- Open Access
On α-ψ-Meir-Keeler contractive mappings
© Karapınar et al.; licensee Springer 2013
- Received: 23 December 2012
- Accepted: 27 March 2013
- Published: 12 April 2013
In this paper, we introduce the notion of α-ψ-Meir-Keeler contractive mappings via a triangular α-admissible mapping. We discuss the existence and uniqueness of a fixed point of such a mapping in the setting of complete metric spaces. We state a number of examples to illustrate our results.
MSC:46N40, 47H10, 54H25, 46T99.
- Meir-Keeler contractive mappings
- triangular α-admissible mappings
- fixed points
Fixed-point theory is one of the most intriguing research fields in nonlinear analysis. The number of authors have published papers and have increased continuously in the last decades. The main reason for this involvement can be observed easily: Application potential. Fixed point theory has an application in many disciplines such as chemistry, physics, biology, computer science and many branches of mathematics. Banach contraction mapping principle or Banach fixed-point theorem is the most celebrated and pioneer result in this direction: In a complete metric space, each contraction mapping has a unique fixed point. Following Banach , many authors give various generalizations of this principle in various space (see e.g. [2–20]). One of the interesting results was given by Samet et al.  by defining α-ψ-contractive mappings via admissible mappings, see also .
In this paper, we introduce an α-ψ-Meir-Keeler contractive mapping in the setting of complete metric spaces via a triangular α-admissible mapping. We prove the existence and uniqueness of a fixed point of such a mapping. We also consider a number of examples to illustrate our results.
implies , ,
implies , .
Example 2 Let , and then f is a triangular α-admissible mapping. Indeed, if then which implies . That is, . Also, if then That is, and so .
Example 3 Let , and . Hence, f is a triangular α-admissible mapping. Again, if then which implies . That is, .
Moreover, if then , and hence, .
Thus, . Now, if then .
Example 5 Let , and . Then f is a triangular α-admissible mapping.
Hence, f is a triangular α-admissible mapping.
Suppose that . Since then from (T2) we have .
Again, since then we deduce .
By continuing this process, we get . □
Denote with Ψ the family of nondecreasing functions continuous in such that
if and only if ,
for all . Then f is called an α-ψ-Meir-Keeler contractive mapping.
for all .
Theorem 10 Let be a complete metric space. Suppose that f is a continuous α-ψ-Meir-Keeler contractive mapping and that there exists such that , then f has a fixed point.
and hence (2.5) holds.
that is, f has a fixed point. □
there exists such that ,
if is a sequence in X such that for all n and as , then for all n.
Then f has a fixed point.
By taking limit as , in the above inequality, we get , that is, . Hence, . □
Next, we give some examples to validate our main result.
and . Clearly, is a complete metric space. We show that f is a triangular α-admissible mapping. Let , if then . On the other hand, for all we have and . It follows that . Also, if and then and hence, . Thus, the assertion holds by the same arguments. Notice that .
That is, the Meir-Keeler theorem cannot applied for this example.
Clearly, by taking the condition (2.1) holds. Otherwise, . Hence, for given we have . Hence, conditions of Theorem 11 holds and f has a fixed point.
Denote with the family of strictly nondecreasing functions continuous in such that
if and only if ,
Then f is called generalized an α--Meir-Keeler contractive mapping.
for all .
Proposition 16 Let be a metric space and a generalized α--Meir-Keeler contractive mapping, if there exists such that . Then .
which is a contradiction since . Then and so . □
Theorem 17 Let be a complete metric space and a orbitally continuous generalized α--Meir-Keeler contractive mapping, if there exist such that . Then, f has a fixed point.
which is a contradiction. We obtained that and so is a Cauchy sequence. Since, X is complete, then there exists such that as . Now, since f is orbitally continuous, then . □
Clearly, by taking the condition (2.8) holds. Otherwise, . Hence, for given , we have . Hence, condition of Theorem 17 is held and f has a fixed point.
for all there exists such that and ,
we obtain the uniqueness of the fixed point of f.
which is a contradiction and so . Similarly, for Theorem 17, we can observe that f has a unique fixed point. □
We can obtain the following corollaries intermediately.
where and . Then f has a unique fixed points.
Then f has unique fixed points.
The authors would like to thank the referees for their valuable comments and suggestions. Also, the second author would like to thank the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No. MRG5580213) for financial support during the preparation of this manuscript. The third author is thankful for support of Astara Branch, Islamic Azad University, during this research.
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