L-Fuzzy Fixed Point Theorems in Metric Spaces - MS ...

L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi

L-Fuzzy Fixed Point Theorems in Metric Spaces M.S Thesis Presentation

Preliminaries Fixed Point Theory Fuzzy Theory

Results Edelstein Type L-fuzzy Fixed Point Theorems L-fuzzy Fixed Point Theorems via βFL -admissible L-fuzzy Fixed Point Theorems via a Rational Inequality

Bibliography

By:

Muhammad Sirajo Abdullahi1,2 Supervised by: Prof. Dr. Akbar Azam1 1 Department of Mathematics COMSATS Institute of Information Technology, Islamabad - Pakistan 2 Department of Mathematics Usmanu Danfodiyo University, Sokoto - Nigeria

COMSATS Institute of Information Technology 1/41

June 03, 2016

Overview L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi



Bibliography


1



1


2




Bibliography



1


2


3

Bibliography



Bibliography



1


2


3

Bibliography



Bibliography



Definition 1 Let (X, d) be a metric space. The map T : X −→ X is said to be a contraction if there exists k ∈ (0, 1) such that



Bibliography


d(T x, T y) ≤ kd(x, y),

for every x, y ∈ X.

(1)




d(T x, T y) ≤ kd(x, y),


(1)


Bibliography


Definition 2 [Banach, 1922]: Let (X, d) be a complete metric space and T : X −→ X be a contraction mapping. Then T has a unique fixed point in X.




d(T x, T y) ≤ kd(x, y),


(1)


Bibliography


Definition 2 [Banach, 1922]: Let (X, d) be a complete metric space and T : X −→ X be a contraction mapping. Then T has a unique fixed point in X. Definition 3 Suppose (X, d) is a metric space. Let T : X −→ X, then a point x ∈ X is said to be a fixed point of T if x = T x.


Preliminaries Fixed Point Theory

Definition 4 [Edelstein, 1962]: Let (X, d) be a compact metric space and T : X −→ X such that d(T x, T y) < d(x, y),

for every x, y ∈ X, x 6= y.

Fuzzy Theory


Bibliography


Then T has a unique fixed point in X.





Fuzzy Theory


Bibliography


Then T has a unique fixed point in X. Definition 5 [Nadler Jr, 1969]: Let (X, d) be a complete metric space and T : X −→ CB(X) be a multi-valued contraction mapping. Then T has a fixed point in X.





Fuzzy Theory


Bibliography


Then T has a unique fixed point in X. Definition 5 [Nadler Jr, 1969]: Let (X, d) be a complete metric space and T : X −→ CB(X) be a multi-valued contraction mapping. Then T has a fixed point in X. Definition 6 Let (X, d) be a metric space and T : X −→ CB(X), then a point x ∈ X is called a fixed point of T if x ∈ T x.


1


2


3

Bibliography



Bibliography





Bibliography


Definition 7 [Zadeh, 1965]: A function A : X −→ [0, 1] is called a fuzzy set.

L-Fuzzy Fixed Point Theorems in Metric Spaces


M. S. Abdullahi



Bibliography


Definition 8 [Heilpern, 1981]: Let (X, d) be a complete metric linear space and A : X −→ W(X) satisfying the following condition: there exists k ∈ (0, 1) such that d(Ax, Ay) ≤ kd(x, y),


Then there exists u ∈ X such that {u} ⊂ Au.

(2)



M. S. Abdullahi



Bibliography


Definition 8 [Heilpern, 1981]: Let (X, d) be a complete metric linear space and A : X −→ W(X) satisfying the following condition: there exists k ∈ (0, 1) such that d(Ax, Ay) ≤ kd(x, y),


(2)

Then there exists u ∈ X such that {u} ⊂ Au. Definition 9 [Goguen, 1967]: A function B : X −→ L is called an L-fuzzy set, where L is a complete distributive lattice containing 1L and 0L .




Bibliography


Remark 1 Whenever L = [0, 1], then the L-fuzzy set is just the fuzzy set in the original sense by [Zadeh, 1965].


Preliminaries


Fixed Point Theory Fuzzy Theory


Bibliography


Definition 10 Suppose (X, d) is a metric space. Let T : X −→ FL (X), then a point x ∈ X is said to be an L-fuzzy fixed point of T if x ∈ [T x]αL for some αL ∈ L\{0L }.


Preliminaries




Bibliography


Definition 10 Suppose (X, d) is a metric space. Let T : X −→ FL (X), then a point x ∈ X is said to be an L-fuzzy fixed point of T if x ∈ [T x]αL for some αL ∈ L\{0L }. Remark 2 If αL = 1L , then x ∈ X is called a fixed point of the L-fuzzy mapping T .


1


2


3

Bibliography



Bibliography


Edelstein Type L-fuzzy Fixed Point Theorems L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi


Definition 11 A mapping T : X −→ FL (X) is called L-fuzzy (globally) contractive if to every x, y ∈ X, x 6= y

Fuzzy Theory


Bibliography


d∞ L (T x, T y) < d(x, y).

(3)

Edelstein Type L-fuzzy Fixed Point Theorems L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi


Definition 11 A mapping T : X −→ FL (X) is called L-fuzzy (globally) contractive if to every x, y ∈ X, x 6= y

Fuzzy Theory

d∞ L (T x, T y) < d(x, y).

Results

(3)

Edelstein Type L-fuzzy Fixed Point Theorems L-fuzzy Fixed Point Theorems via βFL -admissible L-fuzzy Fixed Point Theorems via a Rational Inequality

Bibliography

Definition 12 A mapping T : X −→ FL (X) is called L-fuzzy locally contractive if to each x ∈ X there is an open set U containing x such that if y, z ∈ U, y 6= z d∞ L (T y, T z) < d(y, z).


(4)




Bibliography


Lemma 1 Let A ∈ WL (X). Then pαL (x, A) ≤ d(x, y) + pαL (y, A) for every x, y ∈ X.


Preliminaries




Bibliography


Lemma 2 If A ∈ WL (X) and {x0 } ⊂ A, then pαL (x0 , A) ≤ DαL (A, B) for all B ∈ WL (X).


Preliminaries




Lemma 2 If A ∈ WL (X) and {x0 } ⊂ A, then pαL (x0 , A) ≤ DαL (A, B) for all B ∈ WL (X). Theorem 1

Bibliography

Let (X, d) be a compact metric space. Suppose that T : X −→ FL (X) is an L-fuzzy contractive mapping. Then T has an L-fuzzy fixed point. COMSATS Institute of Information Technology 11/41




Bibliography


For u ∈ X. We have that [T u]αL is nonempty and compact.




Bibliography


For u ∈ X. We have that [T u]αL is nonempty and compact. Define a mapping h : X −→ [0, ∞) as h(u) = pαL (u, T u).


For u ∈ X. We have that [T u]αL is nonempty and compact. Define a mapping h : X −→ [0, ∞) as h(u) = pαL (u, T u). This implies,


Results

h(u) = pαL (u, T u) ≤ d(u, y) + pαL (y, T u)

Edelstein Type L-fuzzy Fixed Point Theorems

≤ d(u, y) + pαL (y, T y) + H([T u]αL , [T y]αL )

L-fuzzy Fixed Point Theorems via βFL -admissible

≤ d(u, y) + pαL (y, T y) + DαL (T u, T y)

L-fuzzy Fixed Point Theorems via a Rational Inequality

≤ d(u, y) + pαL (y, T y) + sup DαL (T u, T y)

Bibliography


≤ d(u, y) + pαL (y, T y) +

αL d∞ L (T u, T y).


For u ∈ X. We have that [T u]αL is nonempty and compact. Define a mapping h : X −→ [0, ∞) as h(u) = pαL (u, T u). This implies,


Results

h(u) = pαL (u, T u) ≤ d(u, y) + pαL (y, T u)

Edelstein Type L-fuzzy Fixed Point Theorems

≤ d(u, y) + pαL (y, T y) + H([T u]αL , [T y]αL )

L-fuzzy Fixed Point Theorems via βFL -admissible

≤ d(u, y) + pαL (y, T y) + DαL (T u, T y)


≤ d(u, y) + pαL (y, T y) + sup DαL (T u, T y)

Bibliography

≤ d(u, y) + pαL (y, T y) +

αL d∞ L (T u, T y).

Thus, by symmetry COMSATS Institute of Information Technology 12/41

|h(u) − h(y)| ≤ d(u, y) + d∞ L (T u, T y).




Bibliography


By (3) and the above inequality h is continuous, since X is compact then h attains it’s minimum say at a point u∗ ∈ X.




Bibliography


By (3) and the above inequality h is continuous, since X is compact then h attains it’s minimum say at a point u∗ ∈ X. Furthermore, by compactness of [T u∗ ]αL we can take u1 ∈ X, so that {u1 } ⊂ T u∗ and d(u∗ , u1 ) = pαL (u∗ , T u∗ ) = h(u∗ ). Then, {u∗ } ⊂ T u∗ .



Results

By (3) and the above inequality h is continuous, since X is compact then h attains it’s minimum say at a point u∗ ∈ X. Furthermore, by compactness of [T u∗ ]αL we can take u1 ∈ X, so that {u1 } ⊂ T u∗ and d(u∗ , u1 ) = pαL (u∗ , T u∗ ) = h(u∗ ). Then, {u∗ } ⊂ T u∗ . Otherwise, h(u1 ) = pαL (u1 , T u1 ) along with Lemma 2 will imply that


Bibliography

h(u1 ) = pαL (u1 , T u1 ) ≤ DαL (T u∗ , T u1 ) ≤ sup DαL (T u∗ , T u1 ) ≤
0 n where ψ is the nth iterate of ψ. Note that ψ(r) < r for any r > 0 and ψ(0) = 0.

L-fuzzy Fixed Point Theorems via a βFL -admissible L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi



Bibliography


Definition 13 Let Ψ be the family of non-decreasing functions P n ψ : [0, ∞) −→ [0, ∞) so that ∞ n=1 ψ (r) < ∞ for any r > 0 n where ψ is the nth iterate of ψ. Note that ψ(r) < r for any r > 0 and ψ(0) = 0. Definition 14 Suppose that (X, d) is a metric space. Let T : X −→ FL (X) and β : X × X −→ [0, ∞). Then T is called βFL -admissible if for any x ∈ X and y ∈ [T x]αL with β(x, y) ≥ 1, it implies that β(y, z) ≥ 1 for every z ∈ [T y]αL , αL ∈ L\{0L }.




Bibliography

Definition 15 Suppose (X, d) is a metric space. Let T : X −→ FL (X) and β : X × X −→ [0, ∞). Then T is called βF∗ L -admissible whenever for any x, y ∈ X, αL ∈ L\{0L }, β(x, y) ≥ 1 implies that β∗ ([T x]αL , [T y]αL ) ≥ 1, such that β∗ ([T x]αL ,[T y]αL ) := inf {β(u, v) : u ∈ [T x]αL and v ∈ [T y]αL }.




Lemma 3 Suppose x ∈ X, B ∈ WL (X), and {x} be an L-fuzzy set with membership function equal to characteristic function of set {x} If {x} ⊂ B, then pαL (x, B) = 0 for αL ∈ L\{0L }.

Fuzzy Theory


Theorem 3 Suppose (X, d) is a complete metric space, αL ∈ L\{0L } and T : X −→ QL (X) be an L-fuzzy mapping. Let there exists ψ ∈ Ψ and β : X × X −→ [0, ∞) so that, for every x, y ∈ X

Bibliography

β(x,y)DαL (T x, T y) ≤ ψ(Ω(x, y)) + K min{pαL (x, T x), pαL (y, T y), pαL (x, T y), pαL (y, T x)}, COMSATS Institute of Information Technology 20/41

where K ≥ 0 and


Theorem 3 (Cont.)

M. S. Abdullahi

Ω(x, y) = Preliminaries Fixed Point Theory Fuzzy Theory

pα (x, T y) + pαL (y, T x) max d(x, y), pαL (x, T x), pαL (y, T y), L . 2


Bibliography

If the following conditions hold, i. if {xn } is a sequence in X so that β(xn , xn+1 ) ≥ 1 and xn → b(n → ∞), then β(xn , b) ≥ 1, ii. there exists x0 ∈ X and x1 ∈ [T x0 ]αL so that β(x0 , x1 ) ≥ 1, iii. T is βFL -admissible or βF∗ L -admissible, iv. ψ is continuous.


Then T has an L-fuzzy fixed point.



Pick x0 ∈ X and x1 ∈ [T x0 ]αL , condition (ii) implies β(x0 , x1 ) ≥ 1. Since [T x0 ]αL is nonempty and compact, then there exists x2 ∈ [T x1 ]αL such that d(x1 , x2 ) = pαL (x1 , T x1 ) ≤ DαL (T x0 , T x1 ).

(5)

Fuzzy Theory


Bibliography

By (5) and condition (ii), we have d(x1 , x2 ) ≤ DαL (T x0 , T x1 ) ≤ β(x0 , x1 )DαL (T x0 , T x1 ) ≤ ψ(Ω(x0 , x1 )) + K min{pαL (x0 , T x0 ), pαL (x1 , T x1 ), pαL (x0 , T x1 ), pαL (x1 , T x0 )} ≤ ψ(Ω(x0 , x1 )) + K min{pαL (x0 , x1 ), pαL (x1 , x2 ), pαL (x0 , x2 ), 0}


= ψ(Ω(x0 , x1 )).


Similarly, for x2 ∈ X we have [T x2 ]αL ∈ C(X), then there exists x3 ∈ [T x2 ]αL such that

Preliminaries

d(x2 , x3 ) = pαL (x2 , T x2 ) ≤ DαL (T x1 , T x2 ).

(6)

d(x2 , x3 ) ≤ ψ(Ω(x1 , x2 )).

(7)


Results

and


Bibliography

Continuing in this pattern, a sequence {xn } is obtained such that, for each n ∈ N, xn ∈ [T xn−1 ]αL with β(xn−1 , xn ) ≥ 1, we have d(xn , xn+1 ) ≤ ψ(Ω(xn−1 , xn )), where


Ω(xn−1 , xn ) ≤ max{d(xn−1 , xn ), d(xn , xn+1 )}.




Hence, for each n ∈ N. d(xn , xn+1 ) ≤ ψ(max{d(xn−1 , xn ), d(xn , xn+1 )}),

(8)

Now, if n0 ∈ N exists so that pαL (xn0 , T xn0 ) = 0 by Lemma 3, we have {xn0 } ⊂ T xn0 , i.e. xn0 ∈ [T xn0 ]αL implying xn0 is an L-fuzzy fixed point of T . Thus, we assume pαL (xn , T xn ) > 0, for any n ∈ N, implying that d(xn−1 , xn ) > 0 for every n ∈ N. So, if d(xn−1 , xn ) < d(xn , xn+1 ) for some n ∈ N, by (8) and Definition 13 it follows that d(xn , xn+1 ) ≤ ψ(d(xn , xn+1 )) < d(xn , xn+1 ), which contradicts our earlier assumption. Thus

Bibliography

d(xn , xn+1 ) ≤ ψ(d(xn−1 , xn )) ≤ ψ(ψ(d(xn−2 , xn−1 )) .. . COMSATS Institute of Information Technology 23/41

≤ ψ n d(x0 , x1 ).

(9)




Now, since ψ ∈ Ψ and continuous, then there exist > 0 and h = h() ∈ N such that X

ψ n d(x0 , x1 ) < .

n≥h

Let m > n > h. By (9) and (10), we obtain d(xn , xm ) ≤

Bibliography

≤

m−1 X k=n m−1 X

d(xk , xk+1 ) ψ k d(x0 , x1 )

k=n

≤ COMSATS Institute of Information Technology 24/41

X n≥h

ψ n d(x0 , x1 ) < .

(10)


Thus, {xn } is Cauchy sequence and since X is complete therefore we have b ∈ X so that xn → b as n → ∞. Now, we show that b ∈ [T b]αL . So consider

Preliminaries

d(b, [T b]αL ) ≤ d(b, xn+1 ) + d(xn+1 , [T b]αL )



≤ d(b, xn+1 ) + H([T xn ]αL , [T b]αL ) ≤ d(b, xn+1 ) + DαL (T xn , T b) ≤ d(b, xn+1 ) + β(xn , b)DαL (T xn , T b) ≤ ψ(Ω(xn , b)) + K min{pαL (xn , T xn ), pαL (b, T b), pαL (xn , T b), pαL (b, T xn )}

≤ ψ max d(xn , b), pαL (xn , T xn ), pαL (b, T b),

Bibliography

pαL (xn , T b) + pαL (b, T xn ) 2 + K min{pαL (xn , T xn ), pαL (b, T b), pαL (xn , T b), pαL (b, T xn )}


= ψ(pαL (b, T b))). (11)


Letting n → ∞ in (11), we have

M. S. Abdullahi

Preliminaries

d(b, [T b]αL ) ≤ ψ(pαL (b, T b))) < pαL (b, T b))


= d(b, [T b]αL ),


Bibliography


a contraction. Hence, b ∈ [T b]αL ,

αL ∈ L\{0L }.


Letting n → ∞ in (11), we have

M. S. Abdullahi

Preliminaries

d(b, [T b]αL ) ≤ ψ(pαL (b, T b))) < pαL (b, T b))


= d(b, [T b]αL ),


a contraction. Hence, b ∈ [T b]αL ,

αL ∈ L\{0L }.

Bibliography

Suppose (X, ) is a POSET, then Z := {(u, v) ∈ X × X : u v or v u}. COMSATS Institute of Information Technology 26/41




Bibliography


Definition 16 Suppose (X, ) is a POSET, whenever (un , u) ∈ Z for every n ∈ N, with a sequence un → u(n → ∞) and (un , un+1 ) ∈ Z for every n ∈ N. Then, (X, ) is said to have an ordered sequential limit property.




Bibliography


Definition 16 Suppose (X, ) is a POSET, whenever (un , u) ∈ Z for every n ∈ N, with a sequence un → u(n → ∞) and (un , un+1 ) ∈ Z for every n ∈ N. Then, (X, ) is said to have an ordered sequential limit property. Definition 17 Suppose that (X, ) is a POSET and αL ∈ L\{0L }. Then the L-fuzzy mapping T : X −→ QL (X) is called comparative, if for any x ∈ X and y ∈ [T x]αL where (x, y) ∈ Z, it implies that (y, z) ∈ Z for every z ∈ [T y]αL .



Theorem 4 Suppose (X, d, ) is a complete partially ordered metric space, αL ∈ L\{0L } and T : X −→ QL (X) be an L-fuzzy mapping. Let there exists ψ ∈ Ψ so that, for every (x, y) ∈ Z

Fuzzy Theory


Bibliography

DαL (T x, T y) ≤ ψ(Ω(x, y)) + K min{pαL (x, T x), pαL (y, T y), pαL (x, T y), pαL (y, T x)}, where K ≥ 0 and Ω(x, y) =

max d(x, y), pαL (x, T x), pαL (y, T y), COMSATS Institute of Information Technology 28/41

If the following conditions hold,

pαL (x, T y) + pαL (y, T x) . 2




Bibliography


Theorem 4 (Cont.) I. X has the OSL property, II. there exists x0 ∈ X and x1 ∈ [T x0 ]αL so that (x0 , x1 ) ∈ Z, III. T is comparative L-fuzzy mapping, IV. ψ is continuous. Then T has an L-fuzzy fixed point. Remark 4 i. If we consider L = [0, 1] in Theorem 3 and 4 above, we get Theorem 1(2) and 3 of [Phiangsungnoen et al., 2014] respectively; ii. If αL = 1L in Theorem 3 and 4 then by Remark 2 the L-fuzzy mapping T has a fixed point.


1


2


3

Bibliography



Bibliography


L-fuzzy Fixed Point Theorems via a Rational Inequality L-Fuzzy Fixed Point Theorems in Metric Spaces M. S. Abdullahi



Bibliography


Lemma 3 [Nadler Jr, 1969]: Suppose (X, d) is a metric space. Let Y, Z ∈ CB(X). If y ∈ Y then d(y, Z) ≤ H(Y, Z).


Preliminaries




Bibliography


Lemma 4 [Nadler Jr, 1969]: Suppose (X, d) is a metric space. Let Y, Z ∈ CB(X) and ψ ∈ R(> 0). Then, for any a ∈ Y there exists y ∈ Z such that d(a, y) ≤ H(Y, Z) + ψ.


Preliminaries




Bibliography

Lemma 4 [Nadler Jr, 1969]: Suppose (X, d) is a metric space. Let Y, Z ∈ CB(X) and ψ ∈ R(> 0). Then, for any a ∈ Y there exists y ∈ Z such that d(a, y) ≤ H(Y, Z) + ψ. In the sequel, suppose Tˆ is an induced mapping as defined below: For x, y, t ∈ X. Tˆ(x)(t) = {y : T (x)(y) = max T (x)(t)}. t





Bibliography


Lemma 5 Suppose (X, d) is a metric space and z ∈ X. Let T : X −→ FL (X) be an L-fuzzy mapping so that Tˆu ∈ C(X) for every u ∈ X. Then z ∈ Tˆ(z) ⇐⇒ T (z)(z) ≥ T (z)(u) for any u ∈ X.




Bibliography


Lemma 5 Suppose (X, d) is a metric space and z ∈ X. Let T : X −→ FL (X) be an L-fuzzy mapping so that Tˆu ∈ C(X) for every u ∈ X. Then z ∈ Tˆ(z) ⇐⇒ T (z)(z) ≥ T (z)(u) for any u ∈ X. Lemma 6 Suppose (V, d) is a complete metric linear space, z0 ∈ V and T : V −→ WL (V ) be an L-fuzzy mapping. Then there exists z1 ∈ X such that {z1 } ⊂ T z0 .



Results Edelstein Type L-fuzzy Fixed Point Theorems L-fuzzy Fixed Point Theorems via βFL -admissible

Theorem 5 Suppose (X, d) is a complete metric space. Let S, T : X −→ FL (X) be L-fuzzy mappings, such that for any x ∈ X, there is αLS (x) , αLT (x) ∈ L\{0L } where [Sx]αLS (x) , [T x]αLT (x) ∈ CB(X). If H([Sx]αLS (x) , [T y]αLT (y) ) ≤ λ1 d(x, y) + λ2 d(x, [Sx]αLS (x) )


+ λ3 d(y, [T y]αLT (y) ) +

λ4 d(x, [Sx]αLS (x) )d(y, [T y]αLT (y) ) 1 + d(x, y)

Bibliography

and λ3 + COMSATS Institute of Information Technology 32/41

λ4 d(x, [Sx]αLS (x) ) 1 + d(x, y)

< 1,

λ2 +

λ4 d(y, [T y]αLT (y) ) 1 + d(x, y)

< 1,

,




Theorem 5 (Cont.) for every x, y ∈ X with λ1 , λ2 , λ3 , λ4 ∈ R+ where λ1 + λ2 + λ3 + λ4 < 1. Then, there exists x∗ ∈ X so that T x∗ ∈ [Sx∗ ]αLS (x∗ ) [T x∗ ]αLT (x∗ ) . Corollary 1 Suppose (X, d) is a complete metric space. Let S, T : X −→ FL (X) be L-fuzzy mappings so that ˆ Tˆx ∈ CB(X). If Sx,

Bibliography


ˆ Tˆy) ≤ λ1 d(x, y) + λ2 d(x, Sx) ˆ + λ3 d(y, Tˆy) H(Sx, ˆ λ4 d(x, Sx)d(y, Tˆy) + , 1 + d(x, y)


Corollary 1 (Cont.) Preliminaries Fixed Point Theory Fuzzy Theory

and


Bibliography


λ3 +

ˆ λ4 d(x, Sx) < 1, 1 + d(x, y)

λ2 +

λ4 d(y, Tˆy) < 1, 1 + d(x, y)

for every x, y ∈ X with λ1 , λ2 , λ3 , λ4 ∈ R+ where λ1 + λ2 + λ3 + λ4 < 1. Then, there exists u ∈ X so that S(u)(u) ≥ S(u)(x) and T (u)(u) ≥ T (u)(x) for every x ∈ X.


Corollary 2 Suppose (X, d) is a complete metric space. Let K, L : X −→ CB(X) be multi-valued mappings. If



Bibliography


H(Kx, Ly) ≤ λ1 d(x, y) + λ2 d(x, Kx) + λ3 d(y, Ly) λ4 d(x, Kx)d(y, Ly) + , 1 + d(x, y) and λ3 +

λ4 d(x, Kx) < 1, 1 + d(x, y)

λ2 +

λ4 d(y, Ly) < 1, 1 + d(x, y)

for every x, y ∈ X with λ1 , λ2 , λ3 , λ4 ∈ R+ where λ1 + λ2 + λ3 + λ4 < 1. Then, there is x∗ ∈ X so that x∗ ∈ Kx∗ ∩ Lx∗ .


Corollary 3 Suppose (X, d) is a complete metric space. Let S, T : X −→ EL (X). If



Bibliography


d∞ L (Sx, T y) ≤ λ1 d(x, y) + λ2 p(x, Sx) + λ3 p(y, T y) λ4 p(x, Sx)p(y, T y) + , 1 + d(x, y) and λ3 +

λ4 p(x, Sx) < 1, 1 + d(x, y)

λ2 +

λ4 p(y, T y) < 1, 1 + d(x, y)

for every x, y ∈ X with λ1 , λ2 , λ3 , λ4 ∈ R+ where λ1 + λ2 + λ3 + λ4 < 1. Then, there exists u ∈ X so that {u} ⊂ Su and {u} ⊂ T u.


Corollary 4 Suppose (V, d) is a complete metric linear space. Let S, T : V −→ WL (V ). If



Bibliography


d∞ L (Sx, T y) ≤ λ1 d(x, y) + λ2 p(x, Sx) + λ3 p(y, T y) λ4 p(x, Sx)p(y, T y) + , 1 + d(x, y) and λ3 +

λ4 p(x, Sx) < 1, 1 + d(x, y)

λ2 +

λ4 p(y, T y) < 1, 1 + d(x, y)

for every x, y ∈ V with λ1 , λ2 , λ3 , λ4 ∈ R+ where λ1 + λ2 + λ3 + λ4 < 1. Then, there exists b ∈ V so that {b} ⊂ Sb and {b} ⊂ T b.




Bibliography


Corollary 5 Suppose (V, d) is a complete metric linear space. Let T : V −→ WL (V ) be an L-fuzzy mapping such that d∞ L (T x, T y) ≤ βd(x, y). for every x, y ∈ V , where 0 ≤ β < 1. Then, there exists b ∈ V so that {b} ⊂ T b.




Bibliography

Remark 5 i. If we consider L = [0, 1] in Theorem 5 and Corollary 1, 2, 3, 4 and 5 above, we get Theorem 2.1, 3.1, 3.2, 4.1, 4.2 and Corollary 4.3 of [Azam, 2011] respectively; ii. If L = [0, 1] in Corollary 5, then the result reduces to Theorem 3.1 of [Heilpern, 1981]; iii. If αL = 1L in Theorem 5 and Corollary 1, 2, 3 and 4, then by Remark 2 the L-fuzzy mappings S and T have a common fixed point; iv. If αL = 1L in Corollary 5, then by Remark 2 the L-fuzzy mapping T has a fixed point.


Bibliography L-Fuzzy Fixed Point Theorems in Metric Spaces

Azam, A. (2011). Fuzzy fixed points of fuzzy mappings via a rational inequality. Hacettepe Journal of Mathematics and Statistics, 40(3).

M. S. Abdullahi

Azam, A., Arshad, M., and Beg, I. (2009). Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos, Solitons and Fractals, 42(5):2836–2841.


Banach, S. (1922). Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math, 3(1):133–181.

Fuzzy Theory


Bibliography

Edelstein, M. (1962). On fixed and periodic points under contractive mappings. Journal of the London Mathematical Society, 37(1):74–79. Goguen, J. A. (1967). l-fuzzy sets. Journal of Mathematical Analysis and Applications, 18(1):145–174. Heilpern, S. (1981). Fuzzy mappings and fixed point theorems. Journal of Mathematical Analysis and Applications, 83(2):566–569. Nadler Jr, S. B. (1969). Multi-valued contraction mappings. Pacific J. Math, 30(2):475–488. Phiangsungnoen, S., Sintunavarat, W., and Kumam, P. (2014). Fuzzy fixed point theorems for fuzzy mappings via β-admissible with applications. Journal of Uncertainty Analysis and Applications, 2(20):1–11.


Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3):338–353.



Questions


Bibliography


Observations Suggestions




Bibliography


Thank You!!!