Approximate Fixed Point Theorems For Weak Contractions On Neutrosophic Normed Spaces

In this paper, we define concept of approximate fixed point property of a function and a set in Neutrosophic normed space. Furthermore, we give Neutrosophic version of some class of maps used in fixed point theory and investigate approximate fixed point property of these


Introduction
Fuzzy Set [FS] initiated by Zadeh [21] has influenced profoundly every one of the logical fields since 1965. It is seen that the idea play a vital role to solve many real life problems, but it is not enough to address certain issues. Atanassov [1] introduced Intuitionistic Fuzzy Set [IFS] for such cases. After defining the IFS, it extends the results those are studied over FS. Neutrosophic Set [NS], defined by Smarandache [16], is another variant of the crisp set which is Neutrosophy theory was published in the year 1998 and included in the literature [17]. Park [15] derived Intuitionistic Fuzzy Metric Space [IFMS] as a generalization of Fuzzy Metric Space. Branzei et al [2] further extended these results to multifunctions in Banach spaces. Berinde [3] obtained approximate fixed point theorems for operators satisfying Kannan, Chatterjea and Zamfirescu type of conditions on metric spaces.
After a while, Smarandache introduced the notion of NS, which is the different kind of the notation of the classical set theory by adding an intermediate membership function. This set is a formal setting trying to measure the truth, indeterminacy and falsehood. Quite recently, Jeyaraman et al. [10] introduced the notion of Neutrosophic normed space and statistical convergence. Since Neutrosophic Normed Space [NNS] is a natural generalization of IFNS and statistical convergence.
In this paper, we define concept of approximate fixed point property of a function and a set in NNS. Furthermore, we give Neutrosophic version of some class of maps used in fixed point theory and investigate approximate fixed point property of these maps.

Main Results
Firstly, we define approximate fixed property, diameter of a set in NNSs and give examples.
Definition 3.1 Let (X, µ, ν, ω, * , ♦, ⊗) be a NNS and f : X → X be a function. Given > 0. It is said that x 0 ∈ X is an neutrosophic -fixed point or approximate for all t > 0. We denote the set of neutrosophic -fixed points of f with F (µ,ν,ω) (f ).
As known, f has not any fixed point on (0, 1). We investigate neutrosophic approximate fixed point of f . For every > 0 and t > 0, there exists x ∈ (0, 1) such that x satisfies So, f has the NAFPP, since F (µ,ν,ω) (f ) is not empty for every > 0.
for every x ∈ X and t > 0.

Proof
: In this case, for every > 0 there exists k 0 ( , t) ∈ N such that This shows that y 0 is neutrosophic approximate fixed point of f .
Theorem 3.11 Let (X, µ, ν, ω, * , ♦, ⊗) be a NNS having partial order relation denoted by , where a * b = min{a, b} and a♦b = max{a, b} and f : X → X be a neutrosophic Kannan operator satisfying x f (x) for every x ∈ X. Assume that ⊂ X × X holds one of the following conditions: is subvector space, (ii) X is a totally ordered space.
Theorem 3.14 Let (X, µ, ν, ω, * , ♦, ⊗) be a NNS having partial order relation denoted by , where a * b = min{a, b} and a♦b = max{a, b}, and f : X → X be a neutrosophic Chatterjea operator satisfying x f (x) for every x ∈ X. Assume that ⊂ X × X holds one of the following conditions: (i) is subvector space (ii) X is a totally ordered space.

Proof:
By taking into consideration assumption of theorem, we get Corollary 3.15 In the Theorem (3.14), if x f (x) for neutrosophic Chatterjea operator f and µ(., t) is non-increasing, ν(., t) is non-decreasing and ω(., t) is non-decreasing for every t ∈ (0, ∞), x θ, f has still NAFPP. Definition 3.16 Let X be an NNS. A mapping f : X → X is called neutrosophic Zamfirecsu operator if there exists at least a ∈ (0, 1), k ∈ 0, 1 2 , c ∈ 0, 1 2 such that atleast one of the followings is true for every x, y ∈ X and t > 0: and and Theorem 3.17 Let (X, µ, ν, ω, * , ♦, ⊗) be a NNS having partial order relation denoted by , where a * b = min{a, b} and a♦b = max{a, b}, and f : X → X be a neutrosophic Zamfirescu operator satisfying x f (x) for every x ∈ X. Assume that ⊂ X × X holds one of the following conditions: is subvector space, (ii) X is a totally ordered space.

Proof:
The proof is clear from Theorem (3.11) and Theorem (3.14).
Definition 3.18 Let X be a NNS. If there exists a ∈ (0, 1) and L ≥ 0 such that and for every x, y ∈ X and t > 0, then f : X → X is called neutrosophic weak contraction operator.
Since t a k → ∞ for k → ∞, by means of (NNS-8),  and (NNS-20) properties of NN, we see neutrosophic weak contraction map has approximate fixed point property by Theorem (3.7). In the following, we give definition of approximate fixed point property of a set. Furthermore, we prove that a dense set of neutrosophic Banach space has approximate fixed point property.
Theorem 3.21 Let X be a NNS having NAFPP, K be dense subset of X. Then K has NFAFPP.

Proof:
Let f : X → X be a neutrosophic nonexpansive mapping. We prove that Let y ∈ X. There exists a sequence (y k ) in K such that y k (u,v,ω) −→ y for all y ∈ X because of K is dense. We know that for each k ∈ N and t, s > 0, sup{µ(x − f (x), t) : x ∈ K} ≥ µ(y k − f (y k ), t)