On Continuity of Complex Fuzzy Functions

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1 Mahemaical Theory and Modeling On Coninuiy of Complex Fuzzy Funcions Pishiwan O. Sabir Deparmen of Mahemaics Faculy of Science and Science Educaion Universiy of Sulaimani Iraq Absrac In his paper Some imporan heorems on fuzzy ype I-coninuous and II-coninuous of complex fuzzy funcions mapping generalized recangular valued bounded closed complex complemen normalized fuzzy numbers ino iself are proved. Keywords: Fuzzy Complex Numbers Fuzzy Complex Funcions Fuzzy Coninuiy. 1. Inroducion I is well known ha fuzzy complex numbers and fuzzy complex analysis were firs inroduced by (Buckley 1989; Buckley and Qu ). Scholars did series research abou he properies of fuzzy complex number from various aspecs (Quan 1996; Ma e al. 2009; Zheng and Ha 2009). Bu hese achievemens were very absrac and i did no consummae unil oday. In view of (Buckley 1989) Guangquan (1992) discussed he limi heory of he sequence of fuzzy complex numbers in deail giving a series of resuls abou limi heory which are he counerpars of well-known resuls valid for real numbers in classical mahemaics analysis. Buckley (1989) suggesed ha inroducing a meric on he space of fuzzy complex numbers provide o sudy convergence coninuiy and differeniaion of fuzzy complex funcion (Chun and Ma 1998; Qiu e al ; Ousmane and Congxin 2003; Shengquan 2006; Cai 2009; Sabir 2012). On he basis of Buckley s work some auhors coninued research and have exensively sudied he heory of fuzzy complex numbers and fuzzy complex analysis (Wu and Qiu 1999; Zengai and Shengquan 2006; Qiu and Shu 2008; Sun and Guo 2010; Sabir e al. 2012b). Sabir e al. (2012a) giving he definiions of he complemen normalized fuzzy numbers (CNFNs) bounded closed complex CNFNs (BCCCNFNs) generalized recangular valued BCCCNFNs (GRVBCCCNFNs) and discussed some of heir basic properies. In secion wo we firs review he definiions and characerizaions relaed o fuzzy complex ses. We will also presen he noaions needed in he res of he paper. In he las secion some heorems on he coninuiy of complex fuzzy funcions are proved. 2. Priliminaries A fuzzy se defined on he universal se is a funcion 01. Frequenly we will wrie () insead of. The family of all fuzzy ses in X is denoed by F(). The α level of a fuzzy se denoed by is he non-fuzzy se of all elemens of he universal se ha belongs o he fuzzy se a leas o he degree 01. The weak α level of a fuzzy se F() is he crisp se ha conains all elemens of he universal se whose membership grades in he given se are greaer han bu do no include he specified value of α. The larges value of for which he α-level is no empy is called he heigh of a fuzzy se denoed. The core of a fuzzy se is he non-fuzzy se of all poins in he universal se X a which! () is essenially aained. Le # F(). Then he union of fuzzy ses # denoed $ # # is defined by $ % % ()=! % ()= % () he inersecion of fuzzy ses # denoed ( # # is defined by (% % ()=)* % ()= % () and he complemen of # denoed # is defined by % () % ()=1 for all in he universal se. A fuzzy number a0 is a fuzzy se defined on he se of real numbers 1 2 characerized by means of a membership funcion 0 (): which saisfies: (1) 40 is upper semiconinuous (2) 0 ()=0 ouside some inerval 56 (3) There are real numbers 47 such ha and 0 () is increasing on ca 0 () is decreasing on 76 0 ()= We denoe he se of all fuzzy numbers by F. A fuzzy complex number ;< is defined by is membership funcions = >(?) which is a mapping from he se of ordinary complex numbers ino [01] if and only if = >(?) is coninuous; ;< is open bounded and conneced; and 2 ;< is nonempy compac and arcwise conneced. We use F o he se of all fuzzy complex numbers. Le (?? ) =A be any mapping from C C ino C. Buckley (1989) exend o F F ino F and wrie ; D ; E =F D if G D(A) = I(JKJKK)LM ( = D(? ) = E(? )). One obains FD =; D ; E or FD =; D ; E by using ; D ; E =; D ; E or ; D ; E =; D ; E respecively. 3. Properies of Coninuous Complex Fuzzy Funcions In his secion we give he coninuiy of complex fuzzy funcion mapping GRVBCCCNFNs ino iself. Mos resuls definiions and sandard noaions on fuzzy complex analysis which are used in his secion can be found 41

2 Mahemaical Theory and Modeling in Sabir e al. (2012a). Some of he resuls in his secion are wihou proofs owing o he simpliciies. Definiion 3.1. Le gr F U and V be a mapping from gr o he se of all GRVBCCCNFNs. If for arbirary ; gr here exiss unique WFXY F U make V;=WFXY we call V a complex fuzzy funcion defined on gr. Le WFXY F U we say V; is fuzzy coninuous a WFXY if for all Z[>0] here exiss ^[ >0] such ha _`V;VWFXYa Z[ as _;WFXY ^[. Definiion 3.2. We say V; is fuzzy ype I-coninuous (resp. II-coninuous) a WFXY F U if for each Z[> 0] here is ^[ > 0] such ha V; VWFXYZ[(resp. VWFXY V;Z[) whenever ; WFXY ^[. Theorem 3.3. Le V and d boh are fuzzy coninuous a GRVBCCCNFN WFX Y hen so is Vd for e / i. Proof: We only prove for = he proof of he res are similar. By hypohesis for any Z[>0] here exiss ^[ > 0] when _;WFXY ^[ _`V;VWFXYa Z[/2 and Γ`GXZGXWWXYa ε]/2. Therefore we have Γ`FGXZ FWWXYGXWWXYa =Γ`FZFWWXYGXWWXY GXZa Γ`FZFWWXYaΓ`FWWXYFWWXYGXWWXY GXZa =Γ`FZFWWXYaΓ`GXWWXYGXZa ε]. Theorem 3.4. Le V be fuzzy coninuous a WFXY F U. Then here exis GRVBCCCNFNs ; r = 0 r )0 2 and ; 2 =Ws r Y)Ws 2 Y saisfy 0 r < `ReV;a < Ws r Y and 0 2 < `ImV;a < Ws 2 Y when _;)WFXY ^[ for ^[ > 0]. Theorem 3.5. Le V be fuzzy coninuous funcion a WFXY F U ; r F U and here exiss ^[ >0] such ha `ReV;a < Re; r (resp. `ReV;a > Re; r ) and `ImV;a < Im; r (resp. `ImV;a `ReVWFX Ya > Im; r ) when _;) WFX Y ^[ hen `ReVWFX Ya > Re; r ) and `ImVWFX Ya < Im; r (resp. `ImVWFX Ya < Re; r > Im; r ). Theorem 3.6. Le V and d are boh fuzzy ype I-coninuous (resp. II-coninuous) and ew Y i. Then 1. Vd is also fuzzy ype I-coninuous (resp. II-coninuous) such ha `ReV;a z `ImV;a z `Red;a `Imd;a `ReV;a `ImV;a `Red;a z `Imd;a z are all greaer han zero for all } ~ 2 r. 2. Vd is also fuzzy ype I-coninuous (resp. II-coninuous). 3. d is fuzzy ype II-coninuous (resp. I-coninuous). 4. 1/d is also fuzzy ype I-coninuous (resp. II-coninuous) such ha `Red;a z `Red;a z `Red;a `Imd;a `Red;a `Imd;a `Imd;a z `Imd;a z are all posiive for any } ~ 2 r. Proof: By hypohesis for any Z[>0] here is a ^ > 0 ]]]] ^ >0 ] such ha V; VWFXYZ[(resp. VWFXY V;Z[) whenever ; WFX Y ^ and d; dwfxyz[ (resp. dwfxy d;z[) whenever ; WFXY ^ ]]]. 1. Le ^[ =^ ^ ]]] and } ~ 2 r we have `ReV;a z < `ReVWFXYa z Z resp. `ReVWFXYa z < `ReV;a z `ReV;a z < `ReVWFXYa z Z resp. `ReVWFXYa z < `ReV;a z (resp. 42

3 Mahemaical Theory and Modeling `ImV;a z < `ImVWFXYa z Z resp. `ImVWFXYa z < `ImV;a z `ImV;a z < `ImVWFXYa z `ReV;a `ReV;a `ImV;a `ImV;a Z resp. `ImVWFXYa z Z resp. `ReVWFXYa Z resp. `ReVWFXYa Z resp. `ImVWFXYa Z resp. `ImVWFXYa < `ImV;a z `Red;a z < `RedWFXYa z Z resp. `RedWFXYa z < `Red;a z `Red;a z < `RedWFXYa z Z resp. `RedWFXYa z < `Red;a z `Imd;a z < `ImdWFXYa z Z resp. `ImdWFXYa z < `Imd;a z `Imd;a z < `ImdWFXYa z `Red;a < `RedWFXYa `Red;a < `RedWFXYa `Imd;a < `ImdWFXYa `Imd;a < `ImdWFXYa `ReV;a z `Red;a z Z resp. `ImdWFXYa z Z resp. `RedWFXYa Z resp. `RedWFXYa Z resp. `ImdWFXYa Z resp. `ImdWFXYa < `Imd;a z < `Red;a < `Red;a < `Imd;a < `Imd;a Hence < `ReVWFXYa z `RedWFXYa z `ReVWFXYa z `RedWFXYa z Z Z < `ReVWFXYa z `RedWFXYa z r `ReVWFXYa z r `RedWFXYa z Z Z = `ReVWFXYa z `RedWFXYa z Z resp. `ReVWFXYa z `RedWFXYa z ˆ < `ReV;a z `Red;a z `ReV;a z `Red;a z Z Z < `ReV;a z `Red;a z r `ReV;a z r `Red;a z Z Z =ˆ `ReV;a z `Red;a z Z. `ReV;a z `Red;a z < `ReVWFXYa z RedFŠ<ReV;ŠRed;ŠZ. `RedWFXYa z Z resp. `ReVWFXYa z 43

4 Mahemaical Theory and Modeling `ImV;a z `Imd;a z < `ImVWFXYa z `ImdWFXYa z Z resp. `ImVWFXYa z ImdFŠ <ImV;Š Imd;Š Z. `ImV;a z `Imd;a z ImdFŠ<ImV;ŠImd;ŠZ `ReV;a `Red;a RedF <ReV; Red; Z. `ReV;a `Red;a RedF <ReV; Red; Z. `ImV;a `Imd;a ImdF <ImV; Imd; Z. `ImV;a resp. `ImVWFXYa `Imd;a `ImdWFXYa < `ImVWFXYa z `ImdWFXYa z Z resp. `ImVWFXYa z `RedWFXYa Z resp. `ReVWFXYa `RedWFXYa `ImdWFXYa `ImdWFXYa Z resp. `ReVWFXYa Z resp. `ImVWFXYa Z `Imd;a Z. 2. For any } ~ 2 r we have `ReV;a z `Red;a z < `ReVWFXYa z `RedWFXYa z resp. `ReVWFXYa z RedFŠ <ReV;Š Red;Š. `ReV;a z `Red;a z RedFŠ<ReV;ŠRed;Š. < `ReVWFXYa z `RedWFXYa z resp. `ReVWFXYa z `ImV;a z `Imd;a z < `ImVWFXYa z `ImdWFXYa z resp. `ImVWFXYa z ImdFŠ <ImV;Š Imd;Š. `ImV;a z `Imd;a z ImdFŠ<ImV;ŠImd;Š `ReV;a `Red;a resp. `ReVWFXYa `ReV;a `Red;a resp. `ReVWFXYa `RedWFXYa `RedWFXYa < `ImVWFXYa z `ImdWFXYa z `RedWFXYa `Red;a `RedWFXYa resp. `ImVWFXYa z `Red;a.. 44

5 Mahemaical Theory and Modeling `ImV;a `Imd;a resp. `ImVWFXYa `ImV;a `Imd;a resp. `ImVWFXYa `Imd;a `Imd;a.. 3. For any } ~ 2 r we have `Red;a z > `RedWFXYa z Z resp. `RedWFXYa z > `Red;a z `Red;a z > `RedWFXYa z Z resp. `RedWFXYa z > `Red;a z `Imd;a z > `ImdWFXYa z Z resp. `ImdWFXYa z > `Imd;a z `Imd;a z `Red;a `Red;a `Imd;a `Imd;a 4. Obvious. > `ImdWFXYa z > `RedWFXYa > `RedWFXYa > `ImdWFXYa > `ImdWFXYa Z resp. `ImdWFXYa z Z resp. `RedWFXYa Z resp. `RedWFXYa Z resp. `ImdWFXYa Z resp. `ImdWFXYa > `Imd;a z > `Red;a > `Red;a > `Imd;a > `Imd;a Theorem 3.7. If V and d are a fuzzy ype I-coninuous and II-coninuous such ha `ReV;a `Red;a and `ImV;a coninuous funcion ŒX saisfy `ImŒX;a `Imd;a. `Imd;a `ReV;a for every ; F U and } ~ 2 r hen here is a fuzzy `ReŒX;a `Red;a and `ImV;a Here an open problem is presened for furher invesivigaions: One can sudy ha V is boh fuzzy ype I- coninuous and II-coninuous if and only if V is fuzzy coninuous. References Buckley J.J. and Qu Y. (1991); (1992) Fuzzy complex analysis I: Differeniaion; II: Inegraion Fuzzy Ses and Sysems 41; 49 Elsevier ; Buckley J.J. (1989) Fuzzy complex numbers Fuzzy Ses and Sysems 33 Elsevier Cai Q.P. (2009) :The coninuiy of complex fuzzy funcion Adv. in In. and Sof Compuing Chun C. and Ma S. (1998) The differeniaion of complex fuzzy funcions Proc. of he 9 h NCFMFS Baoding Hebei U. Press Guangquan Z. (1992) Fuzzy limi heory of fuzzy complex numbers Fuzzy Ses and Sysems 46 Elsevier Ma S.Q. Cai Q.P. and Peng D.J. (2009) Some correlaive concepion and properies of bounded closed fuzzy complex number se Adv. in In. and Sof Compuing Ousmane M. and Congxin W. (2003) Semi coninuiy of complex fuzzy funcions Tsinghua Science and Technology Qiu D. and Shu L. (2008) Noes on On he resudy of fuzzy complex analysis: Par I and Par II Fuzzy Ses and Sysems 159 Elsevier Qiu J. Wu C. and Li F. (2000); (2001) On he resudy of fuzzy complex analysis: Par I. The sequence and series of fuzzy complex numbers and heir convergences; Par II. The coninuiy and differeniaion of fuzzy complex funcions Fuzzy Ses and Sysems 115 Elsevier ;

6 Mahemaical Theory and Modeling Quan M.S. (1996) Fuzzy complex numbers and some operaional properies J. of Lanzhou Universiy 32 NSE Sabir P.O. (2012) On fuzzy complex analysis Ph.D. Thesis Universiy of Sulaimani Sulaimani. Sabir P.O. Adil K.J. Munir A.A. (2012) a The coninuiy and differeniaion of complex fuzzy funcions for new fuzzy quaniies Asian Transacions on Science & Technology 2(4) Sabir P.O. Adil K.J. Munir A.A. (2012) b On fuzzy complex inegrals Asian Transacions on Science & Technology 2(4) Shengquan M. (2006) The series of complex fuzzy valued and is convergence Journal of Fuzzy Mahemaics Sun J. and Guo S. (2010) The soluion algorihm of complex fuzzy valued funcion inegral by fuzzy elemen ACFIE Wu C. and Qiu J. (1999) Some remarks for fuzzy complex analysis Fuzzy Ses and Sysems 106 Elsevier Zengai G. and Shengquan M. (2006) The research advances in fuzzy complex analysis Mah. in Prac. and Theory Zheng L. and Ha M. (2009) Furher discussions on recangular fuzzy complex numbers Proc. of he 8 h ICMLC Baoding

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