Random Generalized Bi-linear Mixed Variational-like Inequality for Random Fuzzy Mappings Hongxia Dai

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1 Ro Geeralzed B-lear Mxed Varaoal-lke Iequaly for Ro Fuzzy Mappgs Hogxa Da Depare of Ecooc Maheacs Souhweser Uversy of Face Ecoocs Chegdu 674 P.R.Cha Absrac I h paper we roduce sudy a ew class of ro geeralzed b-lear xed varaoal-lke equaly for ro fuzzy appgs. By usg he ax equaly exedg auxlary prcple we prove a exece uqueess heore of he soluo for he ro geeralzed b-lear xed varaoal-lke equaly. Keywords: Ro geeralzed b-xed varaoal-lke equaly Max equaly auxlary prcple. Iroduco I well kow ha varaoal equaly heores are very effecve powerful ools for sudyg a wde class of lear olear probles arg ay dverse felds of pure appled sceces such as echacs physcs opzao corol olear prograg ecoocs rasporao equlbru egeerg sceces ec. I rece years classcal varaoal equaly heores have bee geeralzed appled varous drecos he readers refer o []-[3] he refereces here. A useful pora geeralzao of varaoal equales he xed varaoal-lke equales whch have poeal sgfca applcaos opzao heory [45] srucural aalys [6] ecoocs [78]. I oed ha here are ay effecve uercal ehods for fdg approxae soluos of varous varaoal equales. Aog hese ehods he projeco ehod s vara fors he os effecve uercal echque. However he projeco ype echque cao be used o sudy xed varaoal-lke equales sce o possble o fd he projeco of he soluo. These facs ovaed Glowsk e al. [7] o sugges aoher echque whch does o deped o he projeco. The echque called he auxlary prcple echque. Very recely Huag e al. [9] Dg [] exed he auxlary prcple echque o sudy geeralzed olear xed varaoal-lke equales. O he oher h 989 Chag Zhu [] roduced he cocep of varaoal equaly for fuzzy appgs whch was exeded by Lassode [8] Shh Ta []. Recely he ro varaoal equales have bee roduced suded (see [3 45]-[7]. Ispred ovaed by rece works [89467] we roduce sudy a class of ro geeralzed b-lear xed varaoal-lke equaly for ro fuzzy appg. By usg he ax equaly exedg auxlary prcple we prove he exece uqueess heore of he soluo for he ro geeralzed b-lear xed varaoal-lke equaly. Our resuls prove geeralze ay kow correspodg resuls preseed [349].. Prelares Throughou h paper le H be a real Hlber space wh or er produc deoed by < > respecvely D be a oepy H closed covex subse of H. We deoe by CB (H he fales of all he oepy subses he fales of he oepy bouded closed subses of H respecvely. H( represes he Hausdorff erc o CB (H. Le ( Ω Σ be a easurable space where Ω a se Σ σ algebra of subses of Ω. We deoe by β ( H he class of Borel σ felds H. Defo.. A appg f : Ω H sad o be easurable f for ay C β ( H f ( C = { Ω: f( C} Σ. Defo.. A appg f : Ω H H called a ro operaor f for ay w H f( w = w( easurable. A ro operaor f : Ω H H sad o be couous f for ayω he appg f (: H H couous. Defo.3. A ulvalued appg A : Ω CB( H sad o be easurable f for ay C β ( H A ( C = { Ω: A( C Φ} Σ. Defo.4. A appg u: Ω H called a easurable seleco of he ulvalued easurable appg A : Ω CB( H f u a easurable appg Ω u( A(. Defo.5. A appg T : Ω H CB( H called a ro ulvalued appg f for ay w H T( w easurable. A ro ulvalued appg T : Ω H CB( H sad o be Ĥ couous f for ay Ω T( couous he Hausdorff erc. Le F( H be a colleco of fuzzy ses over

2 H. A appg F ~ fro Ω o F( H called a fuzzy appg. If F ~ a fuzzy appg o H for ay Ω F % ((deoe by F % he sequel a fuzzy se o H F% ( z he ebershp fuco of z F %. Le M F( H q [] he he se ( M = { u H : M( u q} q called a q -cu se of M. Defo.6. A fuzzy appg F% : Ω F( H called easurable f for ay a []( F% ( a : Ω H a easurable ulvalued appg. Defo.7. A fuzzy appg F% : Ω H FH ( called a ro fuzzy appg f for ay w H F % ( w: Ω F( H a easurable fuzzy appg. Clearly he ro fuzzy appg cludes ul-valued appgs ro ulvalued appgs fuzzy appgs as he specal cases. Le A % T % : Ω H F( H be wo ro fuzzy appgs safyg he followg codo (I: f here ex wo appgs ac : H [] such ha ( w Ω H( A% w aw ( CBH ( ( T% w cw ( CB( H. By usg he ro fuzzy appgs A % T % we ca defe wo ro ul-valued appgs A T as follows: ( w Ω H A: Ω H CB( H( w ( A % w aw ( T : Ω H CB( H( w ( T % w cw (. So A T are called he ro ul-valued appgs duced by he ro fuzzy appgs A % T % respecvely. Gve appgs ac : H [] he ro fuzzy appgs A% T% : Ω H F( H safy he codo (I. Le N η : H H H be wo appgs. Le b: H H ( ] be a real-valued fuco. We shall sudy he followg proble: Fd easurable appgs uxy : Ω H such ha A% u ( ( x( a( u( T% u ( ( c( u( < Nx ( ( y ( vu ( > bu ( ( v bu ( ( u ( (. for all Ω v H where he fuco b ( odffereal safes he followg codos: ( for ay wv H bwv ( le he frs argue; ( for each w H b( w a covex fuco; ( for ay wv H bwv ( bouded ha here exs a cosaγ > such ha bwv ( γ w v; (v for all wvz H bwv ( bwz ( bwv ( z. Reark.. ( for ay wv H b( w v =b( w v b( w v γ w v hold fro codo ( ( respecvely. So bwv ( γ w v. ( for ay wvz H bwv ( bwz ( γ w v z fro codo ( (v. So bwv ( couous wh respec o secod argue. Iequaly (. called ro geeralzed b-lear xed varaoal-lke equaly for ro fuzzy appgs. The se of easurable appgs ( uxy called a ro soluo of he ro geeralzed b-lear xed varaoal-lke equaly. Specal cases: ( If N( x( = P( x( F( η ( vu ( = v gu ( ( where F = f g P f g: Ω H H buv ( = φ( v for all uv H he proble (. reduces o he followg ro geeralzed olear xed varaoal clusos for ro fuzzy appgs: Fd easurable appgs uxyw : Ω H such ha for all Ω u( H A% u ( ( x( au ( ( T% u ( ( c( u( Su ( ( w( du ( ( g( w( I do( φ Φ < P( x ( { fy ( ( gw ( ( } v gw ( ( > φ( gw ( ( φ( v v H. (. The proble(. was suded by Ahad Baza [3]. ( If N( x( = f ( x ( py ( ( η ( vu ( = v gu ( ( buv ( = φ( v for all uv H he proble (. reduces o he followg ro geeralzed olear varaoal clusos for ro fuzzy appgs: Fd easurable appgs uxy : Ω H such ha for all Ω u( H A% u ( ( x( a( u( T% ( c( u( u ( < f ( x ( py ( ( v gu ( ( >

3 φ( gu ( ( φ( v v H. (.3 The proble(.3 was suded by Huag [4]. % % (3 If Baach space le AT : Ω D B be easurable appgs N( = T( A( he real-valued fuco buv ( = f( v uv D he he proble (. reduces o for all he followg ro xed varaoal-lke equaly proble: v B < Tu ( ( Au ( ( vu ( > f ( u ( f( v. (.4 The proble (.4 was cosdered by Dg []. (4 If reflexve Baach spaces le A % T % : D B η : D D B be appgs he he proble (. reduces o he followg olear xed varaoal-lke equaly: for a gve w B fd u D such ha < NTuAu ( w vu > buv ( buu ( v B. (.5 The proble (.5 was cosdered by Dg []. I oed ha he probles (. (.5 are specal cases of he proble (.. I bref proble (. he os geeral ufyg oe whch also oe of he a ovaos of h paper. Defo.8. Le D be a oepy closed covex subse of H a appg η : D D D called σ Lpschz couous f here exs a cosa σ > such ha uv σ u v uv D. Defo.9. Le D be a oepy closed covex subse of H leη : D D D N(: D D D be wo appgs. ( N( sad o be Lpschz couous frs argue f here exs a cosa r > such ha Nu ( Nv ( r u v uv D. ( N( sad o η srogly oooe frs argue wh respec o he ro ulvalued appg A f here exs a cosa δ > such ha for ay Ω < N( x N( x u u > δ uu u u H x Au ( x Au (. Slarly we ca defe Lpschz couy he η srogly ooocy of N( secod argue wh respec o he ro ul-valued appgst. Defo.. Le A T : Ω H CB( H be wo ro ul-valued appgs duced by he ro fuzzy appgs A % T % respecvely be appg. The appgs η : D D D u N ( x( η are sad o have dagoally cocave relao f for ay Ω he fuco φ : Ω D D ( ] defed by φ ( v u = N ( x( u v has dagoally cocave v where x( Au ( y ( T( u.e. for ay Ω ay fe se v v... v} D u = λ v { ( λ = λ = λφ ( v u. = = 3. Exece uqueess heore A frs we gve he followg Leas. Lea 3.. [] Le T : Ω H CB( H be a Ĥ couous ro ulvalued appg he for easurable appg u: Ω H he ulvalued appg T( u(: Ω CB( H easurable. Lea 3.. [] Le ( Ω Σ be a easurable space D be a oepy covex subse of a opolog- cal vecor space. Le ϕ : Ω D D [ ] be a real-valued fuco such ha ( for each ( vu D D ϕ( vu easurable appg; ( for each ( v Ω D u ϕ( vu couous o each oepy copac subse of D ; (3 for each ( u Ω D v ϕ( vu lower secouous o each oepy copac subse of D ; (4 for each Ω each oepy fe se { v v... v} D for each u = λv ( = = λ λ = ( v u ϕ ; (5 for each Ω here exs a oepy copac covex subse D of D a oepy copac subse K of D such ha for each u D\ K here a v co( D { u} wh ϕ ( v u >. The here exs a easurable appg u: Ω D such ha ϕ( vu ( for all v D Ω. Now we ow sae he a resul of h paper. Theore 3.. Le ( Ω Σ be a easurable space D be a oepy covex subse of H. Le ro fuzzy appgs A% T% : Ω H F( H

4 safy he codo (I A T be he ro ul-valued appgs duced by he ro fuzzy appgs A % T % respecvely. Le N η : D D D be wo appgs. Le b: D D ( ] be a real-valued fuco such ha ( for each Ω he appg A( T( Ĥ couous wh cosa < λ λ respecvely; ( he appg η Lpschz couous wh cosa σ > ; he appg u v couous frs argue secouous secod argue for all uv D uv = vu ; (3 he appg N( Lpschz couous η srogly oooe wh respec o he ra- do ul-valued appg A frs argue wh cosa k > k respecvely. > N( Lpschz couous η srogly oooe wh respec o he ro ul-valued appg T secod argue wh cosa k > k > respecvely oo; (4 for each Ω he appgs u N( x( η have he dagoally cocave relao; (5 he fuco b( safes codos ( (v where γ ( k k. The he proble (. has a uque ro soluo u ˆ( Dx ˆ( Au ( ˆ( y ˆ( Tu ( ˆ(.e. N( x ˆ( y ˆ( η ( vu ˆ( bu ( ˆ( v bu ( ˆ( u ˆ( v D Ω. Proof. Frsly we prove ha for each fxed u ( D here exs a uque uˆ ( D xˆ( yˆ( T ( such ha N( xˆ( yˆ( v b( u ( v bu ( ( u ˆ( v D Ω (3. For ay fxed u D we defe a fuco ϕ : Ω D D ( ] by ϕ( vu = Nx ( ( y ( uv bu ( u bu ( v vu D Ω where x( A( u T( u. Sce A T are he ro ul-valued appgs duced by he ro fuzzy appgs A % T % respecvely.e. for each u D A( u T( u are easurable appgs so for ay fxed ( vu D D ϕ( vu easureable. For ay v D he appg u u v couous. The for each v D ay sequece { u } D wh u u we have u v u v (. Sce for each Ω he appgs A( T( are Ĥ couous follows for ay fxed ( v Ω D ha N( x ( y ( u v N( x( u v N( x( y( N( x( u v N( x( u v u v Nx ( ( Nx ( ( u v N( x( y( N( x( u v N( x( u v u v N( x ( y ( N( x( y ( u v N( x( y ( N( x( u v N( x( u v u v k x( x( u v k y( u v N( x( u v u v kλ u u u v kλ u u u v N( x( u v u v (. Therefore for each fxed ( v Ω D he fuco u N ( x( u v couous o D where x( A( u. Tu (. Sce he fuco u b( u u couous covex o D by he reark. ( so for each fxed ( v Ω D u ϕ( v u couous o D. Sce he fuco v bu ( v couous o D for ay u D v u v secouous so for each fxed ( u Ω D v ϕ( v u secouous o D. Thus we ca cofr ha he fuco ϕ( vu safes he codos ((( Lea 3.. Now we prove ha he fuco ϕ ( vu safes he codo (v Lea 3.. We suppose ha he fuco ϕ( vu safes he codo (v of Lea 3.. If o rue here exs Ω a fe se v v... v} D u = = { λ v ( λ λ = = ϕ ( v u > for all... = ha such ha

5 N( x( u v b( u u b( u v >. I follows ha = λ N( x( u v b( u u = λ bu ( v > Nog ha b ( u v covex he secod argue ha = we have λ b( u v b( u λ v = b( u u = = λ N( x( u v >. (3. Sce for ay Ω he appgs u N( x( y ( η have he dagoally cocave relao v so for ay Ω = λ N( x( u v whch coradcs (3.. Therefore for ay Ω ay fe se { v v... v} D u = = λ v ( λ = λ we have = ϕ( v u (... =. Thus codo (v of lea 3. holds. For each Ω le θ = ( α N( x ( y ( γ u k k K = { u D : u u θ} D = { u } he K D are boh copac covex subses of D. By assupos ( (4 of he heore for u Co( D {} u x ( A( u y ( T( u such ha each u D\ K here ex ϕ( u u = Nx ( ( y ( uu bu ( u bu ( u = Nx ( ( y ( Nx ( ( y ( uu N( x ( N( x ( y ( u u N( x ( y ( u u bu ( u bu ( u k u u k uu α N( x ( y ( uu γ u u u u u = [( k k uu α N( x ( y ( γ u ] >. Hece codo (5 of Lea 3. also safed. By Lea 3. here exs a easurable appg u $ : Ω D such ha ϕ( vu $ ( for all v DΩ. We kow he appg N( Lpschz couous frs argue secod argue he appgs A ( T( are Ĥ couous. Based o Lea3. we oba ha for he easurable appg u $ : Ω D here ex xˆ( A( yˆ( T( such ha N( x ˆ( y ˆ( vu ˆ( bu ( ( v bu ( ( u ˆ(. v D Ω By u( v = η ( v u( we have N( xˆ( yˆ( v b( u ( bu ( ( v v D Ω Th ples ha for ay Ω for each fxed easurable appg u ( D he easurable appg uˆ : Ω D xˆ( A( yˆ( Tu ( ˆ( a ro soluo of he Auxlary proble (3.. Now we prove ha for ayω he easureable appg uˆ ( xˆ( A( yˆ( Tu ( ˆ( a uque ro soluo of he auxlary proble (3.. Supposg he easurable appgs u( D x( A( u( y( Tu ( ( u ( D x ( u ( y ( T ( u ( are wo ro soluos of proble (3. we have he cocluso ha for all v D Ω N( x( y( η ( v u( bu ( ( v bu ( ( u( (3.3 N( x( η ( v u( bu ( ( v bu ( ( u( (3.4 Takg v = u ( (3.3 v = u ( (3.4 addg wo equales by he assupo o he fuco b we oba N( x( y( η ( u( u( N( x( u( u( Sce for all uv D uv = vu we have N( x( N( x( y( u( u( Nog ha N( η srogly oooe wh respec o he ro ul-valued appg A frs argue wh cosa k > η srogly oooe wh respec o he ro ul- valued appg T secod argue wh cosa k > we ge

6 k u ( u( ( k Nx ( ( Nx ( ( u( u ( N( x ( y ( N( x ( y ( u ( u (. Sce k k > we have u( = u(. Furher le x ( A( u ( x ( A( u ( y ( Tu ( ( Tu ( ( we have x( x( H( A( u( A( u( λ u( u( y( HTu ( ( ( Tu ( ( λ u( u(. So we ge x ( = x ( y ( = y ( whch ply ha for ay Ω he easurable appg u ( D he appgs u $ ( D xˆ ( yˆ( T ( (deoe by w he sequel a uque ro soluo of he auxlary proble (3.. Thus we have proved ha for each Ω he easurable appg u ( D here exs a uque soluo w safyg (3.. Defg a appg F : D D by u ( w( u ( we wll prove ha he appg F a coraco appg. Ideed for ay u ( u ( D here ex uque w = Fu ( ( w = F( u ( for all v D Ω such ha N( x ( y ( ( v u ( η bu ( ( v bu ( ( u( (3.5 N( x( η ( v u( bu ( ( v bu ( ( u(. (3.6 v = u ( (3.5 v = u ( (3.6 Takg addg wo equales we have N( x( y( η ( u( u( N( x( η ( u( u( bu ( ( u ( u( bu ( ( u ( u (. By uv vu b ( we have ( k k u ( u ( = he assupo o N( x( y( N( x( y( u( u( Nx ( ( y ( N( x ( y ( u ( u ( b ( u ( u ( u ( b( u ( u ( u( γ u ( u ( u ( u ( whch derves γ k k u ( u ( u ( u ( (3.7 x ( x( H( Au ( ( Au ( ( λ γ u ( u ( k k (3.8 HTu ( ( ( Tu ( ( λ γ u ( u (. k k (3.9 The equales (3.7 (3.8 (3.9 ogeher wh γ ( k k < λ λ resul ha F a coraco appg. Hece here exs a uque po u ˆ( D such ha ˆ = ˆ u ( Fu ( ( N( xˆ( yˆ( η ( v b( v bu ( ˆ( u ˆ( v D Ω Now we kow uˆ ( D xˆ( yˆ( T ( he uque soluo of he proble (.. Th coplees he proof. Refereces [] M.A. Noor Soe rece advaces varaoal equales Par I: basc coceps New Zeal J. Mah. 6: [] M.A. Noor Soe rece advaces varaoal equales Par II: oher coceps New Zeal J. Mah. 6: [3] G. Ta Geeralzed quas varaoal-lke equaly proble Mah. Oper. Res. 8: [4] J. C. Yao The geeralzed quas varaoal equaly proble wh applcaos J. Mah. Aal. Appl. 58 : [5] P. D. Paagoopoulos G. E. Savroulak New ypes of varaoal prcples based o he oo of quasdffereably Aca Mech 94: [6] P. Cubo Exece of soluos for lower secouous quas equlbru probles copu. Mah. Appl. 3: [7] R. Glowsk J. L. Los R. Treoleres Nuercal aalys of varaoal equales Norh-Holl Aserda 98. [8] M. Lassode O he use of KKM

7 ulfuco.xed po heory relaed opcs J. Mah. Aal. Appl. 97: [9] N.J. Huag C.X Deg Auxlary prcple erave algorhs for geeralzed se-valued srogly olear xed varaoal-lke equales J. Mah. Aal.Appl. 56: [] X.P. Dg Exece Algorh of Soluos for Nolear Mxed Varaoal-lke Iequales Baach spaces J. Copu. Appl. Mah. 57: [] S.S. Chag Y. Zhu O varaoal equales for fuzzy appgs Fuzzy Se. Sys. 3: [] M.H. Shh K.K. Ta Geeralzed quas-varaoal equales locally covex spaces J. Mah. Aal. Appl. 8: [3] R. Ahad F. F. Baza A erave algorh for ro geeralzed olear xed varaoal clusos for ro fuzzy appgs Appl. Mah. Copu. 67: [4] N.J. Huag Ro geeralzed olear varaoal clusos for ro fuzzy appgs Fuzzy Se. Sys. 5: [5] S.S. Chag N.J. Huag Ro geeralzed se-valued quas-copleeary probles Aca Mah. Appl. Sca 6: [6] T. Husa E. Tarafdar X.Z. Yua Soe resuls o ro geeralzed gaes ro quas-varaoal equales Far Eas. J. Mah. Sc. : [7] N.J. Huag Ro geeralzed se-valued plc varaoal equales J. Laog Noral Uv. 8: [8] X.P.Dg Geeral Algorhs of Soluos for Nolear Varaoal-lke Iequales Reflexve Baach Spaces Ida.J.Pure.Appl.Mah.9(: [9] X.H.Che Y.F.Lu O he Geeralzed Nolear Varaoal-lke Iequales Reflexve Baach Spaces J.Najg.Uv.Mah.Bquar. 8(: [] X.P. Dg Exece Uqueess of soluos for Ro Mxed Varaoal-lke Iequales Baach spaces J. Schua Noral. Uversy (Naural Scece. : [] S.S. Chag Fxed Po Theory wh Applcaos Chogqg Publhg House Chogqg 984.