1. Introduction

Shoaib et al. ^[40] proved significant results concerning the existence of fixed points of the dominated self mappings satisfying some contractive conditions on a closed ball in a 0-complete quasi-partial metric space. Other results on closed ball can be seen in ^{[5, 6, 7, 8, 25]}. Over the years, Fixed Point Theory has been generalized in different ways by several mathematicians (see [2,3,4,9-15,17-21,23-27]).

For and is a closed ball in

Definition 1 ^[18] Let be a metric space. Let be self mappings and be two functions. is called -continuous if for given and sequence with

In 2012, Wardowski ^[42] introduce a concept of F-contraction as follows:

Definition 2 ^[33] Let be a metric space. A self mapping is said to be an contraction if there exists such that

(1.1)

where is a mapping satisfying the following conditions:

(F1) is strictly increasing, i.e. for all such that

(F2) For each sequence of positive numbers, if and only if

(F3) There exists such that

We denote by , the set of all functions satisfying the conditions (F1)-(F3).

Furthermore, it was done by different investigators see [1,3,15,16,23,27,28,29,31,32,33,34,37,38,39].

Hussain et al. ^[18] introduced the following family of new functions.

Let denotes the set of all functions satisfying:

(G) for all with there exists such that

2. Banach Fixed Point Theorem for F-Contraction on a Closed Ball

In this section, we introduce Banach fixed point theorem for modified F-contraction on a closed ball in complete metric spaces.

Now we state our main result.

Theorem 3 Let T be a continuous self mapping in a complete metric space and be an arbitrary point in Assume that and for all with such that

(2.1)

Morever

(2.2)

Then there exist a point in such that

Proof. Choose a point in such that continuing in this way, so we get for all and this implies that is a nonincreasing sequence. First we show that for all by using mathematical induction. Since from (2.2), we have

(2.3)

thus, Suppose for some Thus from (2.1), we obtain

As F is strictly increasing, we have

(2.4)

Now,

Thus Hence for all Continuing this process, we get

This implies that

(2.5)

From (2.5), we obtain Since we have

(2.6)

From (F3), there exists such that

(2.7)

From (2.5), for all we obtain

(2.8)

By using (2.6), (2.7) and letting in (2.8), we have

(2.9)

We observe that from (2.9), then there exists such that for all we get

(2.10)

Now, such that Then, by the triangle inequality and from (2.10) we have

(2.11)

The series is convergent. By taking the limit as in (2.11), we have Hence is a Cauchy sequence. Since is a complete metric space there exists such that as is a continuous then as That is, Hence is a fixed point of To prove uniqueness, let and be any two fixed point of then from (2.1), we have

we obtain

which is a contradiction. Hence, Therefore, has a unique fixed point in

Example 4 Let and Then is a complete metric space. Define the mapping by,

If and then

If then

This implies that

If then

Hence the contraction does not satisfy on

3. Fixed Point Theoreminvolving GF-Contraction on a Closed Ball

In this section, we define a new contraction called -GF-contraction on a closed ball and obtained Banach fixed point theorems for such contraction in the setting of complete metric spaces. We define -GF-contraction on a closed ball as follows:

Definition 5 Let be a self mapping in a metric space and an arbitrary point in with Also suppose that two functions. We say that is called -GF-contraction on a closed ball if for all with and we have

(3.1)

and

(3.2)

where and

Definition 6 (^[36]). Let and be two functions. We say that is -admissible mapping with respect to if implies that

Theorem 7 Let be a complete metric space. Let be GF-contraction mapping on a closed ball satisfying the following assertions:

(i) T is an -admissible mapping with respect to

(ii) there exists such that

(iii) is an continuous.

Then there exist a point in such that

Proof. Let in such that For we construct a sequence such that Continuing this way, for all Now since, is an -admissible mapping with respect to then By continuing in this process we have,

(3.3)

If there exists such that there is nothing to prove. So, we assume that with

First we show that for all Since be a GF-contraction on closed ball, we have

(3.4)

thus, Suppose for some such that

which implies

(3.5)

Now by definition of

so there exists such that,

Therefore

(3.6)

Rest of the proof follows the similar lines of Theorem 3. Since is a complete metric space there exists such that as is an continuous and for all then as That is, Hence is a fixed point of

Example 8 Let and d be the usual metric on define and by

for all and with then

If then On the other hand, for all Then with clearly Hence we have

Consequently,

If or then either

On the other hand the contraction does not satisfy.

4. Conclusion

This research focus on introducing new idea of F-contraction on a closed ball which is different from F-contraction given in ^{[18, 33, 42]}. Therefore a generalization of results is very useful so far as it requires the F-contraction mapping only on a closed ball rather than the whole space. This new idea however guides the researcher towards further investigations and applications. At the same time, it will be interesting to apply these concepts in a various spaces.

Conflict of Interests

The author declare that he has no competing interests.

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