1. Introduction

Notion of dislocated metric spaces was introduced by Hitzler and Seda in 2000 as a generalization of metric space. They generalized the Banach Contraction Principle in such spaces. These metrics play a very important role not only in topology but also in other branches of science involving mathematics especially in logic programming and electronic engineering. Fixed point theory has been a subject of growing interest of many researchers for various types of well known contractions in these spaces.In 2003 Kirk et al ^[18] introduced cyclic contractions in metric spaces and investigated the existence of proximity points and fixed points for cyclic contraction mappings, Since then many authors has given results in this field.

In this paper we introduced the notion of Geraghty type - cyclic contraction and derive the existence of common fixed point theorems in the framework of dislocated metric spaces. Our main theorem extends and unifies existing results in the recent literature.

Definition 1.1 ^{[12, 20]} Let be a non-empty and let be a function, called a distance function if for all , satisfies:

If d satisfies the condition , then d is called a metric on . If it satisfies the conditions, and it is called a quasi-metric space. If d satisfies conditions,and it is called a dislocated metric (or simply d-metric). If d satisfies only and then d is called a dislocated quasi-metric (or simply dq-metric) on .

Definition 1.2 ^[20] A sequence in a -metric space dislocated quasi-converges ( for short, -converges ) to if . In this case is called a -limit of and we write x_n →x.

Definition 1.3 ^[20] A sequence in a -metric space is said to be Cauchy if for every , such that , and .

Definition 1.4 ^[20] A -metric space is complete if every Cauchy sequence in it is -convergent in .

Example 1.5 Let and. Then the pairis a dislocated metric space, but it is not a metric space.

Lemma 1.6 ^[20] Every subsequence of -convergent sequence to a point is -convergent to .

Definition 1.7 ^[20] Let be a -metric space. A mapping is called contraction if there exists such that: for all .

Lemma 1.8 ^[20] -limit in a -metric space is unique.

Definition 1.9 ^[7] Let and be nonempty subsets of a metric spaceand. is called a cyclic map iffand.

Definition 1.10 ^[18] Let and be nonempty subsets of a metric space. A cyclic mapis said to be a cyclic contraction if there exists such thatfor alland.

Definition 1.11^[4] Let and be nonempty subsets of a metric space. A cyclic map is called a Kannan type cyclic contraction if there exists such that for alland.

In ^[4] Karapinar et al has been shown that Kannan type cyclic contraction and cyclic contraction are independent of each other.

Definition 1.12 ^[4] Let and be nonempty subsets of a metric space. A cyclic mapis called a Chatterjee type cyclic contraction if there exist k ∈ (0, 1/2) such that for alland.

Definition 1.13 ^[18] Let and be nonempty subsets of a dislocated metric space. A cyclic mapis called a -cyclic contraction if there exists such thatfor alland.

Example 1.14 Let and . Then,is a dislocated metric space, but not a metric space. Let and defineby forand for. Thenis a -cyclic contraction in the dislocated metric space. We note that in the usual metric the self mapis not cyclical contraction because for and the cyclic contraction fails.

Hence the class of d- cyclical contraction in dislocated metric space is larger than the class of cyclical contraction in usual metric.

2. Main Results

Theorem 2.1 Let and be nonempty subsets of a complete dislocated quasi-metric space. Let be a cyclic mapping that satisfies the condition

(1)

for allandand .

Then, has a unique fixed point in.

Proof. Taking a point(fix) and using contractive condition of theorem, we have

In the same way we have,

If we put, then from two inequalities above we have,

(2)

(3)

Using (2) and (3) we get, and .

Inductively, using this process for all we have and for all .

Let with, using the triangular inequality, we obtain:

Since, as , we get . Thus is a Cauchy sequence.

Sinceis complete, we have -converges to some.We note, that is a sequence inand is a sequence in in a way that both sequences tend to same limit.

Since and are closed have that . Hence .

We claim that .

Considering the condition (1) we have:

Taking limit asin above inequality, we have

This implies thatsince.

Similarly considering (1) have,

Taking limit as and since, we obtain.

Henceandis a fixed point of.

We shall prove thatis the unique fixed point of. Clearly from (1) if u and v be fixed points of we have .

Then we have,

Since this implies. Hence the proof is completed.

For following theorem we denotes with the class of those real functions that satisfy the condition implies . Examples of those functions exist in the corresponding literature. Using this class of functions we give this definition in the framework of dislocated metric spaces.

Definition. Let and be nonempty subsets of a dislocated quasi-metric space. A cyclic mapis called a Geraghty type-cyclic contraction if there exists such thatfor alland.

Theorem 2.2 Let and be nonempty closed subsets of a dislocated metric spaceandbe a cyclic mapping that satisfies the Geraghty type condition:

(4)

for all and where .

Then has a unique fixed point in.

Proof. Fix a point. If for some, then and so converges to some . Suppose. Using condition (4) we have:

Also we have

Inductively in general we have . Thus the sequence is decreasing and bounded from below, thus it converges to some If we suppose that , then from (4) have

Taking limit as, we get

By property of follows that . In a similar way we obtain So our supposition fail from this contradiction. Hence . To proceed further we define then for all . Using the main condition of theorem we obtain:

Similarly,

As a result get.

Thus in general we get for with, using the triangular inequality, we obtain:

Since, as , we get . This proves that is a Cauchy sequence. Sinceis complete, we have -converges to some. Note that is a sequence inand is a sequence in in a way that both sequences tend to same limit.

Considering the condition (4) we have:

Taking limit asin above inequality, we haveand in similar have as a result .

Uniqueness: Let u and v be two fixed points of .

Then:

from those inequalities we get and also, by propertyhave .

Example 3.3 Let andbe given as . Let . Define the function by . The function defined as , for and . We note thatis a dislocated metric onand the mapis cyclic on and.

Considering all cases and general cases if for all we have,

Clearly all conditions of theorem 3.2 are satisfied and is the unique fixed point of.

For the following theorems and corollaries we consider the set of all continuous functions [some examples for these functions see in 10] with the following properties:

a). is non-decreasing in respect to each variable.

b). , for .

Theorem 2.4 Let and be nonempty closed subsets of a dislocated quasi-metric space and be a cyclic mapping that satisfies the following condition:

(5)

for all and, and , where.

Then has a unique fixed point in.

Proof. Let be a fixed point in. By condition (5) and properties of we have:

Similarly we have

Generally from the above inequalities have:

for.

Since we obtain for that . In the same way we can show that .

Easily as in the above theorems we can show that the sequence is a Cauchy sequence in complete dislocated metric space. So there existssuch thatdislocated quasi converges to . Note that, is a sequence inand is a sequence in in a way that both sequences tend to same limit. For proving that z is a fixed point of we use again the contractive condition (5),

In this inequality passing in limit asand since is non decreasing and continuous we get, and since we obtain . Again from (5) get . As a result .

Uniqueness Let suppose that and are two fixed points of whereand.

From condition of theorem we have,

(6)

If we replace in (6) then we obtain,

Thus from and since, we get. Similarly we have that. Therefore using condition (5) we have:

And also,

So from this inequality we have and property implies . Hence fixed point is unique.

Example 2.5 Let andbe given as . Let and . Define the function by . We note thatis a dislocated quasi-metric onand the mapis cyclic on.

If we consider from the function we see:

Then clearly have,

So for constant the mapsatisfies the condition (5) of theorem 3.4 and is the unique fixed point of .

From general character of theorem 3.4 we can give many corollaries as follows using functions

Corollary 2.6 Let and be nonempty closed subsets of a dislocated quasi-metric space and be a cyclic mapping that satisfies the following condition:

for all and, and .

Then has a unique fixed point in.

Corollary 2.7 Let and be nonempty closed subsets of a dislocated quasi-metric space and be a cyclic mapping that satisfies the following condition:

for all and, and .

Then has a unique fixed point in.

Corollary 2.8 Let and be nonempty closed subsets of a dislocated quasi-metric space and be a cyclic mapping that satisfies the following condition:

for all and, and .

Then has a unique fixed point in.

Further as common applications of fixed point theorems we are giving some corollaries for cyclic maps for integral type contraction. (taking)

Corollary 2.9 Let be a complete dislocated quasi-metric space and be a mapping such that for any ,

where the function , the constant and is Lesbegue-integrable mapping satisfyingfor. Then, has a unique fixed point in .

Remark 2.10 Our Theorem 3.4 generalizes and unifies results for Kannan type cyclic contraction, Chatterjea cyclic contraction, C cyclical contraction, Zamfirescu contraction and some existing results in dislocated-metric spaces ^{[2, 3, 10, 11, 15]}. Statements of many theorems and results can be obtained by taking.

Acknowledgements

The authors would like to thank the referees, who have made valuable comments and suggestions which have improved the manuscript.

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