Approximate Fixed point property in Intuitionistic Fuzzy normed sapce

arXiv:1509.02755v1 [math.FA] 9 Sep 2015

APPROXIMATE FIXED POINT PROPERTY IN INTUITIONISTIC FUZZY NORMED SPACE ¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN Abstract. In this paper, we define concept of approximate fixed point property of a function and a set in intuitionistic fuzzy normed space. Furthermore, we give intuitionistic fuzzy version of some class of maps used in fixed point theory and investigate approximate fixed point property of these maps.

1. Introduction Fuzzy theory was introduced by Zadeh [23] and was generalized by Atanassov [3] as intuititonistic fuzzy theory. This theory is used many branches of science. Using idea of intuitionistic fuzzy, Park [18] defined intuitionistic fuzzy metric space, later Saadati and Park [20] introduced intuitionistic fuzzy normed space. Intuitionistic fuzzy analogous of many concepts used in functional analysis were studied via intuitionistic fuzzy metric and norm ([11], [16],[13], [15], [8],[21], [14]). Fixed point theory is one of fields studied intuitionistic fuzzy version. Some of works related intuitionistic fuzzy fixed point theory can be found in [1], [19], [4], [10], [9]. On the other hand, there are many problems which can be solved with fixed point theory. But in most cases, it is enough finding an approximate solution. So, the existence of fixed point may not be necessary for solution of a problem. A reason of being attractive of this approach is addition of strong conditions for the solution of problem. To find approximate solution of problem may be easier putting less requirement. Hence, it is natural to define approximate fixed point of a function and to produce theory related to this concept. It is meant that x is close to f (x) with approximate fixed point of f (x) . There are several studies related to this concept( [6], [17], [2], [5], [12], [7]). In this study, we define and study the concept of approximate fixed point property of a function and a set which is used fixed point theory in intuitionistic fuzzy normed space by inspiring studies of Berinde [5] and Anoop [2]. We give examples related this concept in intuitionistic fuzzy normed space. Firstly, we mention some concept used in our article. Definition 1 (see [22]). A binary operation ∗ : [0, 1] × [0, 1] is a continuous t-norm if it satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) a ∗ 1 = a for all a ∈ [0, 1]; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. Definition 2 (see [22]). A binary operation ⋄ : [0, 1]×[0, 1] is a continuous t-conorm if it satisfies the following conditions: (i) ⋄ is associative and commutative; (ii) ⋄ 2010 Mathematics Subject Classification. 05A15, 05C38, 15A15, 15A18. Key words and phrases. Approximate fixed point property; intuitionistic fuzzy normed space; classes of contractions. 1

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

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is continuous; (iii) a ⋄ 0 = a for all a ∈ [0, 1]; (iv) a ⋄ b ≤ c ⋄ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. Definition 3 (see [20]). Let ∗ be a continuous t-norm, ⋄ be a continuous t-conorm and X be a linear space over the field IF(R or C). If µ and ν are fuzzy sets on X × (0, ∞) satisfying the following conditions, the five-tuple (X, µ, ν, ∗, ⋄) is said to be an intuitionistic fuzzy normed space and (µ, ν) is called an intuitionistic fuzzy norm. For every x, y ∈ X and s, t > 0, (i) µ (x, t) + ν (x, t) ≤ 1, (ii) µ (x, t) > 0, (iii) µ (x, t) = 1 ⇐⇒  x=  0, t (iv) µ (ax, t) = µ x, |a| for each a 6= 0, (v) µ (x, t) ∗ µ (y, s) ≤ µ (x + y, t + s) (vi) µ (x, .) : (0, ∞) → [0, 1] is continuous, (vii) lim µ (x, t) = 1 and lim µ (x, t) = 0, t→∞

t→0

(viii) ν (x, t) < 1,ν (ix) ν (x, t) = 0 ⇐⇒  0,  x=

t for each a 6= 0, (x) ν (ax, t) = ν x, |a| (xi) ν (x, t) ⋄ ν (y, s) ≥ ν (x + y, t + s), (xii)ν (x, .) : (0, ∞) → [0, 1] is continuous, (xiii) lim ν (x, t) = 0 and lim ν (x, t) = 1, t→∞

t→0

we further assume that  (X, µ, ν, ∗, ⋄) satisfies the following axiom: a⋄a=a (xiv) for all a ∈ [0, 1] . a∗a=a We use IFNS instead of intuitionistic fuzzy normed space for the sake of abbreviation. Lemma 1 (see [20]). Let (µ, ν) be intuitionistic fuzzy norm. For any t > 0,the following hold: (i) µ (x, t) and ν (x, t) are nondecreasing and nonincreasing with respect to t, respectively. (ii) µ (x − y, t) = µ (y − x, t) and ν (x − y, t) = ν (y − x, t). Definition 4 (see [20]). A sequence (xk ) in (X, µ, ν, ∗, ⋄) converges to x if and only if µ (xk − x, t) → 1 and ν (xk − x, t) → 0 (µ,ν)

as k → ∞, for each t > 0. We denote the convergence of (xk ) to x by xk → x. Definition 5 (see [20]). Let (X, µ, ν, ∗, ⋄) be an IFNS. (X, µ, ν, ∗, ⋄) is said to be complete if every Cauchy sequence in (X, µ, ν, ∗, ⋄) is convergent. Definition 6 (see [16]). Let X and Y be two IFNS. f : X → Y is continuous at x0 ∈ X if (f (xk )) in Y convergences to f (x0 ) for any (xk ) in X converging to x0 . If f : X → Y is continuous at each element of X, then f : X → Y is said to be continuous on X. Definition 7 (see [13]). Let (X, µ, ν, ∗, ⋄) be an IFNS. A ⊂ X is dense in (X, µ, ν, ∗, ⋄) (µ,ν)

if there exists a sequence (xk ) ∈ A such that xk → x for all x ∈ X.

APPROXIMATE FIXED POINT PROPERTY IN IFNS

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Definition 8 (see [1]). (X, µ, ν, ∗, ♦) be an intuitionistic fuzzy metric space. We call the mapping f : X → X intuitionistic fuzzy contraction map, if there exists a ∈ (0, 1) such that µ (f (x) , f (y) , at) ≥ µ (x, y, t) and ν (f (x) , f (y) , at) ≤ µ (x, y, t) for all x, y ∈ X and t > 0. Definition 9 (see [4]). (X, µ, ν, ∗, ♦) be an intuitionistic fuzzy metric space. We call the mapping f : X → X intuitionistic fuzzy nonexpansive, if µ (f (x) , f (y) , t) ≥ µ (x, y, t) and ν (f (x) , f (y) , t) ≤ µ (x, y, t) for all x, y ∈ X and t > 0. Lemma 2 (see [1]). Let (X, µ, ν, ∗, ♦) be an intuitionistic fuzzy metric space. a) If lim xk = x and lim yk = y k→∞

k→∞

µ (x, y, t) ≤ lim inf µ (xk , yk , t) and ν (x, y, t) ≥ lim sup ν (xk , yk , t) k→∞

k→∞

for all t > 0. b) If lim xk = x and lim yk = y k→∞

k→∞

µ (x, y, t) ≥ lim sup µ (xk , yk , t) and ν (x, y, t) ≤ lim inf ν (xk , yk , t) k→∞

k→∞

for all t > 0. 2. Main Results Firstly, we define approximate fixed point property, diameter of a set in intitionistic fuzzy normed space and give examples. Definition 10. Let (X, µ, ν, ∗, ♦) be an IFNS and f : X → X be a function. Given ǫ > 0. It is said that x0 ∈ X is an intuitionistic fuzzy ǫ−fixed point or approximate fixed point (ifafp) of f if µ (f (x0 ) − x0 , t) > 1 − ǫ and ν (f (x0 ) − x0 , t) < ǫ for all t > 0. We denote the set of intuitionistic fuzzy ǫ−fixed points of f with (µ,ν) Fǫ (f ). Definition 11. It is said that f has the intuitionistic fuzzy approximate fixed point (µ,ν) (f ) is not empty for every ǫ > 0. property (ifafpp) if Fǫ Example 1. Let f : R → R be a function given by f (x) = x + 21 . For all x ∈ R and every t > 0, (R, µ, ν, ∗, ♦) is an intuitionistic fuzzy normed space with µ (x, t) =

t |x| and ν (x, t) = t + |x| t + |x|

where |.| is usual norm on R, a∗b = a.b and a♦b = min {a + b, 1} for all a, b ∈ [a, b] . 1 This function has not any fixed point. For ǫ > 2t+1 and t > 0, every x ∈ R is ǫ−fixed point since |f (x) − x| t > 1 − ǫ and ν (f (x) − x, t) = , µ (f (x) − x, t) = t + |f (x) − x| t + |f (x) − x| that is ǫt 1 < . 2 1−ǫ

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

4 (µ,ν)

But Fǫ

(f ) = ∅ for ǫ
0.

Example 2. Consider f (x) = x2 defined on (0, 1) . ((0, 1) , µ, ν, ∗, ♦) is an intuitionistic fuzzy normed space with respect to (µ, ν) in the Example 1. As known, f has not any fixed point on (0, 1) . We investigate intuitionistic fuzzy approximate fixed point of f. For every ǫ > 0 and t > 0,there exists x ∈ (0, 1) such that x satisfies µ (f (x) − x, t) =

|f (x) − x| t > 1 − ǫ and ν (f (x) − x, t) = t + |f (x) − x| t + |f (x) − x|

that is 2 x − x
0.

(f )

Definition 12. Let K be nonempty subset of (X, µ, ν, ∗, ♦) .We say that (δµ (K) , δν (K)) is intuitionistic fuzzy diameter of K with respect to t where δµ (K) = inf {µ (x − y, t) : x, y ∈ K} and δν (K) = sup {ν (x − y, t) : x, y ∈ K} for t > 0. Theorem 1. Let X be a intuitionistic fuzzy normed space, and f : X → X be a function. We suppose that (µ,ν)

(i)Fǫ (f ) 6= ∅ (ii) There exist ϑ (ǫ1 ) , ϑ (ǫ2 ) such that µ (x − y, t) ≥ ǫ1 ∗ µ (f (x) − f (y), t1 ) ⇒ µ (x − y, t) ≥ ϑ (ǫ1 ) , ∀x, y ∈ Fǫ(µ,ν) (f ) ν (x − y, t) ≤ ǫ2 ♦ν (f (x) − f (y), t2 ) ⇒ ν (x − y, t) ≤ ϑ (ǫ2 ) , ∀x, y ∈ Fǫ(µ,ν) (f ) (µ,ν)

for x, y ∈Fǫ  (f ) andǫ1 , ǫ2∈ (0, 1) .  (µ,ν) (µ,ν) (f ) = (ϑ (ǫ1 ) , ϑ (ǫ2 )) . (f ) , δν Fǫ Then δµ Fǫ (µ,ν)

Proof. Let ε > 0 and x, y ∈ Fǫ

(f ). Then

µ (f (x) − x, t) > 1 − ǫ and ν (f (x) − x, t) < ǫ and µ (f (y) − y, t) > 1 − ǫ and ν (f (y) − y, t) < ǫ. It can be written µ (x − y, t) ≥ ≥ =

      t t t µ f (x) − x, ∗ µ f (x) − f (y), ∗ µ f (y) − y, 3 3 3   t ǫ ∗ ǫ ∗ µ f (x) − f (y), 3   t ǫ ∗ µ f (x) − f (y), 3

APPROXIMATE FIXED POINT PROPERTY IN IFNS

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and       t t t ♦ν f (x) − f (y), ∗ ν f (y) − y, ν (x − y, t) ≤ ν f (x) − x, 3 3 3   t ≤ ǫ♦ǫ♦ν f (x) − f (y), 3   t = ǫ♦ν f (x) − f (y), . 3 (µ,ν)

By (ii), for every x, y ∈ Fǫ

(f ), we get

µ (x − y, t) ≥ ϑ (ǫ1 ) and ν (x − y, t) ≤ ϑ (ǫ2 ) . Hence, 

    δµ Fǫ(µ,ν) (f ) , δν Fǫ(µ,ν) (f ) = (ϑ (ǫ1 ) , ϑ (ǫ2 )) .



Now, we introduce intuitionistic fuzzy asymptotic regularity to investigate intuitionistic fuzzy approximate fixed point property of some operators. Definition 13. Let (X, µ, ν, ∗, ♦) be an IFNS and f : X → X be a function. It is said that f is intuitionistic fuzzy asymptotic regular if

lim µ f k+1 (x) − f k (x) , t = 1 and lim ν f k+1 (x) − f k (x) , t = 0 k→∞

k→∞

for every x ∈ X and t > 0.

Theorem 2. Let (X, µ, ν, ∗, ♦) be an IFNS and f : X → X be a function. If f has intuitionistic fuzzy asymptotic regular, then there are intuitionistic fuzzy approximate fixed points of f . Proof. Let x0 be arbitrary element of X. Since f is intuitionistic fuzzy asymptotic regular,

lim µ f k+1 (x0 ) − f k (x0 ) , t = 1 and lim ν f k+1 (x0 ) − f k (x0 ) , t = 0. k→∞

k→∞

In this case, for every ǫ > 0 there exists k0 (ǫ) ∈ N such that

µ f k+1 (x0 ) − f k (x0 ) , t > 1 − ǫ and ν f k+1 (x0 ) − f k (x0 ) , t < ǫ

for every k ≥ k0 (ǫ) . If we denote f k (x0 ) by y0 , we have


µ f k+1 (x0 ) − f k (x0 ) , t = µ f f k (x0 ) − f k (x0 ) , t = µ (f (y0 )) − y0 , t) > 1 − ǫ


and ν f k+1 (x0 ) − f k (x0 ) , t = ν f f k (x0 ) − f k (x0 ) , t = ν (f (y0 )) − y0 , t) < ǫ. This shows that y0 is intuitionistic fuzzy approximate fixed point of f .



Now, we introduce intuitionistic fuzzy analogous of mapping such as Kannan, Chatterjea, Zamfrescu and weak contraction and investigate that these maps have approximate fixed point under certain conditions. Firstly, we show that intuitionistic fuzzy contraction map has approximate fixed point property. Theorem 3. Let (X, µ, ν, ∗, ♦) be an IFNS, and f : X → X be a intuiuionistic (µ,ν) fuzzy contraction. Then every ǫ ∈ (0, 1) , Fǫ (f ) 6= ∅.

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¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

Proof. Let x ∈ X and ǫ ∈ (0, 1) , t > 0.



µ f k (x) − f k+1 (x), t = µ f f k−1 (x) − f f k (x) , t   t k−1 k ≥ µ f (x) − f (x), a   t k−2 k−1 ≥ µ f (x) − f (x), 2 a ≥ ...   t ≥ µ x − f (x), k a

and

ν f k (x) − f k+1 (x), t

For a ∈ (0, 1) , k → ∞ ⇒ fuzzy norm

t ak







= ν f f k−1 (x) − f f k (x) , t   t k−1 k ≤ ν f (x) − f (x), a   t k−2 k−1 ≤ ν f (x) − f (x), 2 a ≤ ...   t ≤ ν x − f (x), k a

→ ∞, by properties (vii) and (xiii) of intuitionistic

µ f k (x) − f k+1 (x), t ν f

k+1

By Theorem 2, it follows that

k




(x) − f (x) , t

(µ,ν) Fǫ (f )




→ 1 → 0.

6= ∅ for every ǫ ∈ (0, 1).



Example 3. The open interval (0, 1) is an intuitionistic fuzzy normed space with intuitionistic fuzzy norm, t-norm and t-conorm given in Example 1. Consider f : (0, 1) → (0, 1) given by f (x) = 12 x. This map has not any fixed point in (0, 1) . Furthermore, f is an intuitionistic fuzzy contraction map since   t t = t x2 y = µ (x − y, t) , µ f (x) − f (y), 2 2 + 2 − 2 x y   − t = t 2 x 2 y = ν (x − y, t) ν f (x) − f (y), 2 2 + 2 − 2 ǫt from for every x, y ∈ (0, 1) and t > 0. We write 12 x < 1−ǫ   x µ (x − f (x), t) = µ x − , t 2 x  t = µ >1−ǫ ,t = 2 t + x2

and

ν (x − f (x), t) = =

 x  ν x − ,t 2 x x  ,t = 2 ν 2 t+

x 2

< ǫ.

APPROXIMATE FIXED POINT PROPERTY IN IFNS

For every ǫ ∈ (0, 1) and t > 0 there exists x ∈ (0, 1) such that 21 x < intuitionistic fuzzy approximate fixed point property.

7 ǫt 1−ǫ .

So, f has

Definition 14. Let X be an intuitionistic fuzzy normed space. If there exist a ∈
0, 12 such that µ (f (x) − f (y), at) ≥ µ (x − f (x), t) ∗ µ (y − f (y), t) ν (f (x) − f (y), at) ≤ ν (x − f (x), t) ♦ν (y − f (y), t) for every x, y ∈ X and t > 0, then f : X → X is called intuitionistic fuzzy Kannan operator. Theorem 4. Let (X, µ, ν, ∗, ♦) be an IFNS havig partial order relation denoted by , where a ∗ b = min {a, b} and a♦b = max {a, b}, and f : X → X be an intuitionistic fuzzy Kannan operator satisfying x  f (x) for every x ∈ X. Assume that ⊂ XxX hold following conditions: (i)  is subvector space. or (ii) X is a complete ordered space. If µ (., t) is non-decreasing, ν (., t) is non-increasing for every t ∈ (0, ∞) and for (µ,ν) (f ) 6= every x  θ (θ is unit elemant in vector space X), then every ǫ ∈ (0, 1) , Fǫ ∅.

Proof. Let x ∈ X and ǫ ∈ (0, 1) , t > 0. We can write from x  f (x) for every x ∈ X, x  f (x)  f 2 (x)  f 3 (x)  ...  f k (x)  . . . . Considering assumptions, we have µ f k+1 (x) − f k (x), t




= ≥ ≥ ≥ = = =



µ(f f k (x) − f f k−1 (x) , t) t t µ(f k (x) − f k+1 (x), ) ∗ µ(f k−1 (x) − f k (x), ) a a t t µ(f k (x) − f k−1 (x), ) ∗ µ(f k−1 (x) − f k+1 (x), ) ∗ µ(f k−1 (x) − f k (x), 2a 2a t t µ(f k (x) − f k−1 (x), ) ∗ µ(f k−1 (x) − f k+1 (x), ) ∗ µ(f k−1 (x) − f k (x), 2a 2a t t µ(f k (x) − f k−1 (x), ) ∗ µ(f k−1 (x) − f k+1 (x), ) 2a 2a   t t k k−1 k+1 k−1 min µ(f (x) − f (x), ), µ(f (x) − f (x), ) 2a 2a t µ(f k (x) − f k−1 (x), ) 2a

t ) a t ) 2a

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

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≥ ≥

≥

= = =

≥ = ≥ ≥ ≥ = =

    t t µ f k−1 (x) − f k (x), 2 ∗ µ f k−2 (x) − f k−1 (x), 2 2a 2a   t µ f k−1 (x) − f k−2 (x), 2 4a     t t ∗µ f k−2 (x) − f k (x), 2 ∗ µ f k−2 (x) − f k−1 (x), 2 4a 2a   t µ f k−1 (x) − f k−2 (x), 2 4a     t t k−2 k k−2 k−1 ∗µ f (x) − f (x), 2 ∗ µ f (x) − f (x), 2 4a 4a     t t k−1 k−2 k−2 k µ f (x) − f (x), 2 ∗ µ f (x) − f (x), 2 4a 4a      t t k k−2 k−1 k−2 (x), 2 min µ f (x) − f (x), 2 , µ f (x) − f 4a 4a   t µ f k−1 (x) − f k−2 (x), 2 4a .. .   t k−(k−2) k−(k−1) µ f (x) − f (x), k−1 k−1 2 a   t µ f 2 (x) − f (x), k−1 k−1 2 a     t t 2 µ f (x) − f (x), k−1 k ∗ µ x − f (x), k−1 k 2 a 2 a       t t t 2 µ f (x) − x, k k ∗ µ x − f (x), k k ∗ µ x − f (x), k−1 k 2 a 2 a 2 a     t t µ f 2 (x) − x, k k ∗ µ x − f (x), k k 2 a 2 a      t t 2 min µ x − f (x), k k , µ x − f (x), k k 2 a 2 a   t µ x − f (x), k k 2 a

and ν f k+1 (x) − f k (x), t




= ≤ ≤ ≤ =



ν(f f k (x) − f f k−1 (x) , t) t t ν(f k (x) − f k+1 (x), )♦ν(f k−1 (x) − f k (x), ) a a t t ν(f k (x) − f k−1 (x), )♦ν(f k−1 (x) − f k+1 (x), )♦ν(f k−1 (x) − f k (x), 2a 2a t t ν(f k (x) − f k−1 (x), )♦ν(f k−1 (x) − f k+1 (x), )♦ν(f k−1 (x) − f k (x), 2a 2a t t ν(f k (x) − f k−1 (x), )♦ν(f k−1 (x) − f k+1 (x), ) 2a 2a

t ) a t ) 2a

APPROXIMATE FIXED POINT PROPERTY IN IFNS

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  t t max ν(f k (x) − f k−1 (x), ), ν(f k+1 (x) − f k−1 (x), ) 2a 2a t = ν(f k (x) − f k−1 (x), ) 2a     t t k−2 k−1 k−1 k (x) − f (x), 2 ≤ ν f (x) − f (x), 2 ♦ν f 2a 2a   t ≤ ν f k−1 (x) − f k−2 (x), 2 4a     t t k−2 k−1 k−2 k (x) − f (x), 2 ♦ν f (x) − f (x), 2 ♦ν f 4a 2a   t ≤ ν f k−1 (x) − f k−2 (x), 2 4a     t t k−2 k−1 k−2 k (x) − f (x), 2 ♦ν f (x) − f (x), 2 ♦ν f 4a 4a     t t k−1 k−2 k−2 k = ν f (x) − f (x), 2 ♦ν f (x) − f (x), 2 4a 4a      t t k k−2 k−1 k−2 (x), 2 = max ν f (x) − f (x), 2 , ν f (x) − f 4a 4a   t = ν f k−1 (x) − f k−2 (x), 2 4a .. .   t k−(k−2) k−(k−1) ≤ ν f (x) − f (x), k−1 k−1 2 a   t = ν f 2 (x) − f (x), k−1 k−1 2 a     t t 2 ♦ν x − f (x), ≤ ν f (x) − f (x), k−1 k 2 a 2k−1 ak       t t t ≤ ν f 2 (x) − x, k k ♦ν x − f (x), k k ♦ν x − f (x), k−1 k 2 a 2 a 2 a     t t ≤ ν f 2 (x) − x, k k ♦ν x − f (x), k k 2 a 2 a      t t = max ν x − f 2 (x), k k , ν x − f (x), k k 2 a 2 a   t = ν x − f (x), k k 2 a
t 0, 21 . Using Now, if we take limit for k → ∞, (2a) k tends to infinity for a ∈ prepoerties (vii) and (xiii) of intuitionistic fuzzy norm, !
t k k+1 =1 lim µ f (x) − f (x), t ≥ lim µ x − f (x), k k→∞ k→∞ (2a) !
t k k+1 lim ν f (x) − f (x), t ≤ lim ν x − f (x), = 0. k k→∞ k→∞ (2a) =

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¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

This means that intuitioonistic fuzzy Kannan operator is intuitionistic fuzzy asymptotic regular. That is, intuitioonistic fuzzy Kannan operator has approximate fixed point.  Corollary 1. In the Theorem 4 , if x  f (x) for intuitionistic fuzzy Kannan operator f and µ (., t) is non-increasing, ν (., t) is non-decreasing for every t ∈ (0, ∞) and for every x  θ, f has still intuitionistic fuzzy approximate fixed point property. Definition 15. Let X be an intuitionistic fuzzy normed space. If there exist a ∈
0, 21 such that µ (f (x) − f (y), at) ≥ µ (x − f (y), t) ∗ µ (y − f (x), t) ν (f (x) − f (y), at) ≤ ν (x − f (y), t) ♦ν (y − f (x), t)

for every x, y ∈ X and t > 0, then f : X → X is called intuitionistic fuzzy Chatterjea operator. Theorem 5. Let (X, µ, ν, ∗, ♦) be an IFNS havig partial order relation denoted by  , where a ∗ b = min {a, b} and a♦b = max {a, b}, and f : X → X be an intuitionistic fuzzy Chatterjea operator satisfying x  f (x) for every x ∈ X. Assume that ⊂ XxX hold following conditions: (i)  is subvector space. or (ii) X is a complete ordered space. If µ (., t) is non-decreasing, ν (., t) is non-increasing for every t ∈ (0, ∞) and for (µ,ν) every x  θ (θ is unit elemant in vector space X), then every ǫ ∈ (0, 1) , Fǫ (f ) 6= ∅. Proof. By taking into consideration assumption of theorem, we get µ f k+1 (x) − f k (x), t




≥ = ≥ = = ≥ =

t t µ(f k (x) − f k (x), ) ∗ µ(f k−1 (x) − f k+1 (x), ) a a t t 1 ∗ µ(f k−1 (x) − f k+1 (x), ) = µ(f k−1 (x) − f k+1 (x), ) a a t t µ(f k−1 (x) − f k (x), ) ∗ µ(f k (x) − f k+1 (x), ) 2a 2a   t t k−1 k k+1 k min µ(f (x) − f (x), ), µ(f (x) − f (x), ) 2a 2a t µ(f k−1 (x) − f k (x), ) 2a     t t k−1 k−1 k−2 k (x) − f (x), 2 µ f (x) − f (x), 2 ∗ µ f 2a 2a     t t k−2 k k−2 k (x) − f (x), 2 µ f (x) − f (x), 2 ∗ 1 = µ f 2a 2a

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    t t ≥ µ f k−2 (x) − f k−1 (x), 2 ∗ µ f k−1 (x) − f k (x), 2 4a 4a      t t = min µ f k−2 (x) − f k−1 (x), 2 , µ f k (x) − f k−1 (x), 2 4a 4a   t = µ f k−2 (x) − f k−1 (x), 2 4a .. .   t k−(k−2) k−(k−3) ≥ µ f (x) − f (x), k−2 k−2 2 a   t = µ f 2 (x) − f 3 (x), k−2 k−2 2 a     t t 2 2 3 ≥ µ f (x) − f (x), k−2 k−1 ∗ µ f (x) − f (x), k−2 k−1 2 a 2 a     t t 3 3 = µ f (x) − f (x), k−2 k−1 ∗ 1 = µ f (x) − f (x), k−2 k−1 2 a 2 a     t t 2 3 2 ≥ µ f (x) − f (x), k−1 k−1 ∗ µ f (x) − f (x) , k−1 k−1 2 a 2 a      t t 2 3 2 = min µ f (x) − f (x), k−1 k−1 , µ f (x) − f (x), k−1 k−1 2 a 2 a   t = µ f (x) − f 2 (x), k−1 k−1 2 a     t t 2 ≥ µ x − f (x), k−1 k ∗ µ f (x) − f (x), k−1 k 2 a 2 a     t t 2 2 = µ x − f (x), k−1 k ∗ 1 = µ x − f (x), k−1 k 2 a 2 a     t t ≥ µ x − f (x), k k ∗ µ f (x) − f 2 (x), k k 2 a 2 a      t t 2 = min µ x − f (x), k k , µ f (x) − f (x), k k 2 a 2 a   t = µ x − f (x), k k 2 a and

ν f k+1 (x) − f k (x), t




≤ = ≤ =

t t ν(f k (x) − f k (x), )♦ν(f k−1 (x) − f k+1 (x), ) a a t t 0♦ν(f k−1 (x) − f k+1 (x), ) = ν(f k−1 (x) − f k+1 (x), ) a a t t ν(f k−1 (x) − f k (x), )♦ν(f k (x) − f k+1 (x), ) 2a 2a   t t k+1 k k−1 k (x) − f (x), ) max ν(f (x) − f (x), ), ν(f 2a 2a

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

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= ≤ = ≤ = =

≤ = ≤ = ≤ = = ≤ = ≤ = =

t ν(f k−1 (x) − f k (x), ) 2a     t t ν f k−2 (x) − f k (x), 2 ♦ν f k−1 (x) − f k−1 (x), 2 2a 2a     t t k−2 k k−2 k (x) − f (x), 2 ν f (x) − f (x), 2 ♦0 = ν f 2a 2a     t t k−2 k−1 k−1 k ν f (x) − f (x), 2 ♦ν f (x) − f (x), 2 4a 4a      t t k k−1 k−2 k−1 (x), 2 max µ f (x) − f (x), 2 , ν f (x) − f 4a 4a   t ν f k−2 (x) − f k−1 (x), 2 4a .. .   t k−(k−2) k−(k−3) ν f (x) − f (x), k−2 k−2 2 a   t ν f 2 (x) − f 3 (x), k−2 k−2 2 a     t t 3 2 2 ν f (x) − f (x), k−2 k−1 ♦ν f (x) − f (x), k−2 k−1 2 a 2 a     t t 3 3 ν f (x) − f (x), k−2 k−1 ♦0 = ν f (x) − f (x), k−2 k−1 2 a 2 a     t t 2 2 3 ν f (x) − f (x), k−1 k−1 ♦ν f (x) − f (x) , k−1 k−1 2 a 2 a      t t 3 2 2 max ν f (x) − f (x), k−1 k−1 , ν f (x) − f (x), k−1 k−1 2 a 2 a   t ν f (x) − f 2 (x), k−1 k−1 2 a     t t ν x − f 2 (x), k−1 k ♦ν f (x) − f (x), k−1 k 2 a 2 a     t t ν x − f 2 (x), k−1 k ♦0 = ν x − f 2 (x), k−1 k 2 a 2 a     t t ν x − f (x), k k ♦ν f (x) − f 2 (x), k k 2 a 2 a      t t max ν x − f (x), k k , ν f 2 (x) − f (x), k k 2 a 2 a   t ν x − f (x), k k 2 a

t Since (2a) k → ∞ for k → ∞, by means of (vii) and (xiii) properties of intuitionistic fuzzy norm, we see intuitionistic fuzzy Chatterjea operator has approximate fixed point property. 

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13

Corollary 2. In the Theorem 5, if x  f (x) for intuitionistic fuzzy Chatterjea operator f and µ (., t) is non-increasing, ν (., t) is non-decreasing for every t ∈ (0, ∞) and for every x  θ, f has still intuitionistic fuzzy approximate fixed point property. Definition 16. Let X be an intuitionistic fuzzy normed space. A mapping f : X → X is called intuitionistic

fuzzy Zamfirecsu operator if there exists at least a ∈ (0, 1) , k ∈ 0, 12 , c ∈ 0, 21 such that at least one of the followings is true for every x, y ∈ X and t > 0 : i) µ (f (x) − f (y), at) ≥

µ (x − y), t)

ν (f (x) − f (y), at) ≤

ν (x) − y), t) .

ii) µ (f (x) − f (y), kt) ≥

µ (x − f (x), t) ∗ µ (y − f (y), t)

ν (f (x) − f (y), kt) ≤

ν (x − f (x), t) ♦ν (y − f (y), t) .

iii) µ (f (x) − f (y), ct) ≥

µ (x − f (y), t) ∗ µ (y − f (x), t)

ν (f (x) − f (y), ct) ≤

ν (x − f (y), t) ♦ν (y − f (x), t) .

Theorem 6. Let (X, µ, ν, ∗, ♦) be an IFNS havig partial order relation denoted by  , where a ∗ b = min {a, b} and a♦b = max {a, b}, and f : X → X be an intuitionistic fuzzy Zamfirescu operator satisfying x  f (x) for every x ∈ X. Assume that ⊂ XxX hold following conditions: (i)  is subvector space. or (ii) X is a complete ordered space. If µ (., t) is non-decreasing, ν (., t) is non-increasing for every t ∈ (0, ∞) and for (µ,ν) every x  θ (θ is unit elemant in vector space X), then every ǫ ∈ (0, 1) , Fǫ (f ) 6= ∅. Proof. The proof is clear from Theorem 4 and Theorem 5.



Definition 17. Let X be an IFNS. If there exist a ∈ (0, 1) and L ≥ 0 such that     t t ∗ µ y − f (x), µ (f (x) − f (y), t) ≥ µ x − y, a L     t t ν (f (x) − f (y), t) ≤ ν x − y, ♦ν y − f (x), a L for every x, y ∈ X and t > 0, then f : X → X is called intuitionistic fuzzy weak contraction operator. Theorem 7. Let X be an IFNS, and f : X → X be intuitionistic fuzzy weak contraction. (µ,ν) Then every ǫ ∈ (0, 1) , Fǫ (f ) 6= ∅.

14

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

Proof. Let x ∈ X and ǫ ∈ (0, 1) .



µ f k (x) − f k+1 (x), t = µ f f k−1 (x) − f f k (x) , t     t t ∗ µ f k (x) − f k (x), ≥ µ f k−1 (x) − f k (x), a L     t t ∗ 1 = µ f k−1 (x) − f k (x), = µ f k−1 (x) − f k (x), a a     t t ≥ µ f k−2 (x) − f k−1 (x), 2 ∗ µ f k−1 (x) − f k−1 (x), a L   t = µ f k−2 (x) − f k−1 (x), 2 ∗ 1 a   t ≥ µ f k−2 (x) − f k−1 (x), 2 a ≥ ...     t t = µ f k−(k−1) (x) − f k−(k−2) (x), k−1 = µ f (x) − f 2 (x), k−1 a a     t t ≥ µ x − f (x), k ∗ µ f (x) − f (x), a L     t t ≥ µ x − f (x), k ∗ 1 = µ x − f (x), k a a and ν f k (x) − f k+1 (x), t




= ≤ = ≤ = ≤ ≤ = ≤ ≤




ν f f k−1 (x) − f f k (x) , t     t t k−1 k k k ν f (x) − f (x), ♦ν f (x) − f (x), a L     t t k−1 k k−1 k ♦0 = ν f (x) − f (x), ν f (x) − f (x), a a     t t k−2 k−1 k−1 k−1 ν f (x) − f (x), 2 ♦ν f (x) − f (x), a L   t ν f k−2 (x) − f k−1 (x), 2 ♦0 a   t k−2 k−1 ν f (x) − f (x), 2 a ...     t t k−(k−1) k−(k−2) 2 ν f (x) − f (x), k−1 = ν f (x) − f (x), k−1 a a     t t ν x − f (x), k ♦ν f (x) − f (x), a L     t t ν x − f (x), k ♦0 = µ x − f (x), k a a

Since atk → ∞ for k → ∞, by means of (vii) and (xiii) properties of intuitionistic fuzzy norm, we see intuitionistic fuzzy weak contraction map has approximate fixed point property. 

APPROXIMATE FIXED POINT PROPERTY IN IFNS

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In the following, we give definition of approximate fixed point property of a set. Furthermore, we prove that a dense set of intuitionistic fuzzy Banach space has approximate fixed point property. Definition 18. Let X be IFNS and let K be subset of X. Then K is said to have intuitionistic fuzzy approximate fixed point property (ifafp) if every intuitionistic fuzzy nonexpansive map f : K → K satisfies the property that sup {µ (x − f (x) , t) : x ∈ K} = 1 and inf {ν (x − f (x) , t) : x ∈ K} = 0. Theorem 8. Let X be an intuitionistic fuzzy normed space having ifafp, K be dense subset of X. Then K has ifafpp. Proof. Let f : X → X be an intuitionistic fuzzy nonexpansive mapping. Firstly we prove that sup {µ (x − f (x) , t) : x ∈ K} = sup {µ (y − f (y) , s) : y ∈ X} and inf {ν (x − f (x) , t) : x ∈ K} = inf ν {(y − f (y) , s) : y ∈ X} for t, s > 0. Since K ⊂ X, sup {µ (y − f (y) , s) : y ∈ X} ≥ sup {µ (x − f (x) , t) : x ∈ K} and inf {ν (y − f (y) , s) : y ∈ X} ≤ inf {ν (x − f (x) , t) : x ∈ K} . (µ,ν)

Let y ∈ X. There exists a sequence (yk ) in K such that yk → 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 (yk − f (yk ) , t) ≥ µ (yk − y + y − f (y) + f (y) − f (yk ) , s)       t t t ∗ µ y − f (y) , ∗ µ yk − f (yk ) , ≥ µ yk − y, 3 3 3 and inf {ν (x − f (x) , t) : x ∈ K} ≤ ν (yk − f (yk ) , t) ≤ ν (yk − y + y − f (y) + f (y) − f (yk ) , t)       t t t ≤ ν yk − y, ♦ν y − f (y) , ♦ν yk − f (yk ) , . 3 3 3 Since f is intuitionistic fuzzy nonexpansive mapping, it is intuitionistic fuzzy con(µ,ν)

tinuous. Because, if yk → y, then µ (f (yk ) − f (y) , t) ν (f (yk ) − f (y) , t) (µ,ν)

≥ µ (yk − y, t) → 1 ≤ ν (yk − y, t) → 0.

(µ,ν)

So f (yk ) → f (y) when yk → y. If we take limit above inequalities, we get  s sup {µ (x − f (x) , t) : x ∈ K} ≥ µ y − f (y) , 3 and  s inf {ν (x − f (x) , t) : x ∈ K} ≤ ν y − f (y) , 3 for all y ∈ X and t, s > 0. Thus, if we take 3s = s′ , sup {µ (x − f (x) , t) : x ∈ K} ≥ sup {µ (y − f (y) , s′ ) : y ∈ X}

¨ ¨ MUZEYYEN ERTURK, VATAN KARAKAYA, AND M. MURSALEEN

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and inf {ν (x − f (x) , t) : x ∈ K} ≥ inf {ν (y − f (y) , s′ ) : y ∈ X} . Therefore our claim is proved. Now consider any intuitionistic fuzzy nonexpansive mapping fK : K → K. Since K is dense, there exists a sequence (yk ) in K such (µ,ν)

that yk → y for any y ∈ X. Since an intuitionistic fuzzy nonexpansive mapping is continuous, fK : K → K is intuitionistic fuzzy continuous and it can be extending by defining f (x) = lim (µ, ν) − f (xk ) on X. Hence we can consider f as an intuitionistic fuzzy nonexpansive mapping on X. Because, using Lemma 2, we get µ (f (x) − f (y) , t) = lim sup µ (f (xk ) − f (yk )) , t ≥ lim sup µ (xk − yk , t) = µ (x, y, t) k→∞

k→∞

and . ν (f (x) − f (y) , t) = lim inf ν (f (xk ) − f (yk )) ≤ lim inf ν (xk − yk , t) = ν (x, y, t) k→∞

k→∞

for all x, y ∈ X and t > 0. Then, since X has ifafpp sup {µ (x − f (x) , t) : x ∈ K} = sup {µ (y − f (y) , t) : y ∈ X} = 1 and inf {ν (x − f (x) , t) : x ∈ K} = inf ν {(y − f (y) , t) : y ∈ X} = 0. That is, for given any intuitionistic fuzzy nonexpansive mapping f on K we have sup {µ (x − f (x) , t) : x ∈ K} = 1 and inf {ν (x − f (x) , t) : x ∈ K} = 0 and K has (ifafpp).  Acknowledgement 1. This work is supported by Yildiz Technical University Scientific Research Projects Coordination Unit under the project number BAPK 201207-03-DOP03. References [1] Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 29(5) (2006) 1073-1078. [2] Anoop, S., Ravindran, K.: On approximate fixed point property and fixed points, International Mathematical Forum, 6 (2011) 281-288. [3] Atanassov, K.T.: Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20(1) (1986) 87-96. [4] Bag, T., Samanta, S.K.: Some fixed point theorems in fuzzy normed linear spaces, Information Sci., 177(16) (2007) 3271-3289. [5] Berinde, M.: Approximate fixed point theorems, Studia Univ. ”Babe¸s-Bolyai” Mathematica, LI (1) (2006) 11-15. [6] Dey, D., Laha, A.K., Saha, M.: Approximate coincidence point of two nonlinear mappings, Journal of Mathematics, Volume 2013 (2013), Article ID 962058, 4 pages.14 pages. [7] Dey, D., Saha, M.: Approximate fixed point of Reich operator, Acta Math. Univ. Comenianae, 82(1) (2013) 119-123. [8] Dinda, B., Samanta, T.: Intuitionistic fuzzy continuity and uniform convergence. Int. J. Open Problems Compt. Math 3(1) (2010) 8-26. [9] Ert¨ urk, M., Karakaya, V.: n-tuplet coincidence point theorems in intuitionistic fuzzy normed spaces, Jour. Function Spaces Appl., Volume 2014, 2014, Article number 821342, 14 pages. [10] Jeˇsi´ c, S.N.: Convex structure, normal structure and a fixed point theorem in intuitionistic fuzzy metric spaces, Chaos Solitons Fractals 41(1) (2009) 292-301. [11] Karakaya, V., S ¸ im¸sek, N., Ert¨ urk, M. , G¨ ursoy, F.: Statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces, Abstr. Appl. Anal., Volume 2012 (2012), Article ID 157467, 19 pages. [12] Matouˇskov´ a, E., Reich, S.: Reflexivity and approximate fixed points, Studia Math., 159 (2003) 403-415.

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[13] Mursaleen, M., Karakaya, V., Mohiuddine, S.A.: Schauder basis, separability, and approximation property in intuitionistic fuzzy normed space, Abstr. Appl. Anal., Volume 2010, Article ID 131868, 14 pages. [14] Mursaleen, M., Lohani, Q.M.D.: Intuitionistic fuzzy 2-normed space and some related concepts, Chaos Solitons Fractals, 42(1) (2009) 224-234. [15] Mursaleen, M., Lohani, Q.M.D., Mohiuddine, S.A., Intuitionistic fuzzy 2-metric space and its completion,Chaos Solitons Fractals, 42(2) (2009) 1258-1265. [16] Mursaleen, M., Mohiuddine, S.: Nonlinear operators between intuitionistic fuzzy normed spaces and Fr´ echet derivative, Chaos, Solitons & Fractals, 42(2) (2009) 1010-1015. [17] Pacurar, M., Pacurar, R.V.: Approximate fixed point theorems for weak contractions on metric spaces, Carpathian Journal of Mathematics, 23(1-2) (2007) 149-155. [18] Park, J.H.: Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 22(5) (2004) 10391046. [19] Rafi, M., Noorani, M.: Fixed point theorem on intuitionistic fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 3(1) (2006) 23-29. [20] Saadati, R., Park, J.H.: On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals, 27(2) (2006) 331-344. [21] Saadati, R., Park, J.H.: Intuitionistic fuzzy Euclidean normed spaces, Commun. Math. Anal., 1(2) (2006) 1-6. [22] Schweizer, B., Sklar, A.: Statistical metric spaces, Pacific J. Math., 10 (19601) 313-335. [23] Zadeh, L.A.: Fuzzy sets, Information and Control, 8(3) (1965) 338-353. Department of Mathematics, Adiyaman University, Adiyaman, Turkey E-mail address: [email protected];[email protected] ¨ ˘ bas¸ı Mahallesi, Sahir Kurutluog ˘ lu Cad. No:100, 40100 Ahi Evran Universitesi, Bag Merkez/Krs¸ehir, Turkey E-mail address: [email protected];[email protected] Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address: [email protected]