We define approximate fixed point and fuzzy diameter in fuzzy norm spaces. We prove theorems for various types of well-known generalized contractions on fuzzy norm spaces with the use of two general lemmas that are given regarding approximate fixed points of operators on fuzzy norm spaces.

1. Introduction

In this paper, starting from the article of Berinde [1], we study some well-known types of operators on fuzzy norm spaces, and we give some fuzzy approximate fixed points of such operators.

Fuzzy set was defined by Zadeh [2]. Katsaras [3], while studying fuzzy topological vector spaces, was the first to introduce in 1984 the idea of fuzzy norm on a linear space. In 1992, Felbin [4] defined a fuzzy norm on a linear space with an associated metric of the Kaleva and Seikkala type [5]. A further development along this line of inquiry took place when, in 1994, Cheng and Mordeson [6] evolved the definition of a further type of fuzzy norm having a corresponding metric of the Kramosil and Michálek type [7].

Chitra and Mordeson [8] introduce a definition of norm fuzzy, and thereafter the concept of fuzzy norm space has been introduced and generalized in different ways by Bag and Samanta in [9–11].

Throughout this paper, the symbols ∧ and ∨ mean the min and the max, respectively.

2. Some Preliminary Results

We start our work with the following definitions.

Definition 1.

Let U be a linear space on R. A function N:U×R→[0,1] is called fuzzy norm if and only if for every x,u∈U and for every c∈R, the following properties are satisfied:

N(x,t)=0 for every t∈R-∪{0},

(FN2): N(x,t)=1 if and only if x=0 for every t∈R+,

(FN3): N(cx,t)=N(x,t/|c|) for every c≠0 and t∈R+,

(FN4): N(x+u,s+t)≥min{N(x,s),N(u,t)} for every s,t∈R+,

(FN5): the function N(x,·) is nondecreasing on R, and limt→∞N(x,t)=1.

A pair (U,N) is called a fuzzy norm space. Sometimes, we need two additional conditions as follows:

(FN6): for all t∈R+N(x,t)>0⇒x=0.

(FN7): function N(x,·) is continuous for every x≠0 and on subset
(1){t:0<N(x,t)<1}

is strictly increasing.

Let (U,N) be a fuzzy norm space. For all α∈(0,1), we define α norm on U as follows:
(2)∥x∥α=∧{t>0:N(x,t)≥α}foreveryx∈U.

Then {∥x∥α:α∈(0,1]} is an ascending family of normed on U and they are called α-norm on U corresponding to the fuzzy norm N on U. Some notation, lemmas, and examples which will be used in this paper are given in the following.

Lemma 2 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

Let (U,N) be a fuzzy norm space such that it satisfies conditions (FN6) and (FN7). Define the function N′:U×R→[0,1] as follows:
(3)N′(x,t)={∨{α∈(0,1):∥x∥α≤t}(x,t)≠(0,0),0(x,t)=(0,0).
Then

N′ is a fuzzy norm on U.

N=N′.

Lemma 3 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

Let (U,N) be a fuzzy norm space such that it satisfies conditions (FN6) and (FN7), and {xn}⊆U. Then limn→∞N(xn-x,t)=1 if and only if
(4)limn→∞∥xn-x∥α=0
for every α∈(0,1).

Note that the sequence {xn}⊆U converges if there exists a x∈U such that
(5)limn→∞N(xn-x,t)=1foreveryt∈R+.
In this case, x is called the limit of {xn}.

Example 4 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

Let V be the real or complex vector space and let N be defined on V×R as follows:
(6)N(x,t)={1t>|x|0t≤|x|,
for all x∈V and t∈R. Then (V,N) is a fuzzy norm space, and the function N satisfies conditions (FN6) and ∥x∥α=|x| for every α∈(0,1).

3. Fuzzy Approximate Fixed Point

In the section, we begin with two lemmas which will be used in order to prove all the results given in the same section.

Definition 5.

Let (U,N) be a fuzzy norm space, T:U→U, ϵ>0, and u0∈U. Then u0 is an Fz-approximate fixed point (fuzzy approximate fixed point) of T if for some α∈(0,1),
(7)∧{t>0:N(u0-Tu0,t)≥α}≤ϵ.

Remark 6.

In the rest of the paper we will denote the set of all Fz-approximate fixed points of T, for a given ϵ>0, by
(8)Fϵz(T)={u∈U:∧{t>0:N(u-Tu,t)≥α}≤ϵforsomeα∈(0,1)}.

Definition 7.

Let T:U→U. Then T has the fuzzy approximate fixed point property (f.a.f.p.p.) if
(9)∀ϵ>0Fϵz(T)≠∅.

Lemma 8.

Let (U,N) be a fuzzy norm space such that it satisfies conditions (FN6) and (FN7), T:U→U such that T is asymptotic regular for some α∈(0,1); that is,
(10)limn→∞∧{t>0:N(Tnu-Tn+1u,t)≥α}=0,∀u∈U.
Then T has the fuzzy approximate fixed point property.

Proof.

Suppose u∈U and ϵ>0. Since
(11)limn→∞∧{t>0:N(Tnu-Tn+1u,t)≥α}=0,∃N0>0s.t. for all n≥N0:∧{t>0:N(Tnu-Tn+1u,t)≥α}<ϵ for some α∈(0,1).

If n=N0, then for some α∈(0,1),
(12)∧{t>0:N(TN0u-T(TN0)u,t)≥α}<ϵ.
Therefore, TN0u∈Fϵz(T) and Fϵz(T)≠∅. Hence there exists a fuzzy approximate fixed point in U.

Definition 9.

Let (U,N) be a fuzzy norm space such that satisfies conditions (FN6) and (FN7). We define fuzzy diameter of U≠∅ for some α∈(0,1) as
(13)δα(U)=∨[∧{t>0:N(u-v,t)≥α∀u,v∈U}].

Lemma 10.

Let (U,N) be a fuzzy norm space such that satisfy conditions (FN6) and (FN7), T:U→U, and ϵ>0. One assumes that for some α∈(0,1),

Fϵz(T)≠∅,

for all μ>0 there exists φ(μ)>0 such that
(14)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ⟹∧{t>0:N(u-v,t)≥α}≤φ(μ){t>0:N(u-v,t)(μ)≥α}∀u,v∈Fϵz(T).

Then
(15)δα(Fϵz(T))≤φ(2ϵ). Proof.

Let ϵ>0 and u,v∈Fϵz(T). Then
(16)∧{t>0:N(u-Tu,t)≥α}<ϵ,∧{t>0:N(v-Tv,t)≥α}<ϵ.
We can write
(17)∧{t>0:N(u-v,t)≥α}≤∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(Tu-Tv,t)≥α}+∧{t>0:N(v-Tv,t)≥α}≤∧{t>0:N(Tu-Tv,t)≥α}+2ϵ⟹∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤2ϵ.
Now by (44), it follows that
(18)∧{t>0:N(u-v,t)≥α}≤φ(2ϵ).
So
(19)δα(Fϵz(T))≤φ(2ϵ)
for some α∈(0,1).

Lemma 11.

Let (U,N) be a fuzzy norm space such that it satisfies conditions (FN6) and (FN7), T:U→U, and ϵ>0. One assumes that for some α∈(0,1),

lim
n→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0, for all u∈U,

for all μ>0 there exists φ(μ)>0 such that
(20)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ⟹∧{t>0:N(u-v,t)≥α}≤φ(μ){t>0:N(u-v,t)(μ)≥α}∀u,v∈Fϵz(T).

Then
(21)δα(Fϵz(T))≤φ(2ϵ).Definition 12.

A mapping T:U→U is an a-contraction if there exists a∈(0,1) such that
(22)∧{t>0:N(Tu-Tv,t)≥α}≤a∧{t>0:N(u-v,t)≥α},∀u,v∈U.

Proposition 13.

Let (U,N) be a fuzzy norm space and T:U→U an a-contraction. Then
(23)∀ϵ>0,Fϵz(T)≠∅.

Proof.

Let ϵ>0 and u∈U. Consider
(24)∧{t>0:N(Tnu-Tn+1u,t)≥α}=∧{t>0:N(T(Tn-1u)-T(Tnu),t)≥α}≤a∧{t>0:N(Tn-1u-Tnu,t)≥α}⋮≤an∧{t>0:N(u-Tu,t)≥α}.
But a∈(0,1). Therefore
(25)limn→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0,∀u∈U.
Now by Lemma 8 it follows that Fϵz(T)≠∅, for all ϵ>0.

In 1968, Kannan (see [12, 13]) proved a fixed point theorem for operators which needs not be continuous. We apply it to fuzzy norm space for Fz-approximate fixed points.

Definition 14.

A mapping T:U→U is an F-Kannan operator if there exists a∈(0,1/2) such that
(26)∧{t>0:N(Tu-Tv,t)≥α}≤a[∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(v-Tv,t)≥α}],
for all u,v∈U.

Proposition 15.

Let (U,N) be a fuzzy norm space and T:U→U a Kannan operator. Then
(27)∀ϵ>0,Fϵz(T)≠∅.

Proof.

Let ϵ>0 and u∈U. Consider
(28)∧{t>0:N(Tnu-Tn+1u,t)≥α}=∧{t>0:N(T(Tn-1u)-T(Tnu),t)≥α}≤a[∧{t>0:N(Tn-1u-T(Tn-1u),t)≥α}+∧{t>0:N(Tnu-T(Tnu),t)≥α}]=a∧{t>0:N(Tn-1u-T(Tn-1u),t)≥α}+a∧{t>0:N(Tnu-T(Tnu),t)≥α}.
Therefore
(29)(1-a)∧{t>0:N(Tnu-Tn+1u,t)≥α}≤a∧{t>0:N(Tn-1u-Tnu,t)≥α}.
Then
(30)∧{t>0:N(Tnu-Tn+1u,t)≥α}≤a1-a∧{t>0:N(Tn-1u-Tnu,t)≥α}⋮≤(a1-a)n∧{t>0:N(u-Tu,t)≥α}.
But a∈(0,1/2); hence a/(1-a)∈(0,1). Therefore
(31)limn→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0,∀u∈U.
Now by Lemma 8 it follows that Fϵz(T)≠∅, for all ϵ>0.

In 1972, Chatterjea (see [14]) considered another one which again does not impose the continuity of the operator. We apply it to fuzzy norm space for Fz-approximate fixed points.

Definition 16.

A mapping T:U→U is an F-Chatterjea operator if there exists a∈(0,1/2) such that
(32)∧{t>0:N(Tu-Tv,t)≥α}≤a[∧{t>0:N(u-Tv,t)≥α}+∧{t>0:N(v-Tu,t)≥α}],
for all u,v∈U.

Proposition 17.

Let (U,N) be a fuzzy norm space and T:U→U a Chatterjea operator. Then
(33)∀ϵ>0,Fϵz(T)≠∅.

Proof.

Let ϵ>0 and u∈U. Consider
(34)∧{t>0:N(Tnu-Tn+1u,t)≥α}=∧{t>0:N(T(Tn-1u)-T(Tnu),t)≥α}≤a[∧{t>0:N(Tn-1u-T(Tnu),t)≥α}+∧{t>0:N(Tnu-T(Tn-1u),t)≥α}]=a∧{t>0:N(Tn-1u-Tn+1u,t)≥α}+a∧{t>0:N(Tnu-Tnu,t)≥α}=a[∧{t>0:N(Tn-1u-Tn+1u,t)≥α}].
On the other hand,
(35)∧{t>0:N(Tn-1u-Tn+1u,t)≥α}≤∧{t>0:N(Tn-1u-Tnu,t)≥α}+∧{t>0:N(Tnu-Tn+1u,t)≥α}.
Then
(36)(1-a)∧{t>0:N(Tnu-Tn+1u,t)≥α}≤a∧{t>0:N(Tn-1u-Tnu,t)≥α}.
Hence
(37)∧{t>0:N(Tnu-Tn+1u,t)≥α}≤a1-a∧{t>0:N(Tn-1u-Tnu,t)≥α}⋮≤(a1-a)n∧{t>0:N(u-Tu,t)≥α}.
But a∈(0,1/2); hence a/(1-a)∈(0,1). Therefore
(38)limn→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0,∀u∈U.
Now by Lemma 8 it follows that Fϵz(T)≠∅, for all ϵ>0.

We, by combining the three independent contraction conditions above, obtain another Fz-approximation fixed points result for operators which satisfies the following.

Definition 18.

A mapping T:U→U is an F-Zamfirescu operator if there exist α,β,γ∈R, α∈[0,1[, β∈[0,1/2[, γ∈[0,1/2[ such that for all x,y∈U, at least one of the following is true:
(39)(FZ1):∧{t>0:N(Tu-Tv,t)≥α}≤α1∧{t>0:N(u-v,t)≥α},(FZ2):∧{t>0:N(Tu-Tv,t)≥α}≤β[∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(v-Tv,t)≥α}],(FZ3):∧{t>0:N(Tu-Tv,t)≥α}≤γ[∧{t>0:N(u-Tv,t)≥α}+∧{t>0:N(v-Tu,t)≥α}].

Proposition 19.

Let (U,N) be a fuzzy norm space and T:U→U an F-Zamfirescu operator. Then
(40)∀ϵ>0,Fϵz(T)≠∅.

Proof.

Let x,y∈U.

Supposing that (FZ2) holds, we have
(41)∧{t>0:N(Tu-Tv,t)≥α}≤β[∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(v-Tv,t)≥α}]≤β[∧{t>0:N(v-u,t)≥α}+∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(Tu-Tv,t)≥α}]=2β∧{t>0:N(u-Tu,t)≥α}+β∧{t>0:N(u-v,t)≥α}+β∧{t>0:N(Tu-Tv,t)≥α}.
Thus
(42)∧{t>0:N(Tu-Tv,t)≥α}≤2β1-β∧{t>0:N(u-Tu,t)≥α}+β1-β∧{t>0:N(u-v,t)≥α}.

Supposing (FZ3) holds, we have
(43)∧{t>0:N(Tu-Tv,t)≥α}≤γ[∧{t>0:N(u-Tv,t)≥α}+∧{t>0:N(v-Tu,t)≥α}]≤γ[∧{t>0:N(u-v,t)≥α}+∧{t>0:N(v-Tv,t)≥α}]+γ[∧{t>0:N(v-Tv,t)≥α}]+γ[∧{t>0:N(v-Tv,t)≥α}+∧{t>0:N(Tv-Tu,t)≥α}]=γ∧{t>0:N(Tu-Tv,t)≥α}+2γ∧{t>0:N(v-Tv,t)≥α}+γ∧{t>0:N(u-v,t)≥α}.
Thus
(44)∧{t>0:N(Tu-Tv,t)≥α}≤2γ1-γ∧{t>0:N(v-Tv,t)≥α}+γ1-γ∧{t>0:N(u-v,t)≥α}.

Then
(46)∧{t>0:N(Tu-Tv,t)≥α}≤2γ1-γ∧{t>0:N(u-Tu,t)≥α}+γ1-γ∧{t>0:N(u-v,t)≥α}.

Now looking at (FZ1), (42), (44), and (46), we can denote
(47)η=max{α1,β1-β,γ1-γ},
and it is easy to see that η∈[0,1[.

For T satisfying at least one of the conditions (FZ1), (FZ2), and (FZ3), we have that
(48)∧{t>0:N(Tu-Tv,t)≥α}≤2η∧{t>0:N(u-Tu,t)≥α}+η∧{t>0:N(u-v,t)≥α},∧{t>0:N(Tu-Tv,t)≥α}≤2η∧{t>0:N(v-Tv,t)≥α}+η∧{t>0:N(u-v,t)≥α}
hold.

Using these conditions implied by (FZ1)–(FZ3) and taking x∈U, we have
(49)∧{t>0:N(Tnu-Tn+1u,t)≥α}=∧{t>0:N(T(Tn-1u)-T(Tnu),t)≥α}≤(48)2η∧{t>0:N(Tn-1u-T(Tn-1u),t)≥α}+η∧{t>0:N(Tn-1u-T(Tnu),t)≥α}=3η∧{t>0:N(Tn-1u-Tnu,t)≥α}.
Then
(50)∧{t>0:N(Tnu-Tn+1u,t)≥α}≤⋯≤(3η)3∧{t>0:N((u-Tu),t)≥α}.

Therefore
(51)limn→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0,∀u∈U.
Now by Lemma 8 it follows that Fϵz(T)≠∅, for all ϵ>0.

Definition 20.

A mapping T:U→U is an F-weak contraction operator if there exist α1∈[0,1[ and L≥0 such that(52)∧{t>0:N(Tu-Tv,t)≥α}≤α1∧{t>0:N(u-v,t)≥α}+L∧{t>0:N(v-Tu,t)≥α}
for all x,y∈U.

Proposition 21.

Let (U,N) be a fuzzy norm space and T:U→U an F-weak contraction operator. Then
(53)∀ϵ>0,Fϵz(T)≠∅.

Proof.

Let x∈U. Consider
(54)∧{t>0:N(Tnu-Tn+1u,t)≥α}=∧{t>0:N(T(Tn-1u)-T(Tnu),t)≥α}≤α1∧{t>0:N(Tn-1u-Tnu),t)≥α}+L∧{t>0:N(Tnu-Tnu,t)≥α}=∧{t>0:N(Tn-1u-Tnu,t)≥α}≤⋯≤α1n∧{t>0:N(u-Tu,t)≥α}.
But α1∈[0,1[. Therefore
(55)limn→∞∧{t>0:N(Tnu-Tn+1x,t)≥α}=0,∀u∈U.
Now by Lemma 8, it follows that Fϵz(T)≠∅, for all ϵ>0.

For the same operators, we will now use Lemma 10 in order to obtain new results.

Proposition 22.

Let (U,N) be a fuzzy norm space and T:U→U an a-contraction operator.

Then for some α∈(0,1),
(56)δα(Fϵz(T))≤2ϵ1-a,∀ϵ>0.

Proof.

Let ϵ>0. Condition (1) in Lemma 10 is satisfied, as one can see in the proof of Proposition 13.

We will show now that (2) also holds for a-contractions.

Let μ>0 and x,y∈Fϵz(T) and assume that
(57)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ
and aim to show that there exists an φ(μ)>0 such that
(58)∧{t>0:N(u-v,t)≥α}≤φ(μ).
We have
(59)∧{t>0:N(u-v,t)≥α}≤∧{t>0:N(Tu-Tv,t)≥α}+μ≤a∧{t>0:N(u-v,t)≥α}+μ.

Therefore
(60)(1-a)∧{t>0:N(u-v,t)≥α}≤μ,
which implies ∧{t>0:N(u-v,t)≥α}≤μ/(1-a).

So for every μ>0, there exist φ(μ)=μ/(1-a)>0 such that
(61)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Therefore ∧{t>0:N(Tu-Tv,t)≥α}≤φ(μ).

Now by Lemma 10 it follows that
(62)δα(Fϵz(T))≤φ(2ϵ),∀ϵ>0,
for some α∈(0,1), which means exactly that
(63)δα(Fϵz(T))≤2ϵ1-a,∀ϵ>0,
for some α∈(0,1).

Proposition 23.

Let (U,N) be a fuzzy norm space and T:U→U an F-Kannan operator.

Then for some α∈(0,1),
(64)δα(Fϵz(T))≤2ϵ(1+a),∀ϵ>0.

Proof.

Let ϵ>0. As in the proof of Proposition 22 we only verify that condition (2) in Lemma 10 holds.

Let μ>0 and x,y∈Fϵz(T) and assume that
(65)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Then
(66)∧{t>0:N(u-v,t)≥α}≤a[∧{t>0:N(u-Tu,t)≥α}+∧{t>0:N(v-Tv,t)≥α}]+μ.
As x,y∈Fϵz(T), we know that ∧{t>0:N(u-Tu,t)≥α}<ϵ and ∧{t>0:N(v-Tv,t)≥α}<ϵ.

Then ∧{t>0:N(u-v,t)≥α}≤2aϵ+μ. So for every μ>0 there exist φ(μ)=μ+2aϵ>0 such that
(67)∧{t>0:N(u-v,t)≥α}-a∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Therefore ∧{t>0:N(u-v,t)≥α}≤μ. Now by Lemma 10 it follows that
(68)δα(Fϵz(T))≤φ(2ϵ),∀ϵ>0,
for some α∈(0,1), which means exactly that
(69)δα(Fϵz(T))≤2ϵ(1+a),∀ϵ>0,
for some α∈(0,1).

Proposition 24.

Let (U,N) be a fuzzy norm space and T:U→U an F-Chatterjea operator.

Then for some α∈(0,1),
(70)δα(Fϵz(T))≤2ϵ(1+a)1-2a,∀ϵ>0.

Proof.

Let ϵ>0. We will only verify that condition (2) in Lemma 10 holds.

Let μ>0 and x,y∈Fϵz(T) and assume that
(71)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Then
(72)∧{t>0:N(u-v,t)≥α}≤a[∧{t>0:N(u-Tv,t)≥α}+∧{t>0:N(v-Tu,t)≥α}]+μ≤a∧{t>0:N(u-Tv,t)≥α}+a∧{t>0:N(v-Tu,t)≥α}+μ≤a[∧{t>0:N(u-v,t)≥α}-∧{t>0:N(v-Tv,t)≥α}]+a[∧{t>0:N(v-u,t)≥α}-∧{t>0:N(u-Tu,t)≥α}]+μ.
As x,y∈Fϵz(T), it follows that
(73)∧{t>0:N(u-v,t)≥α}≤2a∧{t>0:N(u-v,t)≥α}2aϵ+μ.
Then (1-2a)∧{t>0:N(u-v,t)≥α}≤2aϵ+μ⇒(74)∧{t>0:N(u-v,t)≥α}≤2aϵ+μ1-2a.

So for every μ>0 there exist φ(μ)=(2aϵ+μ)/(1-2a)>0 such that
(75)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Then ∧{t>0:N(u-v,t)≥α}≤φ(μ). Now by Lemma 10 it follows that
(76)δα(Fϵz(T))≤φ(2ϵ),∀ϵ>0,
for some α∈(0,1), which means exactly that
(77)δα(Fϵz(T))≤2aϵ+μ1-2a,∀ϵ>0,
for some α∈(0,1).

Proposition 25.

Let (U,N) be a fuzzy norm space and T:U→U an F-Zamfirescu operator. Then for some α∈(0,1),
(78)δα(Fϵz(T))≤2ϵ1+η1-η,∀ϵ>0,
where η=max{α1,β/(1-β),γ/(1-γ)}, and α, β, γ as in Definition 18.

Proof.

In the proof of Proposition 19 we have already shown that if f satisfies at least one of the conditions (FZ1), (FZ2), and (FZ3) from Definition 18, then(79)∧{t>0:N(Tu-Tv,t)≥α}≤2η∧{t>0:N(u-Tu,t)≥α}+η∧{t>0:N(u-v,t)≥α},∧{t>0:N(Tu-Tv,t)≥α}≤2η∧{t>0:N(v-Tv,t)≥α}+η∧{t>0:N(u-v,t)≥α}
hold.

Let ϵ>0. We will only verify that condition (2) in Lemma 10 is satisfied, as (1) holds; see the Proof of Proposition 19.

Let μ>0 and x,y∈Fϵz(T) and assume that
(80)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.

Let μ>0 and x,y∈Fϵz(T) and assume that
(81)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Then
(82)∧{t>0:N(u-v,t)≥α}≤∧{t>0:N(Tu-Tv,t)≥α}+μ≤2η∧{t>0:N(u-Tu,t)≥α}+η∧{t>0:N(u-v,t)≥α}+μ⟹(1-η)∧{t>0:N(u-v,t)≥α}≤2ηϵ+μ⟹∧{t>0:N(v-u,t)≥α}≤2ηϵ+μ1-η.

So for every μ>0 there exist φ(μ)=(2ηϵ+μ)/(1-η)>0 such that
(83)∧{t>0:N(u-v,t)≥α}-∧{t>0:N(Tu-Tv,t)≥α}≤μ.
Then ∧{t>0:N(u-v,t)≥α}≤φ(μ). Now by Lemma 10 it follows that
(84)δα(Fϵz(T))≤φ(2ϵ),∀ϵ>0,
for some α∈(0,1), which means exactly that
(85)δα(Fϵz(T))≤2ϵ1+η1-η,∀ϵ>0,
for some α∈(0,1).

Example 26.

Let us consider the linear space U=C[0,1], the space of all continuous real valued functions on [0,1] with the usual linear operations.

Consider the two norms on C[0,1] defined by
(86)∥x∥0={∫x01(x(t))2dt}1/2,∥x∥1=sup0≤t≤1|x(t)|.
Define two fuzzy norms N1 and N2 by
(87)N1(x,t)={1t>∥x∥0,0t≤∥x∥0,N2(x,t)={1t>∥x∥1,12∥x∥0≤t≤∥x∥1,0t≤∥x∥0.
For the fuzzy norm N1, its α-norm ∥·∥α1 is given by ∥·∥α1=∥x∥0 for all α∈(0,1), and for the fuzzy norm N2, its α-norm ∥·∥α2 is given by
(88)∥x∥α={∥x∥00<α≤12,∥x∥112<α<1.
Suppose that T:U→U with T(x)=x+1 for all x∈U. It can easily be seen that for 1/2<α<1 and x,y∈U,
(89)∧{t>0:N(Tx-Ty,t)≥α}≤12[∧{t>0:N(x-y,t)≥α}].
However, T has no any fixed point of course. But by the Proposition 13 for some ϵ>0, T has a fuzzy approximate fixed point in U. That is, there exists u0∈U such that
(90)∧{t>0:N(u0-Tu0,t)≥α}≤ϵ.

4. Conclusions

Nowadays, fixed point and operator theory play an important role in different areas of mathematics and its applications, particularly in mathematics, physics, differential equation, game theory, and dynamic programming. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the fuzzy type of the fixed point and operator theory can also play an important role in the new fuzzy area and fuzzy mathematical physics.

We think that this paper could be of interest to the researchers working in the field fuzzy functional analysis in particular, fuzzy approximate fixed point theory. We proved results about fuzzy approximate fixed points and fuzzy diameter on fuzzy norm spaces, starting from a result presented in [1], but the study may go further to other classes of operators, which will be the subject of future papers.

Acknowledgment

The authors are extremely grateful to the referees for their helpful suggestions for the improvement of the paper.

BerindeM.Approximate fixed point theoremsZadehL. A.Fuzzy setsKatsarasA. K.Fuzzy topological vector spaces. IIFelbinC.Finite dimensional fuzzy normed linear spaceKalevaO.SeikkalaS.On fuzzy metric spacesChengS. C.MordesonJ. N.Fuzzy linear operators and fuzzy normed linear spacesKramosilI.MichálekJ.Fuzzy metrics and statistical metric spacesChitraA.MordesonP. V.Fuzzy linear operators and fuzzy normed linear spacesBagT.SamantaS. K.Finite dimensional fuzzy normed linear spacesBagT.SamantaS. K.Fuzzy bounded linear operatorsBagT.SamantaS. K.Some fixed point theorems in fuzzy normed linear spacesKannanR.Some results on fixed pointsBerindeV.On the approximation of fixed points of weak contractive mappingsChatterjeaS. K.Fixed-point theorems