Microsoft Word - 426-435 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|426    Fuzzy Fixed Point Theorem for Some Types of Fuzzy Jungck Contractive Mappings in Hilbert Space Buthainah A.A.Ahmed Email: Buthainah_altai@scbaghdad.edu Manar Falih Dheyab Email: manarfalih@yahoo.com 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. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|427    1. 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 A is defined by A = {u : A(u) α } if α ∈ 0,1 A = 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 A is compact and convex for each α ∈ 0,1 , and sup ∈ A(u) = 1 .When A is an approximate quantity and A(u )=1 for some u ∈ H , A is identified with an approximate of u . 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. δ (A,B) = inf ∈ , ∈ ‖u v‖ ii. D (A,B) = dis(A , B ) , where “dis” is the Hausdorff distance iii. D(A , B) = sup D (A,B) iv. δ (A , B) = sup δ (A,B). It is to be noted that for any ‘α’, δ 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 u ⊂ Tu is called fuzzy fixed point of M . We shall use the following lemmas due to Helipern. Lemma δ (u,B) ‖u v‖ δ v, B , ∀ u, v ∈ H. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|428    Lemma If u ⊂ A , then δ (u ,B) D A, B , ∀ B ∈ W H . Lemma Let A ∈ W H and u ∈ H, if u ⊂ A then δ (u ,A)= 0, for each α ∈ 0,1 . Lemma Let H be a Hilbert space and M fuzzy mapping from H into W(H) and u ∈ H, then there exists u ∈ H such that u ⊂ Tu . Fuzzy fixed point theorem for types of fuzzy Jungck The scope of the function in this section, Ω the class of all functins Ψ: 0, ∞ → 0, ∞ , where Ψ is non-decreasing and ∑ 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 D T u , T v ξD S u , S v , for all u, v ∈ H. Example Let H =[0 ,1] , let us define M,N :H→ W H by S u s T v s ⎩ ⎪ ⎨ ⎪ ⎧0 , 0 s , s , 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 D S u , T v R‖u v‖ . Example 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 , T(0)(s) = 1, s 0 α, s ∈ 0, , s ∈ , 1 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|429    T(1)(s) = 1, s 0 3α, s ∈ 0, , s ∈ , 1 For all z ∈ 0 , 1 T(z)(s) = 1, s 0 α, s ∈ 0, 0 , s ∈ , 1 When 0 α , then M 0 = M z M 1 ={0} M 0 M z M 1 0 , , M 0 M 1 0 , 1 , M z 0 , and N 0 = N z N 1 ={0} N 0 N z N 1 0 , , N 0 N 1 0 , 1 , N z 0 , . Consequently D N u , M v H N u , M v 0 , ∀ u, v ∈ H, D N u , M v H N u , M v 0 , ∀ u, v ∈ H, D N u , M v H N u , M v 0 , ∀ u, v ∈ 0,1 and u, v ∈ 0,1 , D N u , M v H N u , M v , ∀ v ∈ 0,1 and u ∈ 0,1 . Now, D N u , M v = sup {D N u , M v = D N u , M v R‖u v‖ . 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 . 3. {N , M} are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of M and N . Proof Let u ∈ H, there exists u ∈ Nu ⊂ Mu . In general, choose u such that for n 1, 3,5, … … . u ∈ N u ⊂ Mu and ‖ u u ‖ =δ u , M u D M u , M u So ‖ u u ‖ D M u , M u Since M,N are a fuzzy Jungck contractive mappings , then ‖ u u ‖ D M u , M u ξ D N u , N u IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|430    Sine u ∈ N u and u ∈ N u . Then , ‖ u u ‖ ξ D N u , N u =ξ‖ u u ‖ Whenever ξ ∈ (0 , 1) ‖ u u ‖ ∑ u u ∑ ξ ‖u u ‖ ) ‖u u ‖ If n → ∞ , then ξ converge to 0. Therefore the sequence { u is a Cauchy sequence in H . So by completeness of H, { u converge to u ∈ H . Now, Since N is continuous fuzzy mapping , then N u also converges on H. Finally, we show that δ u , Mu = 0 δ u , M u ‖u u‖ + D N u , Mu Since , {N , M} are fuzzy R∗ weakly commuting .Then δ u , N u ‖u u‖ +R‖ u u‖ . Hence , δ u , N u 0 and u ⊂ Nu. Clearly u 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. D N u , N v Ψ β max m u , v .Such that, β ∈ 0,1 and m(u , v) = D M u , M v , , , , D M v , N v D 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 u is a common fuzzy fixed point of M and N . Proof Let u ∈ H, there exists u ∈ Mu ⊂ Nu . In general, choose u such that for n 1, 3,5, … … . u ∈ M u ⊂ Nu and ‖ u u ‖ = δ u , N u D N u , N u So ‖ u u ‖ D N u , N u By condition 1 , then ‖ u u ‖ D N u , N u Ψ β max m u , u m u , u = D M u , M u , , , , D M u , N u D M u , N u 2 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|431    m u , u = ‖ u u ‖ , ‖ ‖ ‖ ‖ , ‖ u u ‖ ‖ u u ‖ 2 m u , u = ‖ u u ‖ , ‖ ‖ , ‖ u u ‖ 2 Hence, ‖ u u ‖ Ψ β max ‖ u u ‖ , ‖ ‖ , ‖ ‖ ‖ u u ‖ Ψ β‖ u u ‖ . Therefore, ‖ u u ‖ Ψ ‖ u u ‖ Ψ D N u , N u ‖ u u ‖ Ψ ‖ u u ‖ . . . ‖ u u ‖ Ψ ‖ u u ‖ ‖ u u ‖ Ψ ‖ u u ‖ ⋯ … … … Ψ ‖ u u ‖ ‖ u u ‖ ∑ Ψ ‖ u u ‖ . Since, ∑ Ψ ‖ u u ‖ ∞ Therefore, the sequence { u is a Cauchy sequence in H. So by completeness of H, { u converges to x in H. Now, Since M is continuous fuzzy mapping , then M u also converges on H. Finally, we show that δ u , Nu = 0 δ u , N u ‖u u‖ + D N u , Nu ‖u u‖ Ψ β max m u , u m u , u = D M u , M u , , , , D M u , N u D M u , N u 2 Since , {M , N} are fuzzy R∗ weakly commuting .Then δ u , N u ‖u u‖ +Ψ ‖ u u‖ . Hence , δ u , N u 0 and u ⊂ Nu. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|432    Clearly u 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 N , N , N be a fuzzy mappings satisfies the following conditions: 1. D N u , N v β D N u , N u ξ D N v , N v λ D N u , N v γ D N u , N v η D N v , N v For all u, v ∈ H where γ, η, β, λ, ξ 0 with γ η β λ ξ 1 . 2. N and N are continuous fuzzy mapping. 3. N (H) ⊂ N H ∩ N H . 4. {N , N } and {N , N } are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of N , N and N . Proof: Let u ∈ H, there exists u and u such that u ∈ N u ⊂ N u and u ∈ N u ⊂ N u . By induction one can construct a sequence { u in H, such that for n= 1, 3 , 5, ….. u ∈ N u ⊂ N u and u ∈ N u ⊂ N u . And ‖ u u ‖ = δ u , N u D N u , N u So ‖ u u ‖ D N u , N u By condition 1 , then ‖ u u ‖ D N u , N u β D N u , N u ξ D N u , N u λ D N u , N u γ D N u , N u η D N u , N u ‖ u u ‖ β ‖ u u ‖ ξ ‖ u u ‖ λ ‖ u u ‖ γ ‖ u u ‖ η ‖ u u ‖ Hence ‖ u u ‖ ξ ‖ u u ‖ η ‖ u u ‖ β ‖ u u ‖ ‖ u u ‖ ‖ u u ‖ . Putting q = 1 Then, we have ‖u u ‖ q‖u u ‖ Now , for any positive integer m IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|433    ‖u u ‖ q‖u u ‖ ‖u u ‖ . ‖u u ‖ q q q ⋯ q ‖u u ‖ ‖u u ‖ Which implies that ‖u u ‖ → 0 as n → ∞ Hence {u is cauchy sequance in H, but H is a Hilbert space, so {u is converge to u. Now, Since N and N are continuous fuzzy mapping , then N u and N u also converges on H. Finally, we show that δ u , N u = 0 δ u , N u ‖u u‖ + D N u , N u ‖u u‖ β D N u , N u ξ D N u , N u λ D N u , N u γ D N u , N u η D N u , N u . Since , {N , N } and {N , N are fuzzy R∗ weakly commuting .Then δ u , N u λ γ ‖ u u‖ . Consequently , δ u , N u 0 and u ⊂ Nu. Clearly u is a common fuzzy fixed point of the fuzzy mappings N , N , N ,.In particular if α 1, then x is a common fixed point of N , N and N . ∎ Theorem Let H be a Hilbert space and N , N , N be a fuzzy mappings satisfy the following conditions: 1. D N u , N v β D N u , N v . D N v , N v ξ D N v , N v . D N u , N u λ D N u , N v . D N v , N u γ ‖v u‖ For all u, v ∈ H where γ, β, λ, ξ 0 with 2γ 2β λ 2ξ 1 . 2. N and N are continuous fuzzy mapping. 3. N (H) ⊂ N H ∩ N H . IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|434    4. {N , N } and {N , N } are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of N , N and N . Proof Similar to prove theorem 3.7 ∎ Definition Let H be a Hilbert space and N, M:H→ W H . A fuzzy mappings T and S are called fuzzy Jungck like contractive mapping, if there exists ξ ∈ 0 , 1 , such that D N u , N v ξ‖u v‖ Ψ D M u , M v , for all u, v ∈ H. Theorem Let H be a Hilbert space and N, M be a fuzzy Jungck like contractive mappings satisfies the following conditions: 1. M is continuous fuzzy mapping. 2. N(H)⊂ M H . 3. {M , N} are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of N and M . Proof Trivial ∎ Definition Let H be a Hilbert space and N, M:H→ W H . A fuzzy mappings T and S are called fuzzy Jungck generalized like contractive mapping, if there exists ξ ∈ 0 , 1 , such that D N u , N v ξ‖u v‖ Ψ D M u , N u , D N u , M v , for all u, v ∈ H. Theorem Let H be a Hilbert space and N, M be a fuzzy Jungck generalized like contractive mappings satisfies the following conditions: 1. M is continuous fuzzy mapping. 2. N(H)⊂ M H . 3. {M , N} are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of N and M . Proof: Trivial ∎ Definition Let H be a Hilbert space and N, M:H→ W H . A fuzzy mappings T and S are called fuzzy Jungck S- like contractive mapping, if there exists ξ ∈ 0 , 1 , such that D N u , N v ξ m u , v Ψ m u , v , for all u, v ∈ H And m u , v max ‖u v‖ , D N u , M v , D M v , N v , D M u , N v D M v , N u 2 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1816 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|435    Theorem Let H be a Hilbert space and N, M be a fuzzy Jungck S-like contractive mappings satisfies the following conditions: 1. M is continuous fuzzy mapping. 2. N(H)⊂ M H . 3. {M , N} are fuzzy R∗ weakly commuting . Then , there exists u ∈ H such that u is a common fuzzy fixed point of N and M . Proof Trivial ∎ References [1] VD. Estruch and A.Vidal, “A note on fixed fuzzy points for fuzzy mappings”,Rend. Ist. Mat. Univ. Trieste.32,39-45,2001. [2] Jungck, g commuting mappings and fixed points, The American mathematical monthly,.83. 4,1976, 261-263 [3] L.A.zadeh,probability measures of fuzzy events, J.Math.Anal.Appl.23,421- 427,.1968 [4] S.Heilpern, Fuzzy mappings and fixed point theorem, J.Math. Anal.83, 566-569. 1981