and Pal  generalized a fixed point for maps satisfying the following condition:
Later in 1981, Park and Rhoades in  established fixed point theorems for a pair of mappings f, g satisfying a contractive condition that can be reduced to the following generalization of (2) when .
In this article we develop the fixed point theorems for α-contractive type maps introduced recently in  (for the α-ψ-contractive multivalued case, see ) to Meir-Keeler versions and hence generalize the results obtained in  and the references therein. Then, we apply part of our results to the coupled case on the basis of Amini-Harandi .
2 Fixed and common fixed point theorems for generalized Meir-Keeler α-contractive maps and pairs
The first part of the following definition was introduced in .
Definition 1 Let be self-mappings of a set X and be a mapping, then the mapping f is called α-admissible if
and the pair is called α-admissible if
Example 2 Let and
Then the pair is α-admissible but the pair is not α-admissible.
Definition 3 Let be a metric space and be a self-mapping, be a mapping. Then f is called Meir-Keeler α-contractive if, given an , there exists a such that
Definition 4 Let be a metric space and be a self-mapping, be a mapping. Then f is called generalized Meir-Keeler α-contractive if, given an , there exists a such that
Definition 5 Let be a metric space and be self-mappings, be a mapping. Then the pair is called generalized Meir-Keeler α-contractive if, given an , there exists a such that
We write .
Clearly, f is generalized Meir-Keeler α-contractive if and only if is generalized Meir-Keeler α-contractive.
Definition 6 Let X be any set, and f, g be self-maps of X. Define and , . Then is called the -orbit of . If d is a metric on X, then is called -orbitally complete if every Cauchy sequence in the -orbit of is convergent and the map f or g is called orbitally continuous if it is continuous on the orbit.
The proof of the following lemma is immediate.
Lemma 7 Let be self-mappings of a set X, be a mapping and be the -orbit of with . If the pair is α-admissible, then for all .
Theorem 8 Let be an -orbitally complete metric space, where f, g are self-mappings of X. Also, let be a mapping. Assume the following:
is α-admissible and there exists an such that ;
the pair is generalized Meir-Keeler α-contractive.
Then the sequence is monotone decreasing. If, moreover, we assume that
on the -orbit of , we have for all n even and odd and that f and g are continuous on the -orbit of .
Then either (1) f or g has a fixed point in the -orbit of or (2) f and g have a common fixed point p and . If, moreover, we assume that the following condition (H) holds: If is a sequence in X such that for all n and implies for all n, then uniqueness of the fixed point is obtained.
Proof Define for . If for some even integer n, then f has a fixed point. If for some odd integer n, then g has a fixed point. Hence, we may assume that for each n. The fact that the pair is generalized Meir-Keeler α-contractive implies that
Note that assumption (3) implies that . Hence, since is α-admissible, then Lemma 7 implies that for all and hence by (5), we have
whence . □
Similarly, it can be shown that . Thus, is monotone decreasing in n and converges to a limit, say ϱ.
Suppose . Then, for each , there exists a positive integer such that , where N can be chosen even. Thus, from assumption (1) and Lemma 7, we have , a contradiction. Therefore, . To show that is Cauchy, we assume the contrary. Thus, there exists an such that for each integer N, there exist integers such that . Define ϵ by . Choose a number δ, , for which (4) is satisfied. Since , there exists an integer such that for . With this choice of N, pick integers such that
in which it is clear that . Otherwise, , contradicting (7). Without loss of generality, we may assume that n is even since from (7) it follows that . From (7) there exists the smallest odd integer such that
Hence, , and so . Therefore, we have
so that, by (7) and assumption (3), . Then we have
This contradicts the choice of j in (8). Therefore, is Cauchy.
Since X is -orbitally complete, converges to some point . Since f and g are orbitally continuous, then p is a common fixed point of f and g. To prove uniqueness, assume p is the common fixed point obtained as and q is another common fixed point. Then (5) and the condition (H) yield
If we let , then we reach , which implies that .
Corollary 9 Let be an f-orbitally complete metric space, where f is a self-mapping of X. Also, let be a mapping. Assume the following:
f is α-admissible and there exists an such that ;
f is generalized Meir-Keeler α-contractive.
Then the sequence is monotone decreasing. If, moreover, we assume that
on the f-orbit of , we have for all n even and odd.
Then either (1) f has a fixed point in the f-orbit of or (2) f has a fixed point p and . If, moreover, we assume that the following condition (H) holds: If is a sequence in X such that for all n and , then for all n, then uniqueness of the fixed point is obtained.
Since generalized Meir-Keeler α-contractions are Meir-Keeler α-contractions, then Corollary 9 is valid also for Meir-Keeler α-contractions. In the following example, the existence and uniqueness of the fixed point cannot be proved in the category of Meir-Keeler contractions, but can be proved by means of Corollary 9.
Example 10 Let with the absolute value metric . Define by
Then, for , and any , we have implies and hence . Hence, f is not a Meir-Keeler contraction. However, f is a Meir-Keeler α-contraction, where
Indeed, for (the case is trivial, since ), let , then implies that and hence . Also, notice that f is continuous on the orbit of and that for all n, j. Clearly, is the unique fixed point.
Remark 11 Note that the admissibility condition (1) in Theorem 8 is not enough to proceed to guarantee the existence of the fixed point. However, such an admissibility condition was used in obtaining the main result in Theorem 2.2 of .
3 Generalized Meir-Keeler α-f-contractive fixed points
Definition 12 Let f be a continuous self-map of a metric space , , the sequence defined by , , with the understanding that if for some n, then for each is called the f-iteration of under g.
Definition 13 Let f be a self-map of a metric space and . Then g is called a Meir-Keeler α-f-contractive map if for each , there exists a such that for all ,
Definition 14 Let f be a self-map of a metric space and . Then g is called a generalized Meir-Keeler α-f-contractive map if for each , there exists a such that for all ,
Lemma 15 Let f, g be continuous self-maps of a metric space such that . Assume g is a generalized Meir-Keeler α-f-contractive map such that for all n. Then .
Proof Let and . From the definition of the f-iteration of under g and from the fact that g is a generalized Meir-Keeler α-f-contractive map, for each n, we have
Hence, and is monotone decreasing so that . From the assumption that g is a Meir-Keeler α-f-contractive map, for , find such that (10) is satisfied. For the chosen δ, pick N so that . Noting that for and , , we by (10) conclude that . But , a contradiction. □
Theorem 16 Let f, g be continuous self-maps of a metric space such that . Assume for all . If g is a generalized Meir-Keeler α-f-contractive map such that α satisfies the condition (f-H): If is a sequence in X such that for all and , then and for all n. Then f and g have a unique common fixed point.
Proof Let for which its f-iteration under g satisfies the assumptions of the theorem. The proof will be divided into four steps.
Step 1: By Lemma 15, .
Step 2: We find a coincidence point for f and g. That is to find a such that . If there exists an n such that , then , and we are finished. Hence, we may assume that for each n. We claim to show that is Cauchy. Suppose not. Then there exists an and a subsequence of such that . From (10), there exists a δ satisfying for which (10) is true. Since =0, there exists an N such that
Let . We will show that there exists an integer j satisfying such that(11)
First of all, there exist values of j such that . For example, choose . The inequality is also true for . If not, then and hence
a contradiction. There are also values of j such that . For example, choose and . Pick j to be the smallest integer greater than such that . Then , and hence
Thus (11) is established. Now, note that
Then from the choice of j and the fact that , , we reach
On the other hand,
contradicting (11). Therefore, is Cauchy hence convergent to . Since , the continuity of f and g implies that .
Step 3: We show that is a common fixed point for f and g. Assume , then and by the help of the (f-H) condition, we have
If we let above and use continuity and commutativity of f and g, then we reach and hence . Moreover, .
Step 4: Uniqueness of the common fixed point. Assume is our common fixed point for f and g where and ω is another common fixed point. Then, by the (f-H) condition, we have
If we let above and use the continuity of f and g, we conclude that and hence .
Remark 17 Theorem 16 has been proved for commuting maps. It would be interesting to extend it for weakly commuting and compatible mappings and so forth. For example, can we extend the results in [20–22] to α-type contractions?
4 Application to coupled α-Meir-Keeler fixed points
Let be a mapping. We say that is a coupled fixed point of F if and . If we define by , then clearly is a coupled fixed point of F if and only if is a fixed point of T. If , then the F-orbit of means the orbit , where .
If is a metric space, then defined by is a metric on .
Theorem 18 Let be a complete metric space and be a continuous mapping. Also, let be a mapping. Assume the following:
For all , we have
Also, assume there exists such that and ;
For each , there exists such that
Then the sequence is monotone decreasing. If, moreover, we assume that
on the F-orbit of , we have and for all n, j.
Then either (1) F has a coupled fixed point in the F-orbit of or (2) F has a coupled fixed point and . If, moreover, we assume that the following condition (H) holds: If is a sequence in such that for all n and , , then and for all n, then uniqueness of the coupled fixed point is obtained.
Proof The proof will follow by applying Corollary 9, with as above, to the metric space . The controlling function will be given by
In fact, if is given, then by assumption (2), find such that
Let and assume . Then
which leads to
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The author declares that he has no competing interests.
An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2013-110.