Abstract
In this paper, we introduce a new type of a generalized--Meir-Keeler contractive mapping and establish some interesting theorems on the existence of fixed points for such mappings via admissible mappings. Applying our results, we derive fixed point theorems in ordinary metric spaces and metric spaces endowed with an arbitrary binary relation.
MSC:47H10, 54H25.
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1 Introduction
Fixed points and fixed point theorems have countless applications and have become a major theoretical tool in many fields such as differential equations, mathematical economics, game theory, dynamics, optimal control, functional analysis, and operator theory. In particular, the well-known Banach contraction principle is one of the forceful tools in nonlinear analysis, which states that every contraction self-mapping T on complete metric spaces (i.e., for all , where ) has a unique fixed point. Due to its simplicity and importance, this classical principle has been generalized by several authors in different directions.
In 1969 Meir and Keeler [1] established a fixed point theorem in a metric space for mappings satisfying the condition that for each there exists such that
for all . This condition is called the Meir-Keeler contractive type condition. Afterward, many authors extended and improved this condition and established fixed point results for new generalized conditions, see Maiti and Pal [2], Park and Rhoades [3], Mongkolkeha and Kumam [4] and others.
On the other hand, Samet et al. [5] introduced the notions of α, ψ contractive and α-admissible mapping in metric spaces. They also proved a fixed point theorem for α, ψ contractive mappings in complete metric spaces using the concept of α-admissible mapping.
Theorem 1 ([5])
Let be a complete metric space and be -contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
Theorem 2 ([5])
Let be a complete metric space and be -contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as then for all n.
Then, there exists such that .
These results can be used as an efficient tool to study various fixed point results such as fixed point results in partially ordered spaces, fixed point results for cyclic mappings, multidimensional fixed point results (coupled fixed point results, tripled fixed point results, quadrupled fixed point etc.). Moreover, such type of fixed point results are helpful to solve several problems and equations like the boundary value problem, differential equations, nonlinear integral equations etc. In the recent literature, a wide-ranging discussion of fixed point theorems for admissible mappings had the interest of several mathematicians, for example, see [6–12].
In this paper, we introduce new type of contractive mapping based on Meir-Keeler type contractive condition. For such mappings, we study and establish fixed point theorems via admissible mappings. Moreover, we present some applications of our new results.
2 Preliminaries
In the sequel, ℕ denote the set of positive integers. Let Ψ stands for the family of nondecreasing functions such that for each , where is the n th iterate of ψ.
Remark 3 For every function the following holds:
if ψ is nondecreasing, then for each ,
Therefore, if , then for each , and .
Example 4 Let be defined in the following way:
It is clear that . Notice that , are examples of continuous and discontinuous functions in Ψ.
Definition 5 ([5])
Let be a metric space and be a given mapping. We say that T is an -contractive mapping if there exist two functions and such that
for all .
Remark 6 If satisfies the Banach contraction principle in a metric space , then T is an -contractive mapping, where for all and for all , where .
Definition 7 ([5])
Let and . We say that T is α-admissible when if such that then we have .
Example 8 Let . Define and by for all and
Then, T is α-admissible.
Example 9 Let . Define and by for all and
Then T is α-admissible.
Remark 10 ([5])
Every nondecreasing self-mapping T is α-admissible.
3 Main results
In this section, introducing the class of -Meir-Keeler contractive mappings and the class of generalized--Meir-Keeler contractive mappings, we study the existence of fixed points for mappings via admissible mappings.
Definition 11 Let be a metric space and . The mapping T is called an -Meir-Keeler contractive mapping if there exist two functions and satisfying the following condition:
for each there exists such that
Remark 12 It is easily shown that if is an -Meir-Keeler type contraction, then
for all when . Also, if then and thus . Therefore, if is a -Meir-Keeler type contraction, then
for all .
Definition 13 Let be a metric space and . The mapping T is called a generalized--Meir-Keeler contractive mapping if there exist two functions and satisfying the following condition:
for each there exists such that
where
Remark 14 If is a generalized--Meir-Keeler type contraction, then
for all when . Also, if then and thus . Therefore, if is a generalized--Meir-Keeler type contraction, then
for all .
Theorem 15 Let be a complete metric space and be a generalized--Meir-Keeler contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
Proof Let be such that . Note that such a point exists due to condition (ii). We define the sequence in X by for all . If for some , then clearly is a fixed point of T. Hence, throughout the proof, we suppose that for all . Since T is α-admissible, we have
By induction, we obtain
From (3.7) and Remark (14), it follows that for all , we have
If , from (3.8) and Remark 3, we have
which is a contradiction. So we have . From (3.8), we get
for all . Inductively, for each , we obtain
Now we show that is a Cauchy sequence. Take and in such a way that . Let with . Due to the triangle inequality, we have
Hence, we conclude that is a Cauchy sequence in the complete metric space . Thus, there exists such that . Since T is continuous,
that is, u is a fixed point of T. This completes the proof. □
Corollary 16 Let be a complete metric space and be an -Meir-Keeler contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
We obtain the following fixed point result without any continuity assumption for the mapping T.
Theorem 17 Let be a complete metric space and be a generalized--Meir-Keeler contractive mapping such that ψ is continuous. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii′)
if is a sequence in X such that for all n and as then for all n.
Then there exists such that .
Proof Following the proof of Theorem 15, we obtain the sequence in X defined by for all and which converges for some . From (3.7) and condition (iii), we have for all . Next, we suppose that . Applying Remark 12, for each , we have
Letting in the above equality and keeping the continuity of ψ in mind, we get
which is a contradiction. Thus, we have , that is, . Therefore, u is a fixed point of T. This completes the proof. □
In the next corollary, we can omit the continuity hypothesis at every point of ψ.
Corollary 18 Let be a complete metric space and be an -Meir-Keeler contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as then for all n.
Then there exists such that .
Proof Since the proof of the existence of a fixed point is straightforward by following the same lines as in the proof of Theorem 17, in order to avoid repetition, the details are omitted. □
Now, we present the following example in support of our main result.
Example 19 Let with the usual metric . Then becomes a metric on . Define as follows: where is defined by
for all . Let us take
It is clear that is an -Meir-Keeler contractive mapping with . Indeed, for all , we have
On the other hand, by elementary calculations and taking into account that is α-admissible, we get
with .
Remark 20 Note that the main theorem of [11] is not applicable to this example. Hence, our result is stronger than the main result of [11]. Indeed, we take in the statement of the mentioned theorem, we get
On the other hand, by elementary calculations, we get
Thus, we get
which is a contradiction.
4 Applications
4.1 Fixed point results on an ordinary metric space
We have the following fixed point results in ordinary metric space.
Theorem 21 Let be a complete metric space and be continuous mapping and there exists satisfying the following condition:
for each there exists such that
where
Then there exists such that .
Proof Consider the mapping defined by
From the definition of α, it easy to see that T is α-admissible and also it is a generalized--Meir-Keeler contractive mapping. Moreover, all the hypotheses of Theorem 15 (or Theorem 17) are satisfied and so the existence of the fixed point of T follows from Theorem 15 (or Theorem 17). □
Taking , where , we get the following result.
Corollary 22 Let be a complete metric space and be continuous mapping satisfying the following condition:
for each there exists such that
for all , where and
Then there exists such that .
4.2 Fixed point results on a metric space endowed with an arbitrary binary relation
In this section, we present fixed point theorems on a metric space endowed with an arbitrary binary relation. The following notions and definition are needed.
Let be a metric space and ℛ be a binary relation over X. Denote
this is the symmetric relation attached to ℛ. Clearly,
Definition 23 We say that is a comparative mapping if T maps comparable elements into comparable elements, that is,
for all .
Definition 24 Let be a metric space, be a symmetric relation attached to binary relation ℛ over X and . The mapping T is called a generalized-ψ-Meir-Keeler contractive mapping with respect to if there exists a function satisfying the following condition:
for each there exists such that for for which ,
where
Theorem 25 Let be a complete metric space, ℛ be a binary relation over X and be a comparative generalized--Meir-Keeler contractive mapping. Suppose that
-
(i)
T is comparative mapping;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
Proof Consider the mapping defined by
From condition (ii), we get . It follows from T is comparative mapping that T is an α-admissible mapping. Since T is a generalized-ψ-Meir-Keeler contractive mapping with respect to , we have, for all ,
This implies that T is a generalized--Meir-Keeler contractive mapping. Now all the hypotheses of Theorem 15 are satisfied and so the existence of the fixed point of T follows from Theorem 15. □
In order to remove the continuity of T, we need the following condition:
() if is the sequence in X such that for all and it converges to the point , then .
Theorem 26 Let be a complete metric space, be a symmetric relation attached to binary relation ℛ over X and be a generalized--Meir-Keeler contractive mapping with respect to such that ψ is continuous. Suppose that
-
(i)
T is comparative mapping;
-
(ii)
there exists such that ;
-
(iii)
the condition () holds.
Then there exists such that .
Proof The result follows from Theorem 17 by considering the mapping α given by (4.4) and by observing that condition () implies condition (iii′). □
References
Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6
Maiti M, Pal TK: Generalizations of two fixed point theorems. Bull. Calcutta Math. Soc. 1978, 70: 57–61.
Park S, Rhoades BE: Meir-Keeler type contractive conditions. Math. Jpn. 1981,26(1):13–20.
Mongkolkeha C, Kumam P: Best proximity points for asymptotic proximal pointwise weaker Meir-Keeler-type ψ -contraction mappings. J. Egypt. Math. Soc. 2013, 21: 87–90. 10.1016/j.joems.2012.12.002
Samet B, Vetro C, Vetro P: Fixed point theorem for α - ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Agarwal RP, Sintunavarat W, Kumam P: PPF dependent fixed point theorems for an -admissible non-self mapping in the Razumikhin class. Fixed Point Theory Appl. 2013., 2013: Article ID 280
Hasanzade Asl J, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212
Hussain N, Karapınar E, Salimi P, Vetro P: Fixed point results for -Meir-Keeler contractive and -Meir-Keeler contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 34
Kutbi MA, Sintunavarat W: The existence of fixed point theorems via w -distance and α -admissible mappings and applications. Abstr. Appl. Anal. 2013., 2013: Article ID 165434
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24
Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Sintunavarat W, Plubtieng S, Katchang P: Fixed point result and applications on b -metric space endowed with an arbitrary binary relation. Fixed Point Theory Appl. 2013., 2013: Article ID 296
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR for technical and financial support. The authors thank the referees for their valuable comments and suggestions.
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Latif, A., Gordji, M.E., Karapınar, E. et al. Fixed point results for generalized -Meir-Keeler contractive mappings and applications. J Inequal Appl 2014, 68 (2014). https://doi.org/10.1186/1029-242X-2014-68
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DOI: https://doi.org/10.1186/1029-242X-2014-68