Applied Mahemaics ad Physics, 23, Vol, No 3, 39-44 Available olie a hp://pubssciepubcom/amp//3/ Sciece ad Educaio Publishig DOI:269/amp--3- Averagig of Fuzzy Iegral Equaios Naalia V Skripik * Deparme of Opimal Corol ad Ecoomic Cybereics, Odessa Naioal Uiversiy amed afer II Mechikov, Odessa, Ukraie *Correspodig auhor: alie@ukre Received Augus 9, 23; Revised Augus 23, 23; Acceped Augus 26, 23 Absrac I his paper he subsaiaio of he averagig mehod for fuzzy iegral equaio is cosidered Keywor: fuzzy iegral equaio, averagig mehod Cie his Aricle: Naalia V Skripik, Averagig of Fuzzy Iegral Equaios Applied Mahemaics ad Physics, o 3 (23): 39-44 doi: 269/amp--3- Iroducio Iegral equaios are ecouered i various fiel of sciece ad i umerous applicaios, icludig elasiciy, plasiciy, hea ad mass rasfer, oscillaio heory, fluid dyamics, filraio heory, elecrosaics, elecrodyamics, biomechaics, game heory, corol, queuig heory, elecrical egieerig, ecoomics, ad medicie Ofe by cosideraio of he iegral equaios coaiig small parameer, he averagig mehod of Bogolyubov - Krylov [,2] is applied wo approaches are hus used: ) he iegral equaio is reduced o iegrodiffereial by differeiaio, ad he oe of he averagig schemes is applied [3,4,5]; 2) oe of he averagig schemes is applied direcly o he iegral equaio [3,4,5,6,7] Recely he se-valued ad fuzzy iegral equaios ad iclusios bega o be cosidered [6-4] I his paper he subsaiaio of he averagig mehod for fuzzy iegral equaio usig he secod approach is cosidered 2 Prelimiaries Le cov( R ) be a se of all oempy covex compac subses of he space R, (, ) = mi { ( ), ( ) } h AB S A BS B A r r be Hausdorff disace bewee ses A ad B, S ( ) r A is r -eighborhood of he se A Le E be a se of all mappigs u: R [,] such ha u saisfies he followig codiios: i) u is ormal, ie here exiss x R such ha ux ( ) = ; ii) u is fuzzy covex, ie ( λ + ( λ) ) mi { ( ), ( )} u x y ux uy r for ay xy, R ad λ [,] ; iii) u is upper semicoiuous, ie for ay x R ad ε > here exiss such δε (, x ) > ha ux ( ) < ux ( ) + ε wheever x x < δ, x R ; u = closure x R : u( x) > is compac iv) [ ] { } α For α (,] deoe [ u] { x R : ux ( ) α} = he from i)-iv), i follows ha he α -level se α u cov R for all α [,] [ ] ( ) Le ˆ θ be he fuzzy mappig defied by ( x) x ad ˆ θ ( ) = Defie D: E E R + by he relaio ([ ] α α [ ] ) Duv (, ) = sup h u, v α ˆ θ = if he D is a meric i E Furher we kow ha [5]: E, D is a complee meric space; ) ( ) + + = for all uvw,, E ; = for all uv, E ad λ R 2) D( u w, v w) D( u, v) 3) D( λu, λv) λ Duv (, ) Defiiio [5] A mappig F :[, ] E is measurable [coiuous] if for ay α [,] he se- valued mappig F :[, ] cov( R ) [ ] α defied by α Fα () = F () is Lebesgue measurable [coiuous] Defiiio 2 [5] A mappig F :[, ] E is said o be iegrably bouded if here exiss a iegrable fucio h () such ha x () h () for all x () F () Defiiio 3 [5] he iegral of a fuzzy mappig α F : [, ] E is defied levelwise by F() d =
4 Applied Mahemaics ad Physics Fα () d = { f () d : f :[, ] R is a measurable selecio of F :[, ] cov( R )} α, α for all [ ] Defiiio 4 [5] A measurable ad iegrably F :, E is said o be iegrable bouded mappig [ ] over [, ] if F() d E F E is measurable ad Noe ha if : [, ] iegrably bouded, he F is iegrable Furher if F :, E is coiuous, he i is iegrable [ ] Lemma Le FG, :[, ] E be iegrable ad λ R he ) F() + G() d = F() d + G() d ; 2) λf() d = λ F() d ; DF G is iegrable; 3) ( ( ), ( )) 4) D F() d, G() d D( F(), G()) d 3 Mai Resuls Cosider he fuzzy iegral equaio wih a small parameer X ( ) = X + ε F(, s,, () where R + is ime, X: R+ DD, E is a phase variable, ε > is a small parameer Cosider he followig full averaged fuzzy iegral equaio where X ( ) = X + ε F(,, (2) + FX (, ) = lim FsX ( ),, (3) he followig heorem ha proves he closeess of soluios of sysems () ad (2) o he fiie ierval hol: heorem Le i he domai Q= {( sx,, ) :, s R+, X D E } he followig codiios fulfill: ) he fuzzy mappig FsX (,, ) is coiuous ad saisfies he Lipschiz codiio i X wih he cosa λ, ie DFsX ( (,, ), FsY (,, )) λdxy (, ) for all (, sx, ),(, sy, ) Q; 2) he fuzzy mappig F(, s, is equicoiuous o R + for ay soluio X : R+ D of equaio (); 3) uiformly wih respec o, R+, X D limi (3) exiss; 4) he soluio X () of equaio (2) is defied for all ad ε (, σ] ad belogs wih some ρ eighborhood o he domai D he for ay η > ad L > here exiss such ε ( η, L) (, σ], ha for ε (, ε] ad for all [, Lε ] he iequaliy hol DX ( (), X ()) < η, (4) where X () ad X () are he soluios of equaios () ad (2) Proof Firs of all, le us oice ha he fuzzy mappig FX (, ) saisfies he Lipschiz codiio i X wih he cosa λ Really, usig codiio 3) of he heorem we have ha for ay δ > here exiss such ( δ ) ha for > ( δ ) he followig esimae hol: he + D FsX (,, ) FX, (, ) < δ ( (, ), FX (, )) D FX + D FX (, ), FsX (,, ) + + + D F(, s, X ), F(, s, X ) + + D FsX (,, ) FX, (, ) + + < 2 δ + D F(, s, X ) d, F(, s, X ) d 2δ + + 2 δ + λdx (, X ) ( (,, ), (,, )) D FsX FsX d As δ is arbirary small, we ge ( ) D FX (, ), FX (, ) λdx (, X ) Le us esimae he disace bewee he soluios of he iiial ad he averaged fuzzy iegral equaios: DX ( (), X ()) = D X + ε F(, s,, X + ε F(,
Applied Mahemaics ad Physics 4 = ε D F(, s,, F(, ε D F(, s,, F(, + ε D F(,, F(, ε D F(, s,, F(, + ε DFXs ( (, ( )), FXs (, ( ))) ε D F(, s,, F(, + ελ D X() s, X() s ( ) Usig he Growall s lemma we ge DX ( (), X ()) ελ εe D F(, s,, F(, λl εe sup D F(, s,, F(, [, Lε ] (5) Deoe by ϕ : R+ R+ he module of coiuiy of he fuzzy mappig G( ) = F(, s, he D F(, s,, F(, s, < ϕ ( ) for < Divide he ierval [, Lε ] wih he sep γε, ( ) where γε ( ), εϕγε ( ( )) as ε Such choice of he sep is possible i view of he properies of he module of coiuiy, ad exacly as εϕγε ( ( )) ε( + γε ( )) ϕ() ad oe ca ake γε ( ) for example equal o ε 2 Deoe by ( ) i i γ( ε) i m ( m ) γ( ε) L ε m γ( ε) m L = =,, <, = ε he divisio pois, Xi = X ( i) is he soluio of equaio () i he divisio pois Esimae he disace ε D F(, s,, F(, o he ierval [ k, k + ], k =, m : ε D F(, s,, F(, k i+ = ε D F(, s, + F(, s,, i k k i+ i k k i+ i+ D F(, s,, F(, i i ε D F( s F( +,,,, k k ε ε k F(, + F(, i+ i+ D F(, s,, F(, s, Xi ) i i i + i+ + D F(, s, Xi), F(, Xi) i i i+ i+ + D F(, Xi ), F(, i i + D F(, s,, F(, s, X k ) k k + D F(, s, X k ), F(, X k ) k k + D F(, X k ), F(, k k i + DFsXs ( (,, ( )), FsX (,, i )) i k i+ i+ D F( s Xi) F( Xi) +,,,, i i i + + DFXs ( (, ( )), FX (, i )) i + DFsXs ( (,, ( )), FsX (,, k )) k + D FsX (,, k ), F (, X k ) k k + DFXs ( (, ( )), FX (, k )) k
42 Applied Mahemaics ad Physics he i + DFsXs ( (,, ( )), FsX (,, i )) k i i i + = + DFXs ( (, ( )), FX (, i )) i ε k i + i+ D F( s Xi) F( Xi) +,,,, i i + D F(, s, X k), F(, X k) k k k i + 2 λε D( X ( s), X ( i )) i k i+ i+ + D F( s Xi) F( Xi),,,, i i + D F(, s, X k), F(, X k) k k Usig codiio 2) of he heorem we have DXs ( (), Xi ) s i = D F( s,, X( )) d, F( i,, X( )) d εϕ( γ ( ε )), ε ν ν ν ν ν ν k i + 2 λε D( X ( s), X ( i )) i 2 2 λεϕγε ( ( ))( k + ) γε ( ) 2 λlεϕ( γ ( ε )) Accordig o codiio 3) of he heorem here exiss such moooe decreasig fucio θυ, ( ) ha e o as υ, ha for all, υ, R+, X D he ε + υ + υ D F(, s, X ), F(, X ) υθ ( υ) i+ i+ D F(, s, Xi), F(, Xi) γεθγε ( ) ( ( )), i i εd F(, s, X k), F(, X k) γεθγε ( ) ( ( )) k k herefore, So k i+ i+ D F( s, Xi), F( s, Xi) i i ε D F( s X k) F( s X k) +,,, k k mεγεθγε ( ) ( ( )) Lθγε ( ( )) ε D F( s,, F( s, 2 λlεϕγε ( ( )) + Lθγε ( ( )) (6) Deoe by η = mi{ ρη, } Choose ε from he codiio 2 λlεϕγε ( ( )) + Lθγε ( ( )) ηe λl he for all ε (, ε] usig (5) ad (6) we ge DX ( (), X ()) η = mi{ ρη, } ad he saeme of he heorem is proved provided he soluio X () belogs o he domai Q o he ierval [, Lε ], ad i follows from codiio 4) If he fuzzy mappig is periodic i s, oe ca receive he more exac esimae heorem 2 Le i he domai Q= {( sx,, ) :, s R+, X D E } he followig codiios fulfill: ) he fuzzy mappig FsX (,, ) is coiuous, bouded wih he cosa M, periodic i s ad saisfies he Lipschiz codiio i X wih he cosa λ ; 2) he fuzzy mappig F(, s, is equicoiuous o R + for ay soluio X : R+ D of equaio (); 3) he soluio X () of equaio where X ( ) = X + ε F(,, (7) + FX (, ) = FsX ( ),, (8) is defied for all ad ε (, σ] ad belogs wih some ρ eighborhood o he domai D he for ay L > here exiss such ε ( L) (, σ] ad CL ( ) >, ha for ε (, ε] ad for all [, Lε ] he iequaliy hol DX ( (), X ()) < Cε, (9) where X () ad X () are he soluios of equaios () ad (7)
Applied Mahemaics ad Physics 43 he proof of his heorem is similar o he proof above akig he sep γε ( ) equal o Proof Firs of all, le us oice ha he fuzzy mappig FX (, ) saisfies he Lipschiz codiio: ( (, ), FX (, )) D FX + D FX (, ), FsX (,, ) + + + D F(, s, X ), F(, s, X ) + + D FsX (,, ) FX, (, ) + D( FsX ( ) FsX ( )),,,,, λdx (, X ) Le us esimae he disace bewee he soluios of he iiial ad he averaged fuzzy iegral equaios Similar o he proof of heorem we ge DX ( (), X ()) λl εe sup D F(, s,, F(, [, Lε ] () Deoe by ϕ : R+ R+ he module of coiuiy of he fuzzy mappig G( ) = F(, s, he D F(, s,, F(, s, < ϕ ( ) for < Divide he ierval [, Lε ] wih he sep, deoe by i ( ( ) ) m = i i =, m, m < Lε m, = Lε he divisio pois, Xi = X ( i) is he soluio of equaio () i he divisio pois Esimae he disace ε D F(, s,, F(, o he ierval [ k, k + ], k =, m Similar o he proof of heorem we ge ε D F(, s,, F(, k i + 2 λε D( X ( s), X ( )) i i he k i+ i+ + D F( s Xi) F( Xi),,,, i i + D F(, s, X k), F(, X k) k k Usig codiio 2) of he heorem we have DXs ( (), Xi ) s i = D F( s,, X( )) d, F( i,, X( )) d εϕ( ), ε ν ν ν ν ν ν Accordig o (8) k i + 2 λε D( X ( s), X ( i )) i 2 2 λε ϕ ( )( k+ ) 2 λlεϕ ( ) i+ i+ ε D F(, s, Xi), F(, Xi) = i i Usig he codiio ) of he heorem we have ε D F(, s, X k), F(, X k) 2Mε k k herefore, ε D F(, s,, F( s, 2[ λlϕ ( ) + Mε ] () he for all ε (, ε] usig () we ge DX ( (), X ()) Cε, C= 2[ λlϕ ( ) + M] e λl ad he saeme of he heorem is proved provided he soluio X () belogs o he domai Q o he ierval [, Lε ], ad i follows from codiio 3) Also he scheme of parial averagig for he fuzzy iegral equaio wih a small parameer () ca be applied Such varia of he averagig mehod happes o be useful whe he average of some mappigs does o exis or heir presece i he equaio does o complicae is research Cosider he followig parial averaged fuzzy iegral equaio where X ( ) = X + ε F(, s,, (2) + + lim h F(, s, X ), F(, s, X ) = (3)
44 Applied Mahemaics ad Physics he followig heorem ha proves he closeess of soluios of sysems () ad (2) o he fiie ierval hol: heorem 3 Le i he domai Q= {( sx,, ) :, s R+, X D E } he followig codiios fulfill: ) he fuzzy mappigs FsX (,, ), FsX (,, ) are coiuous ad saisfy he Lipschiz codiio i X wih he cosa λ ; 2) he fuzzy mappig F(, s, is equicoiuous o R + for ay ay soluio X : R+ D of equaio (); 3) uiformly wih respec o, R+, X D limi (3) exiss; 4) he soluio X () of equaio (2) is defied for all ad ε (, σ] ad belogs wih some ρ eighborhood o he domai D he for ay η > ad L > here exiss such ε ( η, L) (, σ], ha for ε (, ε] ad for all [, Lε ] he iequaliy hol DX ( (), X ()) < η, where X () ad X () are he soluios of equaios () ad (2) he proof of his heorem is similar o he proof of heorem Similar o heorem 2 he periodic case ca be cosidered 4 Coclusio heorems -3 geeralize he resuls of [6] o subsaiaio of he averagig mehod for fuzzy differeial equaios Whe passig from he fuzzy differeial equaio o he equivale fuzzy iegral equaio we have ha codiios ), 3), 4) of heorems,3 hold owig o he correspodig codiios o he righ had sides of he iiial fuzzy differeial equaio, ad codiio 2) is saisfied because he fuzzy mappig F( s, is Lipschizia i as he righ-had side FsX (, ) of he fuzzy differeial equaio is uiformly bouded Refereces [] Krylov, NM, Bogoliubov, NN, Iroducio o oliear mechaics, Priceo Uiversiy Press, Priceo, 947 [2] Bogoliubov, NN, Miropolsky, YuA, Asympoic meho i he heory of o-liear oscillaios, Gordo ad Breach, New York, 96 [3] Filaov, AN, Averagig i sysems of differeial, iegrodiffereial ad iegral equaios, Fa, ashke, 967 (i Russia) [4] Filaov, AN, Sharova, LV, Iegral iequaliies ad heory of oliear oscillaios, Nauka, Moscow, 976 (i Russia) [5] Miropolsky, YuA, Filaov, AN, Averagig of iegrodiffereial ad iegral equaios, Ukraiia Mah J 24 () 22-37 972 [6] Viyuk, AN, Averagig i Volerra iegral iclusios, Russia Mah (Iz VUZ) 39 (2) 9-23 995 [7] Viyuk, AN, Averagig i mulivalued Volerra iegral equaios, Ukraiia Mah J 47(2) 852-857 995 [8] Friedma, M, Mig, Ma, Kadel, A, O fuzzy iegral equaios, Fudamea Iformaicae 37 89-99 999 [9] Ploikov, AV, Skripik, NV, Exisece ad uiqueess heorem for se iegral equaios, JAdvaced Research Dy Corol Sys 5 (2) 65-72 23 [] Ploikov, AV, Skripik, NV, Exisece ad uiqueess heorem for fuzzy iegral equaio, JMahSciAppl () -5 23 [] Ploikov, AV, Skripik, NV, Exisece ad uiqueess heorem for se-valued Volerra iegral equaios, America J ApplMahSaisics (3) 4-45 23 [2] Hossei Aari, Allahbakhsh Yazdai, A compuaioal mehod for fuzzy Volerra-Fredholm iegral equaios, Fuzzy Iformaio ad Egieerig 3 (2) 47-56 2 [3] Bica, AM, Error esimaio i he approximaio of he soluio of oliear fuzzy Fredholm iegral equaios, Iformaio Scieces, 78, 279-292, 28 [4] Buckley, JJ; Feurig, ; Hayashi, Y, Fuzzy iegral equaios, Nih IEEE Ieraioal Coferece o Fuzzy Sysems 39-323 2 [5] Park, JY, Ha, HK Exisece ad uiqueess heorem for a soluio of fuzzy differeial equaios, Iera J Mah Sci 22 (2) 27-279 999 [6] Ploikov AV, Skripik, NV, Differeial equaios wih clear ad fuzzy mulivalued righ-had side: Asympoical meho, Asropri, Odessa, 29 (i Russia)