Abstract and Applied Analysis
Volume 2018 (2018), Article ID 1327691, 6 pages
https://doi.org/10.1155/2018/1327691

Research Article

Fixed Point Theorems for -Contractions in Generalized Metric Spaces

Seong-Hoon Cho

Department of Mathematics, Hanseo University, Chungnam 356-706, Republic of Korea

Correspondence should be addressed to Seong-Hoon Cho; shcho@hanseo.ac.kr

Received 21 August 2018; Revised 8 October 2018; Accepted 4 November 2018; Published 2 December 2018

Academic Editor: Aref Jeribi

Copyright © 2018 Seong-Hoon Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 that

and so

Let 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 have

By 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 that

which 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 have

Letting 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.

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