Approximate fixed point theorems of cyclical contraction mapping on G-metric spaces

This paper introduce a new class of operators and contraction mapping for a cyclical map T on G-metric spaces and the approximately fixed point properties. Also,we prove two general lemmas regarding approximate fixed Point of cyclical contraction mapping on G-metric spaces. Using these results we prove several approximate fixed point theorems for a new class of operators on G-metric spaces (not necessarily complete). These results can be exploited to establish new approximate fixed point theorems for cyclical contraction maps. Further,there is a new class of cyclical operators and contraction mapping on G-metric space (not necessarily complete)which do not need to be continuous.Finally,examples are given to support the usability of our results.


Introduction
Fixed point theory is a very popular tool in solving existence problems in many branches of Mathematical Analysis and its applications. In physics and engineering fixed point technique has been used in areas like image retrieval, signal processing and the study of existence and uniqueness of solutions for a class of nonlinear integral equations. Some recent work on fixed point theorems of integral type in G-metric spaces, stability of functional difference equation can be found in [19,20] and the references therein.
In 1968, Kannan (see [7] ) proved a fixed point theorem for operators which need not be continuous.
Further, Chatterjea (see [6]), in 1972, also proved a fixed point theorem for discontinuous mapping, which is actually a kind of dual of Kannan mapping. In 1972, by combining the above three independent contraction conditions above, Zamfirescu (see [22]) obtained another fixed point result for operators which satisfy the following. In 2001, Rus (see [21]) defined α−contraction. In [3], the author obtained Vali-e-Asr University, Rafsanjan, Iran, e-mail: amah@vru.ac.ir; mohsenialhosseini@gmail.com a different contraction condition, also he formulated a corresponding fixed point theorem. In 2006, Berinde (see [4]) obtained some result on α−contraction for approximate fixed point in metric space.
Miandaragh et al. [9,10] obtained some result on approximate fixed points in metric space.
On the other hand, in 2006, Mustafa and Sims [17,18] introduced the notion of generalized metric spaces or simply G-metric spaces. Many researchers have obtained fixd point, coupled fixed point, coupled common fixed point results on G-metric spaces (see [2,5,19]).
In 2011, Mohsenalhosseini et al [11], introduced the approximate best proximity pairs and proved the approximate best proximity pairs property for it. Also, In 2012 , Mohsenalhosseini et al [12], introduced the approximate fixed point for completely norm space and map T α and proved the approximate fixed point property for it. In 2014 , Mohsenalhosseini [13] introduced the Approximate best proximity pairs on metric space for contraction maps. Also, Mohsenalhosseini in [14] introduced the approximate fixed point in G-metric spaces for various types of operators. Recently, in 2017 Mohsenialhosseini [15] introduced the approximate fixed points of operators on G-metric spaces. The aim of this paper is to introduce the new classes of operators and contraction maps (not necessarily continuous) regarding approximate fixed point and diameter approximate fixed point for cyclical contraction mapping on G-metric spaces.
Also, we give some illustrative example of our main results.

Preliminaries
This section recalls the following notations and the ones that will be used in what follows. In 2003, kirk et al [8], obtained an extension of Banachs fixed point theorem by considering a cyclical operator.
is called a cyclical operator.
[17] Let X be a nonempty set and let G : X × X × X −→ R + be a function satisfying the following properties: (G1) G(x, y, z) = 0 i f and only i f x = y = z; (G2) 0 < G(x, x, y) f or all x, y ∈ X with x = y; (G3) G(x, x, y) ≤ G(x, y, z) f or all x, y, z ∈ X with z = y; (G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) f or all x, y, z, a ∈ X ( rectangle inequality).
Then, the function G is called generalized metric or, more specifically G − metric on X , and the pair Remark 2.5. [12] In this paper we will denote the set of all ε− fixed points of T , for a given ε, by : ) → 0 as n → ∞ for some k > 0, then T k has an ε− fixed point.
X i a cyclical operator and ε > 0. We assume that: Then:

MAIN RESULT
We begin with two lemmas which will be used in order to prove all the results given in third section.
Let (X , G) be a G − metric space.
Remark 3.2. In this paper we will denote the set of all ε-fixed points of T , for a given ε, by: where T n denotes the nth iterate of T at x.
Proof. The proof of Lemma is the same as the proof of Lemma 2.7 for x ∈ ∪ m i=1 X i .
Proof. The proof of Lemma is the same as the proof of Lemma 2.8 for x ∈ ∪ m i=1 X i . Then: Proof. The proof of Lemma is the same as the proof of Lemma 2.9 for x ∈ ∪ m i=1 X i .

APPROXIMATE FIXED POINT FOR SEVERAL OPERATOR ON G− METRIC SPACES
In this section a series of qualitative and quantitative results will be obtained regarding the properties of approximate fixed point. Also, by using Proposition 2.3 and lemma 2.8 we prove approximate fixed point theorems and diameter approximate theorems for a new class of cyclical operators on G-metric spaces.
Hence by Lemma 3.6 it follows that F ε In 1972, Chatterjea (see [6]) considered another operator in which continuity is not imposed. Now, the appoximate fixed point theorems by using cyclical operators on G-metric spaces are obtained.
On the other hand Then . . .
Example 4.8. Consider the sets: It is easily to be checked that T (A 1 ) ⊆ A 2 and T (A 2 ) ⊆ A 1 . For any x ∈ A 1 and y ∈ A 2 and Proposition

y) + G(y, x, x) + G(T x, Ty, Ty) + G(Ty, T x, T x)].
So T satisfies all the conditions of Theorem 4.7 and thus it has a approximate fixed point.
Example 4.9. Let X be a subset in R endowed with the usual metric. Suppose A 1 =]0, 0.8] and A 2 =]0, 1 2 ]. Define the map T : For any x ∈ A 1 and y ∈ A 2 and Proposition 2.3 we have the chain of inequalities

, Ty, Ty) + G(Ty, T x, T x)].
So T satisfies all the conditions of Theorem 4.7 and thus for every ε > 0, F ε (T )) = / 0. on the other

Hence by by Proposition 2.3, G(T x, x, x) + G(x, T x, T x) ≤ ε. So T has an approximate fixed point which
implies that F ε (T )) = / 0. On the contrary, there is no fixed point of T in ∪ 2 i=1 A i . By combining the three independent contraction conditions: G − α−cyclical contraction, G-Mohseni cyclical, and G-Chatterjea cyclical operators we obtain another approximate fixed point result for operators which satisfy the following.
at least one of the following is true: Then T has an ε−fixed point.
Supposing ii) holds, we have that:
Therfore  Define the map T : It is easily to be checked that T (A 1 ) ⊆ A 2 and T (A 2 ) ⊆ A 1 . For any x, y ∈ ∪ 2 i=1 A i there exists α ∈ (0, 1 2 ) such that holds at least one of the condition Theorem 4.11. Thus by Proposition 2.3 and Theorem 4.11 Theorem 4.14. Let {X i } m i=1 be nonempty subsets of a G − metric space X and Suppose T : . . .
.  1 2 ]. Define the map T : For any x ∈ A 1 and y ∈ A 2 and Proposition 2.3 we have the chain of inequalities So T satisfies all the conditions of Theorem 4.14 and thus for every ε > 0, F ε G (T ) = / 0.

DIAMETER APPROXIMATE FIXED POINT FOR SEVERAL OPERATOR ON G− METRIC SPACES
In this section, using Lemma 3.8, quantitative results for new cyclical operators will be formulated and proved, and some results regarding diameter approximate fixed point of such operators on G− metric spaces were given.
Proof: Let ε > 0. Condition i) in Lemma 3.8 is satisfied, as one can see in the proof of Theorem 4.7 we only verify that condition ii) in Lemma 3.8 holds. Let θ > 0 and x, y ∈ F ε G (T ) and assume that Then: [G(x, y, y) + G(y, x, x)] − [G(T x, Ty, Ty) + G(Ty, T x, T x)] < θ . Then: Therefore As x, y ∈ F ε G (T ), we know that which means exactly that  1 2 ]. Define the map T :  β ∈ (0, 1) and define T : x 4 x ∈ [1 − β , 1) 1−β 4 x ∈ [1, 2] By example 4.12 T : ∪ m i=1 X i → ∪ m i=1 X i is a G-Mohsenialhosseini cyclical operator. So T satisfies all the conditions of Theorem 5.3 and thus for every ε > 0, It is easy to check that T (A 1 ) ⊆ A 2 and T (A 2 ) ⊆ A 1 .