Abstract
In this paper, the notion of -contractions is introduced and a new fixed point theorem for such contractions is established.
1. Introduction and Preliminaries
Branciari [1] introduced the notion of generalized metric spaces and obtained a generalization of the Banach contraction principle, whereafter many authors proved various fixed point results in such spaces, for example, [2–8] and references therein. Also, Suzuki et al. [9] and Abtahi et al. [10] studied -generalized metric spaces and proved the Banach and Kannan contraction principles in such spaces, and Mitrović et al. [11] introduced the notion of -generalized metric spaces and Banach and Reich contraction principles in such spaces.
In particular, Jleli and Samet [12] introduced the notion of -contractions and gave a generalization of the Banach contraction principle in generalized metric spaces, where is a function satisfying the following conditions: is nondecreasing;, :
Also, Ahmad et al. [13] extended the result of Jleli and Samet [12] to metric spaces by applying the following simple condition (4) instead of (3). is continuous on .
Recently, Khojasteh et al. [14] introduced the notion of -contractions by defining the concept of simulation functions. They unified some existing metric fixed point results. Afterward, many authors ([15–19] and references therein) obtained generalizations of the result of [14].
In the paper, we introduce the concept of a new type of contraction maps, and we establish a new fixed point theorem for such contraction maps in the setting of generalized metric spaces.
Let be the family of all mappings such that;;for any sequence with
We say that is a -simulation function.
Note that .
Example 1. Let be functions defined as follows, respectively:(1), where ;(2), where is nondecreasing and lower semicontinuous such that ; , where . Then .
We recall the following definitions which are in [1].
Let be a nonempty set, and let be a map such that for all and for all distinct points , each of them is different from and :(d1) if and only if ;(d2);(d3).
Then is called a generalized metric on and is called a generalized metric space.
Let be a generalized metric space, let be a sequence, and .
Then we say that(1) is convergent to (denoted by ) if and only if ;(2) is Cauchy if and only if ;(3) is complete if and only if every Cauchy sequence in is convergent to some point in .
Let be a generalized metric space.
A map is called continuous at if, for any containing , there exists containing such that , where is the topology on induced by the generalized metric . That is,
If is continuous at each point , then it is called continuous.
Note that is continuous if and only if it is sequentially continuous, i.e., for any sequence with .
Remark 2 (see [6]). If is a generalized metric on , then it is not continuous in each coordinate.
Lemma 3 (see [20]). Let be a generalized metric space, let be a Cauchy sequence, and . If there exists a positive integer such that (1);(2);(3);(4),then .
2. Fixed Point Theorems
We denote by the class of all functions such that conditions (1) and (2) hold.
A mapping is called -contraction with respect to if there exist and such that, for all with ,
Note that if is -contraction with respect to , then it is continuous. In fact, let be a point and let be any sequence such that
Then from () .
It follows from (6) and () that which implies
Since is nondecreasing, we have
and so Hence is continuous.
Now, we prove our main result.
Theorem 4. Let be a complete generalized metric space, and let be a -contraction with respect to .
Then has a unique fixed point, and for every initial point , the Picard sequence converges to the fixed point.
Proof. Firstly, we show uniqueness of fixed point whenever it exists.
Assume that and are fixed points of .
If , then , and so it follows from (6) that Hence which is a contradiction.
Hence , and fixed point of is unique.
Secondly, we prove existence of fixed point.
Let be a point. Define a sequence by .
If for some , then is a fixed point of , and the proof is finished.
Assume that It follows from (6) and (14) that Consequently, we obtain that which implies Hence is a decreasing sequence, and so there exists such that We now show that .
Assume that .
Then it follows from thatand soLet and
From we obtain which is a contradiction.
Thus we have and so We show that We consider three cases.
Case 1.
From (6) and (14) we obtain that and so which implies Hence is decreasing.
In a manner similar to that which proved (22), we have Case 2. There exists such that .
From the first term to the th term shall be removed, and let
Then . By Case 1, we have Case 3. .
We have Hence In all cases, (24) is satisfied.
Now, we show that is bounded.
If is not bounded, then there exists a subsequence of such that and is the minimum integer greater than with for
Then we haveBy letting in the above, we obtain By using (22), (34), and condition (d3), we deduce that It follows from , (34), and (35) that From (6) and (32) we infer thatwhich implies Let Then and .
It follows from 3 that which is a contradiction.
Thus is bounded.
Now, we show that is a Cauchy sequence.
Let Clearly, and so there exists such that Assume that .
It follows from (42) that there exist with So It follows from (6) and (14) that which implies Hence we haveLetting in the above inequality, we obtain Let Then Since , Thus we have which is a contradiction.
Hence , and hence is a Cauchy sequence.
Since is complete, there exists such that Because is continuous, By Lemma 3,
We give an example to illustrate Theorem 4.
Example 5. Let and define as follows:Then is a complete generalized metric space, but not a metric space (see [21]).
Define a map by And define a function by We now show that is a -contraction with respect to , where , .
We have so We have, for and , We deduce that, for all with ,Thus all hypotheses of Theorem 4 are satisfied, and has a fixed point .
Note that Banach’s contraction principle is not satisfied with the usual metric . In fact, if , , then which implies
Also, note that the -contraction condition [13] does not hold.
Let .
Then satisfies conditions (, 2, and .
If then and so . Hence is not -contraction map.
By taking in Theorem 4, we obtain Corollary 6.
Corollary 6. Let be a complete generalized metric space, and let be a mapping such that for all with where and .
Then has a unique fixed point.
Remark 7. Corollary 6 is a generalization of Theorem 2.1 of [12] without condition () and Theorem 2.2 of [13] without condition ().
By taking in Theorem 4, we obtain Corollary 8.
Corollary 8. Let be a complete generalized metric space, and let be a mapping such that for all with where and is nondecreasing and lower semicontinuous such that .
Then has a unique fixed point.
Corollary 9. Let be a complete generalized metric space, and let be a mapping such that for all with where is nondecreasing and lower semicontinuous such that .
Then has a unique fixed point.
Proof. Condition (69) implies is continuous.
Let
From (69) we have that, for all with ,Let , where is nondecreasing and lower semicontinuous such that .
Then is nondecreasing and lower semicontinuous, and It follows from (70) that, for all with , By Corollary 8, has a unique fixed point.
By taking , where in Corollary 8, we obtain the following result.
Corollary 10. Let be a complete generalized metric space, and let be a mapping such that for all with where and is nondecreasing and lower semicontinuous such that .
Then has a unique fixed point.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author express his gratitude to the referees for careful reading and giving variable comments. This research was supported by Hanseo University.