IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
Fuzzy Fixed Point Theorem for Some Types of Fuzzy Jungck Contractive Mappings in Hilbert Space Buthainah A.A.Ahmed Email:
[email protected] Manar Falih Dheyab Email:
[email protected] Dept. of Mathematics / College of Science / University of Baghdad
Abstract In this paper, developed Jungck contractive mappings into fuzzy Jungck contractive and proved fuzzy fixed point for some types of generalize fuzzy Jungck contractive mappings. Keywords: fuzzy mapping , fuzzy fixed point and Jungck contractive .
For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|445
IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
Introduction The concept of fuzzy set was introduced by L.Zadeh [3]in (1965).After that a lot of work has been done regarding fuzzy set and fuzzy mappings. The concept of fuzzy mapping was first introduced by Heilpern [4].In (2001) , Estruch and Vidal [1] proved a fuzzy fixed point theorem for fuzzy contractive mappings .Jungck, G.[2][(1976) introduced Jungck contractive mapping and proved fixed point theorems . In this paper , we introduced Jungck contractive mapping and studied some results of fuzzy fixed point theorems for some types of generalized fuzzy Jungck contractive mapping in Hilbert space.
Preliminaries In this section, we recall some basic definitions and preliminaries that will be needed in this paper . Definition 2.1[3]: Let H be a Hilbert space and F(H) be a collection of all fuzzy sets in H . Let A ∈ F(H) and α ∈ [0, 1] the α − level set of A, denoted by Aα is defined by Aα = {u : A(u) ≥ α } if α ∈ [0,1] A0 = {u : A(u) > α } Where B denotes the closure of a set B. Definition A fuzzy set A is said to be an approximate quantity if and only if Aα is compact and convex for each α ∈ [0,1] , and supu∈X A(u) = 1 .When A is an approximate quantity and A(u0 )=1 for some u0 ∈ H , A is identified with an approximate of u0 . The collection of all fuzzy sets in H is denoted by F(H) and W(H) is the sub collection of all approximate quantities.
Definition Let A , B ∈ W(H) and α ∈ [0,1]. Then i. δα (A,B) = infu∈Aα , v∈Bα ‖u − v‖ ii. Dα (A,B) = dis(Aα , Bα ) , where “dis” is the Hausdorff distance iii. D(A , B) = supα Dα (A,B) iv. δ (A , B) = supα δα (A,B). It is to be noted that for any ‘α’, δα is a non decreasing as well as continuous function. Definition Let A , B ∈ W(H). An approximate quantity A is said to be more accurate than B (denoted by A⊂ B) if and only if A(u) ≤ B(u), ∀ u ∈ H.
Definition A mapping M from the set H into W(H) is said to be fuzzy mapping. Definition The point u ∈ H is called fixed point for the fuzzy mapping M if {u}⊂Tu. If uα ⊂ Tu is called fuzzy fixed point of M . We shall use the following lemmas due to Helipern. For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|446
IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
Lemma δα (u,B) ≤ ‖u − v‖ + δα (v, B), ∀ u, v ∈ H. Lemma If {u0 } ⊂ A , then δα(u0 ,B) ≤ Dα (A, B), ∀ B ∈ W(H). Lemma Let A ∈ W(H) and u0 ∈ H, if{u0 } ⊂ A then δα (u0 ,A)= 0, for each α ∈ [0,1]. Lemma Let H be a Hilbert space and M fuzzy mapping from H into W(H) and u0 ∈ H, then there exists u1 ∈ H such that{u1 } ⊂ Tu0 .
Fuzzy fixed point theorem for types of fuzzy Jungck The scope of the function in this section, [0, ∞), where Ψ is non-decreasing and
Ω the class of all functins Ψ: [0, ∞) →
n n ∑∞ n=1 Ψ (t) < ∞ , for each t > 0 and Ψ is n − th iteration of Ψ and
Ψ(0) = 0.
First of all, we introduce the following definitions and examples.
Definition Let H be a Hilbert space and M, N:H→ W(H). A fuzzy mappings M and N are called fuzzy Jungck contraction mapping if there exists ξ ∈ [0 , 1] , such that D2 (T(u) , T(v)) ≤ ξD2 (S(u), S(v)), for all u, v ∈ H.
Example
Let H =[0 ,1] , let us define M,N :H→ W(H) by 0
S(u)(s) = T(v)(s) =
5 4 3
{4
1
,0 ≤ s < 2 1
,2 ≤ s < ,
u+1 2
u+1 2
≤s≤1
And let ξ = 1 . Then, M and N are fuzzy Jungck contraction mapping .
Definition Let H be a Hilbert space and M, N:H→ W(H). A fuzzy mappings {M , N} are said to be fuzzy R∗ − weakly commuting if for each x, y ∈ H and R ≥ 0 , such that D2 (S(u) , T(v)) = R‖u − v‖2 .
Example For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|447
IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
Let H=[0,1] and define M,N:H→ W(H) be a fuzzy mapping such that M=N for all x ∈ [0,1] , T(u) is a fuzzy set on H given by , 1,
s=0
T(0)(s) = {α,
s ∈ (0, 3]
α
1
1
,
s ∈ (3 , 1]
1,
s=0
3
1
T(1)(s) = {3α, α 3
s ∈ (0, 3] 1
,
s ∈ (3 , 1]
For all z ∈ (0 , 1) 1,
s=0
T(z)(s) = {α,
s ∈ (0, 3]
1
1
0 ,
s ∈ (3 , 1] 1
When 0 ≤ α ≤ , then [M(0)]1=[M(z)]1 = [M(1)]1={0} 3
1
1
[M(0)]α = [M(z)]α = [M(1)]α = [0 , 3] , [M(0)]α = [M(1)]α = [0 , 1], [M(z)]α = [0 , 3] 3
3
3
and [N(0)]1=[N(z)]1 = [N(1)]1={0} 1 [N(0)]α = [N(z)]α = [N(1)]α = [0 , ] , [N(0)]α = [N(1)]α = [0 , 1], 3 3
1
3
[N(z)]α = [0 , 3]. Consequently 3
D12 (N(u) , M(v)) = H 2 ([N(u)]1 , [M(v)]1 ) = 0 , ∀ u, v ∈ H, D2α (N(u) , M(v)) = H 2 ([N(u)]α , [M(v)]α ) = 0 , ∀ u, v ∈ H, D2α (N(u) , M(v)) = H 2 ([N(u)]α , [M(v)]α ) = 0 , ∀ u, v ∈ (0,1) and u, v ∈ {0,1}, 3
3
2
3
2
Dα (N(u) , M(v)) = H ([N(u)]α , [M(v)]α ) = 3
3
3
1 3
, ∀ v ∈ (0,1) and u ∈ {0,1}.
Now, 1 D2 (N(u) , M(v)) = supα {D2α (N(u) , M(v))}=3 D2 (N(u) , M(v)) = R‖u − v‖2 . Then N and M is fuzzy R∗ − weakly commuting . We prove the following theorem:
Theorem Let H be a Hilbert space and M,N be a fuzzy Jungck contractive mappings satisfy the following conditions: 1. N is continuous fuzzy mapping. 2. M(H)⊂ N(H). For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|448
IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
3. {N , M} are fuzzy R∗ − weakly commuting . Then , there exists u ∈ H such that uα is a common fuzzy fixed point of M and N .
Proof Let u0 ∈ H, there exists u1 ∈ [Nu1 ] α ⊂ [Mu0 ] α . In general, choose un such that for n = 1, 3,5, … … . u2n+1 ∈ [N u2n+1 ] α ⊂ [Mu2n ] α and ‖ un − un+1 ‖2=δ2α ( un , M un ) ≤ D2 (M un−1 , M un ) So ‖ un − un+1 ‖2 ≤ D2 (M un−1 , M un ) Since M,N are a fuzzy Jungck contractive mappings , then ‖ un − un+1 ‖2 ≤ D2 (M un−1 , M un ) ≤ ξ D2 (N un−1 , N un ) Sine un−1 ∈ [N un−1 ] α and un ∈ [N un ] α . Then , ‖ un − un+1 ‖2 ≤ ξ D2 (N un−1 , N un )=ξ‖ un−1 − un ‖2 Whenever ξ ∈ (0 , 1) 2 ‖ un − un+m ‖2 ≤ ∑n+m−1 ‖uj+1 − uj ‖ j=n ≤ ∑n+m−1 ( ξj ‖u1 − u0 ‖2 ) j=n ξn
≤ 1−ξ ‖u1 − u0 ‖2 If n → ∞ , then ξn converge to 0. Therefore the sequence { un } is a Cauchy sequence in H . So by completeness of H, { un } converge to u ∈ H . Now, Since N is continuous fuzzy mapping , then [N u2n+1 ] α also converges on H. Finally, we show that δ2α (u , Mu)= 0 δ2α (u , M(u)) ≤ ‖un − u‖2 + D2 (N un , Mu) Since , {N , M} are fuzzy R∗ − weakly commuting .Then δ2α (u , N(u)) ≤ ‖un − u‖2 +R‖ un − u‖2 . Hence , δ2α (u , N(u)) = 0 and uα ⊂ Nu. Clearly uα is a common fuzzy fixed point of the fuzzy mappings N and M. In particular if α = 1, then u is a common fixed point of N and M. ∎
Theorem Let H be a Hilbert space and T,S be a fuzzy mappings satisfy the following conditions: 1. D2 (N(u), N(v)) ≤ Ψ(β max{m(u , v)}) .Such that, β ∈ (0,1) and D2 (M(u),N(v))+D2 (M(v),N(u))
m(u , v) = {D2 (M(u), M(v)), , 2 D2 (M(v) , N(v)) + D2 (M(v) , N(u)) } 2 2. M is continuous fuzzy mapping. 3. N(H) ⊂ M(H). 4. {N , M} are fuzzy R∗ − weakly commuting . Then , there exists u ∈ H such that uα is a common fuzzy fixed point of M and N .
Proof Let u0 ∈ H, there exists u1 ∈ [Mu1 ] α ⊂ [Nu0 ] α . In general, choose For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|449
IHSCICONF 2017
Special Issue
Ibn Al-Haitham Journal for Pure and Applied science
https://doi.org/ 10.30526/2017.IHSCICONF.1816
un such that for n = 1, 3,5, … … . u2n+1 ∈ [M u2n+1 ] α ⊂ [Nu2n ] α and ‖ un − un+1 ‖2= δ2α ( un , N un ) ≤ D2 (N un−1 , N un ) So ‖ un − un+1 ‖2 ≤ D2 (N un−1 , N un ) By condition 1 , then ‖ un − un+1 ‖2 ≤ D2 (N un−1 , N un ) ≤ Ψ(β max{m( un−1 , un )}) D2 (M( u
))+D2 (M( un ),N( un−1 ))
),N( u
n−1 n−1 m( un−1 , un )= {D2 (M( un−1 ), M( un )), 2 D2 (M( un ) , N( un )) + D2 (M( un ) , N( un−1 )) , } 2 2 ‖u − u ‖ +‖ un − un ‖2 m( un−1 , un ) ={ ‖ un−1 − un ‖2 , n−1 n , 2 2 2 ‖ un − un−1 ‖ + ‖ un − un ‖ } 2 2 ‖u −u ‖ m( un−1 , un ) ={ ‖ un−1 − un ‖2 , n−12 n , ‖ un−1 − un ‖2 } 2
Hence , ‖ un − un+1 ‖2 ≤ Ψ (β max{‖ un−1 − un ‖2 ,
‖ un−1 − un ‖2 ‖ un−1 − un ‖2 2
,
2
})
‖ un − un+1 ‖2 ≤ Ψ(β‖ un−1 − un ‖2 ) . Therefore, ‖ un − un+1 ‖2 ≤ Ψ(‖ un−1 − un ‖2 ) = Ψ(D2 (N( un−2 ), N( un−1 ))) ‖ un − un+1 ‖2 ≤ Ψ2 (‖ un−2 − un−1 ‖2 ) . . . 2 ‖ un − un+1 ‖ ≤ Ψn (‖ u0 − u1 ‖2 ) Ψ
n+m−1 ‖
‖ un − un+m ‖2 ≤ Ψn (‖ u0 − u1 ‖2 + ⋯ … … … + u0 − u1 ‖2 )
‖ un − un+m ‖2 ≤ ∑n+m−1 Ψj (‖ u0 − u1 ‖2 ) . j=n n 2 Since, ∑∞ n=1 Ψ (‖ u0 − u1 ‖ ) < ∞
Therefore, the sequence { un } is a Cauchy sequence in H. So by completeness of H, { un } converges to x in H. Now, Since M is continuous fuzzy mapping , then [M u2n+1 ] α also converges on H. Finally, we show that δ2α (u , Nu)= 0 δ2α (u , N(u)) ≤ ‖un − u‖2 + D2 (N un−1 , Nu) For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics|450
IHSCICONF 2017
Ibn Al-Haitham Journal for Pure and Applied science
Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1816
≤ ‖un − u‖2 + Ψ(β max{m( un−1 , u)}) D2 (M( u
),N( u
))+D2 (M(u),N( un−1 ))
n−1 n−1 m( un−1 , u)= {D2 (M( un−1 ), M(u)), 2 D2 (M(u) , N(u)) + D2 (M(u) , N( un−1 )) , } 2 Since , {M , N} are fuzzy R∗ − weakly commuting .Then
δ2α (u , N(u)) ≤ ‖un − u‖2 +Ψ(‖ un−1 − u‖2 ). Hence , δ2α (u , N(u)) = 0 and uα ⊂ Nu. Clearly uα is a common fuzzy fixed point of the fuzzy mappings N and M .In particular if α = 1, then u is a common fixed point of T and S. ∎
Theorem
Let H be a Hilbert space and N1 , N2 , N3 be a fuzzy mappings satisfies the following conditions: 1. D2 (N1 (u) , N1 (v)) ≤ β D2 (N1 (u), N2 (u)) + ξ D2 (N1 (v), N3 (v)) + λ D2 (N1 (u) , N2 (v)) + γ D2 (N1 (u) , N3 (v)) + η D2 (N1 (v) , N2 (v)) For all u, v ∈ H where γ, η, β, λ, ξ ≥ 0 with γ+η+β+λ+ξ
