Some Fixed-Point Results via Mix-Type Contractive Condition

Abstract

We consider a fixed-point problem for mappings involving a rational type and almost type contraction on complete metric spaces. To do this, we are using -contraction and -contraction. We also present an example to illustrate our result.

1. Introduction

The beginning of metrical fixed point theory is related to Banach’s Contraction Principle, presented in 1922 [1], which says that any contraction self-map on has a unique fixed point whenever is complete. Afterwards, the crucial role of the principle in existence and uniqueness problems arising in mathematics has been realized which fact directed the researchers to extend and generalize the principle in many ways (see [2–7]).

In the studies of generalizations and modifications of contractions, an interesting generalization was given by Wardowski [8] using a new concept -contraction. Then, many authors gave some results using this concept in different type metric spaces. One of them is given by Jleli et al. [9] by introducing a family of functions with the certain assumption. Also, you can find this type generalizations in [10–12].

In this paper, we consider a fixed-point problem for mappings involving a rational type contraction and almost contraction. Firstly, we recall some basic on the notions of -contraction and -contraction.

2. Preliminaries

Let be the family of all functions satisfying the following conditions:

(F1) is nondecreasing;

(F2) for every sequence of positive numbers if and only if ;

(F3) there exists such that . ([8])

Definition 1. (see [8]). Let be a metric space and be a mapping. Given we say that is -contraction, if there exists such that Taking in (1) different functions one gets a variety of -contractions, and some of them being already known in the literature. You can see this contractions in [8]. In addition, Wardowski concluded that every -contraction is a contractive mapping, i.e., Thus, every -contraction is a continuous mapping.

Theorem 2. (see [8]). Let be a complete metric space () and let be an -contraction. Then, has a unique fixed point in .
In [9], Jleli et al. introduced a family of functions satisfying the following conditions:
(H1) for all ;
(H2) ;
(H3) is continuous.
Some examples of functions belonging to are given as follows: (i) for all (ii) for all (iii) for all Using a function , the authors of [9] introduced the following notion of -contraction.

Definition 3. (see [9]). Let be a metric space, be a given function, and . Then, is called a -contraction with respect to the metric if and only if for some constant .

Now, we set

Furthermore, we say that is a -Picard operator if and only if the following condition holds

Theorem 4. (see [9]). Let be a , be a given function and . Suppose that the following conditions hold
(A1) is lower semicontinuous (l.s.c.);
(A2) is a -contraction with respect to the metric .
Then, (i) is a -Picard operator(ii)For all and for all , we havewhere .
Recently, Vetro ([13]) generalized Theorem 4 by using --contraction.

Definition 5. (see [13]). Let be a metric space and let be a mapping. The mapping is called an --contraction if there exists , , a real number, and s.t. for all with .
We remark that every -contraction is an --contraction such that defined by for all and defined by for all .

Lemma 6. (see [13]). Let be a metric space and let be an --contraction with respect to the functions , , , and the real number . If is a sequence of Picard starting at , then and hence

Theorem 7. (see [13]). Let be a and be an --contraction with respect to the functions , , the real number , and a l.s.c. function such that (8) holds; that is, for all with . Then, has a unique fixed point such that .

Theorem 8. (see [13]). Let be a and let be a mapping. Assume that there exists a continuous function that satisfies the conditions () and (), a function , a real number , and a l.s.c. function such that (8) holds; that is, for all with . Then, has a unique fixed point such that .

3. Main Results

We first introduce the rational type --contraction.

Definition 9. Let be a metric space and be a mapping. is called a rational type --contraction if there exists , , a real number , and s.t.for all with where

Lemma 10. Let be a metric space and be a rational type --contraction with respect to the functions , , , and the real number . If is a sequence of Picard starting at , then and hence

Proof. By replacing the contradiction in [[13], (29)] with contradiction (13) and following the proof of [[13], Lemma 1], we immediately have the desired result.

Theorem 11. Let be a and let be an rational type --contraction with respect to the functions , , the real number , and a l.s.c. function such that (13) holds for all with . Then, has a unique fixed point such that .

Proof. First, we shall proof the uniqueness. Arguing by contradiction, we assume that there exist such that , , and . The hypothesis ensures, by the property () of the function , that Using (13) with and , we obtain which is a contradiction. So, we have , and the fixed point is unique.
Now, we can show the existence of a fixed point. Take a point and create the sequence starting at . We emphasize that if for some , then ; that is, is a fixed point of such that . In fact, by Lemma 10, and by the property () of the function , we have . So, we can suppose that for every .
In this step, we show that is a Cauchy. By Lemma 10, we say that There exists such that as by he property () of Using (13) with and , we get for all ; that is, From we deduce that This provides that is convergent. By the property () of the function , also, the series is convergent and hence is a Cauchy sequence. Now, since is complete, there exists such that By (13), taking into account that is a l.s.c. function, we have that is, . Now, show that is a fixed point. If there exists a subsequence of such that or , for all , then is a fixed point. Otherwise, we can assume that and for all . So, using (13) with and , we deduce that Since , we obtain and so for all .
Finally, letting in the above calculations and using that is continuous in , we deduce that ; that is, .

Imposing that is a continuous function and relaxing the hypothesis , we can give t Theorem 12.

Theorem 12. Let be a and be a mapping. Assume that there exists a continuous function that satisfies the conditions () and (), a function , a real number , and a l.s.c. function s.t. for all with . Then, has a unique fixed point such that .

Proof. Following the similar arguments as in the proof of Theorem 11, we obtain easily the uniqueness of the fixed point. The existence of a fixed point, we take a point and create the sequence starting at . Clearly, if for some , then ; that is, is a fixed point of such that (see the proof of Theorem 11), and so we have already done.
So, we can suppose that for every . Now, showing that is a Cauchy. Let us admit the opposite. Then, there exists a positive real number and two sequences and such that By Lemma 10, we say that , , as . This implies Now, the hypothesis that ensures that Using the continuity of , we have Using again (29), with and , we get for all . Letting in the previous inequality, since the function is continuous, we get which leads to contradiction. It follows that is a Cauchy sequence.
Now, since is complete, there exists some such that By (29), using lower semicontinuity of , we get that is, . Now, show that is a fixed point of . Clearly, is a fixed point of if there exists a subsequence of such that or , for all . Otherwise, we can assume that and for all . Then, the property () of the function ensures that for all . So, using (29) with and , we deduce that Since , we conclude that and so for all . Finally, letting and using that is continuous in , we deduce that ; that is, .

Definition 13. Let be a metric space and let be a mapping. The mapping is called almost --contraction if there exists a function , , a real number , and and a l.s.c. function such that for all with .

Theorem 14. Let be a and let be an almost --contraction with respect to the functions , , the real number , and and a l.s.c. function s.t. for all with . Then, has a fixed point such that .

Proof. The existence of a fixed point we take a point and create the sequence starting at . We stress that if for some , then ; that is, is a fixed point of such that . In fact, by Lemma 10, and by the property () of the function , we have . So, we can suppose that for every .
Now, showing that is a Cauchy. By Lemma 10, we say that The property () of the function ensures that there exists such that as . Using (42), with and , we get for ; that is, From we deduce that This ensures that the series is convergent. By the property () of the function , also, the series is convergent, and hence is a Cauchy sequence. Now, since is complete, there exists some such that By (42), using lower semicontinuity of , we get that is, . We assert that is a fixed point of . Clearly, is a fixed point of if there exists a subsequence of such that or , for all . Otherwise, we can assume that and for all . So, using (42) with and , we deduce that Since , this inequality leads to and so for all .
Finally, letting in the above calculations and using that is continuous in , we deduce that ; that is, .

Example 15. Let endowed with the standart metric for all Consider the mapping defined by Clearly, is not a contraction but is an almost --contraction with respect to the functions defined by for all defined by for all the real number and , and a l.s.c. function for all , indeed.

Case 1. we have

Case 2. we have

Case 3. with we have

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.