International Journal of Pure and Applied Mathematics Volume 56 No ,

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1 Inernaional Journal of Pure and Applied Mahemaics Volume 56 No , THE GENERALIZED SOLUTIONS OF THE FUZZY DIFFERENTIAL INCLUSIONS Andrej V. Plonikov 1, Naalia V. Skripnik 2 1 Deparmen of Numerical Analysis, SAPR Odessa Sae Academy of Civil Engineering and Archiecure 4, Didrihsona Sr., Odessa, 65029, UKRAINE a-plonikov@ukr.ne 2 Deparmen of Opimal Conrol and Economics Cyberneics Odessa Naional Universiy I.I. Mechnikov 2, Dvoryanskaya Sr., Odessa, 65026, UKRAINE alie@ukr.ne Absrac: In his aricle we inroduce he definiion of he generalized soluion of he fuzzy differenial inclusion and find he condiions when his soluion is equivalen o he ordinary one. AMS Subjec Classificaion: 34A60, 03E72 Key Words: fuzzy differenial inclusion 1. Inroducion When a real world problem is ransferred ino a deerminisic iniial value problem of ordinary differenial equaions (ODE), namely x = f(,x), x( 0 ) = x 0, we canno usually be sure ha he model is perfec. If he underlying srucure of he model depends upon subjecive choices, one way o incorporae hese ino he model, is o uilize he aspec of fuzziness, which leads o he consideraion of fuzzy differenial equaions (FDE). The inricacies involved in incorporaing fuzziness ino he heory of ODE pose a cerain disadvanage and oher possibiliies are being explored o address his problem. One of he approaches is o connec FDE o mulivalued differenial equaions and examine he inercon- Received: Augus 3, 2009 c 2009 Academic Publicaions Correspondence auhor

2 166 A.V. Plonikov, N.V. Skripnik necion beween hem [10], [11], [12], [19], [20], [24], [26]. The oher approach is o ransform FDE ino differenial inclusion wih he fuzzy righ-hand sides so as o employ he exising heory of differenial inclusions [2], [4], [5], [8], [16], [17]. In his aricle we consider fuzzy differenial inclusions. 2. Preliminaries Le conv(r n ) be he family of all nonempy compac convex subses of R n wih he Hausdorff meric h(a,b) = max{max min a A b B a b, max min b B a A a b }, where denoes he usual Euclidean norm in R n. I is clear ha (conv(r n ),h) becomes a meric space. Le E n be he family of mappings x : R n [0,1] saisfying he following condiions: (i) x is normal, i.e. here exiss an y 0 R n such ha x(y 0 ) = 1; (ii) x is fuzzy convex, i.e. x(λy + (1 λ)z) min{x(y),x(z)} whenever y,z R n and λ [0,1]; (iii) x is upper semiconinuous, i.e. for any y 0 R n and ε > 0 exiss δ(y 0,ε) > 0 such ha x(y) < x(y 0 ) + ε whenever y y 0 < δ, y R n ; (iv) [x] 0 = cl{y R n : x(y) > 0 } is compac. Denoe [x] α = {y R n : x(y) α } for α (0,1]. Then from (i) (iv) follows ha he α-level se [x] α conv(r n ) for all α [0,1]. Le 0 be he fuzzy mapping defined by 0(y) = 0 if y 0 and 0(0) = 1. Define D : E n E n R + by he equaion D(x,y) = sup h([x] α,[y] α ). I α [0,1] is easy o show ha D is a meric in E n. Definiion 1. (see [19]) A mapping f : I E n is called measurable if for any α [0,1] he mulivalued mapping f α () = [f()] α is Lebesgue measurable. Definiion 2. (see [19]) A mapping f : I E n is called coninuous a poin 0 I provided for any ε > 0 here exiss δ > 0 such ha D(f(),f( 0 )) < ε whenever 0 < δ, I. Definiion 3. (see [9]) A mapping f : I E n is called differeniable a

3 THE GENERALIZED SOLUTIONS OF poin I if for any α [0,1] he mulivalued mapping f α () is Hukuhara differeniable a poin (see [7]) and he family {D H f α () : α [0,1]} defines a fuzzy number f () E n (which is called he fuzzy derivaive of f() a poin ). Definiion 4. (see [9]) The elemen g E n is called he inegral of f : I E n over I if [g] α = (A) f α ()d for any α (0,1], where (A) f α ()d I I is he Aumann inegral, see [3]. Definiion 5. A mapping f : I E n is called absoluely coninuous on I if here exiss an inegrable map g : I E n such ha f() = f( 0 ) + 0 g(s)ds, 0 I for every I. Le comp(e n ) (conv(e n )) be he family of all subses F of he space E n such ha for any α [0,1] he family of all α level ses of he elemens from F is he nonempy compac (and convex) elemen in comp(r n ) (ha is he elemen of cc(r n ) (cocc(r n )) (see [15])) wih meric d(a,b) = max{sup inf D(a,b),sup inf D(a,b)}. b B a A a A Define also he disance from he elemen x E n o he se A comp(e n ) : b B ρ(x,a) = min a A D(x,a). We inroduce in comp(e n ) he usual algebraic operaions: addiion : X + Y = {x + y : x X, y Y }; muliplicaion by scalars λ: λx = {λx : x X}. The following properies hold (see [25]): 1) A + B comp(e n ); 4) αa comp(e n ); 2) A + B = B + A; 5) α(βa) = (αβ)a; 3) (A + B) + C = A + (B + C); 6) α(a + B) = αa + αb; 4) A + {0} = A; 7) 1 A = A. Definiion 6. A mapping F : I comp(e n ) is called measurable on I if he se { I : F() G } is measurable for every G comp(e n ). Definiion 7. A mapping F : I conv(e n ) is called coninuous a poin 0 I provided for any ε > 0 here exiss δ > 0 such ha d(f(),f( 0 )) < ε whenever 0 < δ, I.

4 168 A.V. Plonikov, N.V. Skripnik Define an inegral of F : I comp(e n ) over I : F()d = f()d : f() F() almos everywhere on I. I I 3. Main Resuls Consider he fuzzy differenial inclusion where F : I E n comp(e n ), 0 I, x 0 E n. x F(,x), x( 0 ) = x 0, (1) Definiion 8. The absoluely coninuous mapping x( ) is called he ordinary soluion of he differenial inclusion (1) if x () F(,x()) almos everywhere on I. Le O(F) be he se of all ordinary soluions of he differenial inclusion (1). Definiion 9. The coninuous mapping x( ) is called he generalized soluion of he differenial inclusion (1) if he inegral inclusion x( ) x( ) + F(,x())d (2) fulfills for all, I such ha <. Le G(F) be he se of all generalized soluions of he differenial inclusion (1). Definiion 10. The mulivalued mapping F : I D comp(e n ),D E n saisfies he Caraheodory condiions if: 1) for every fixed x D he mulivalued mapping F(,x) is measurable on he inerval I; 2) for almos every fixed I he mulivalued mapping F(, ) is coninuous on D; 3) here exiss a Lebesgue summable funcion m : I R such ha d(f(,x), { 0}) m() for almos every (,x) I D. Using he ideas of (see [13]) we can proof he following lemma. Lemma 1. Le he mulivalued mapping F : I D comp(e n ) saisfy he Caraheodory condiions, ϕ : I D is measurable. Then he mulivalued

5 THE GENERALIZED SOLUTIONS OF mapping F(, ϕ()) is summable on I. Theorem 11. Le F : I D conv(e n ) saisfy he Caraheodory condiions. Then O(F) = G(F). Proof. Le x : I D be he ordinary soluion of he inclusion (1) and Φ() = F(,x()). The mulivalued mapping Φ : I conv(e n ) is summable on I by Lemma 1. According o he definiion 8 x( ) is absoluely coninuous on I and x () Φ() for almos every I. In addiion D(x (), 0) d(φ(), { 0}) m(). So x ( ) is a summable single-valued selecion of he mulivalued mapping Φ( ) and according o he definiion of he inegral we have x ()d Φ()d for all, I such ha <. I means ha he inclusion x( ) x( ) + is rue for all, I such ha <. Φ()d Therefore x( ) is he generalized soluion of he differenial inclusion (1), ha is O(F) G(F). Le us prove he inverse inclusion. Le x( ) be he generalized soluion of he inclusion (1). Then x( ) is coninuous and x( ) x( ) + F(,x())d for all, I such ha <. Using he properies of he inegral of he mulivalued mappings we have D(x( ),x( )) d F(s,x(s))ds, { 0} for all, I such ha <. d(f(s,x(s)), { 0})ds m(s)ds, (3)

6 170 A.V. Plonikov, N.V. Skripnik Consider he funcion ϕ() = m(s)ds. As m( ) is summable on I, hen 0 ϕ( ) is absoluely coninuous on I as he inegral wih a variable op limi. Therefore for any ε > 0 exiss δ(ε) > 0 such ha for every m N and every i < i, i = 1,m such ha m ( ) < δ fulfills he inequaion i=1 i i m (ϕ( i ) ϕ( i)) < ε. i=1 Thereby using (3) we have m D(x( i ),x( i )) < ε. i=1 Moreover according o (2) for all, exiss r(, ) E n such ha x( ) = x( )+r(, ). Then similarly o [1] we have ha he mapping x() is absoluely coninuous and here exiss x( ) x( ) for all, such ha >. Using he properies of he meric we ge 0 ρ(x (),F(,x())) D (x (), 1η ) (x( + η) x()) + d 1 η for any η > 0 and I. +η F(s,x(s))ds,F(,x()) = A 1 (,η) + A 2 (,η), (4) The absoluely coninuous mapping x( ) is differeniable almos everywhere, so A 1 (,η) 0 when η 0 for almos every I. By [14] for almos every we have A 2 (,η) = d 1 +η F(s,x(s))ds,F(,x()) η = d 1 η +η 1 η F(s,x(s))ds, 1 η +η +η F(,x())ds d(f(s,x(s)),f(,x())ds 0 when η 0. Le η in (4) go o 0, hen we have ρ(x (),F(,x()) = 0 almos everywhere

7 THE GENERALIZED SOLUTIONS OF on I. So x () F(,x()) almos everywhere on I. I means ha x( ) is he ordinary soluion of he differenial inclusion (1), ha is G(F) O(F). Remark. This resul generalizes he resuls of M. Dawidowski [6] for he ordinary differenial inclusions and A.V. Plonikov [15], [21] for he differenial inclusion wih he Hukuhara derivaive. References [1] Z. Arsein, On he calculus of closed se-valued funcions, Indiana Univ. Mah. J., 24, No. 5 (1974), [2] J.-P. Aubin, Fuzzy differenial inclusions, Probl. Conrol Inf. Theory, 19, No. 1 (1990), [3] R.J. Aumann, Inegrals of se-valued funcions, J. Mah. Anal. Appl., 12 (1965), [4] V.A. Baidosov, Differenial inclusions wih fuzzy righ-hand side, Sovie Mahemaics, 40, No. 3 (1990), [5] V.A. Baidosov, Fuzzy differenial inclusions, J. of Appl. Mah. and Mechan., 54, No. 1 (1990), [6] M. Dawidowski, On some generalizaion of Bogolubov averaging heorem, Func. e Approxim. (RPL), No. 7 (1979), [7] M. Hukuhara, Inegraion des applicaions mesurables don la valeur es un compac convexe, Func. Ekvacioj, 10 (1967), [8] E. Hullermeier, An approach o modelling and simulaion of uncerain dynamical sysem, Inerna. J. Uncerain. Fuzziness Knowledge - Based Sysems, 7 (1997), [9] O. Kaleva, Fuzzy differenial equaions, Fuzzy Ses and Sysems, 24, No. 3 (1987), [10] O. Kaleva, The Cauchy problem for fuzzy differenial equaions, Fuzzy Ses and Sysems, 35 (1990), [11] O. Kaleva, The Peano Theorem for fuzzy differenial equaions revisied, Fuzzy Ses and Sysems, 98 (1998), [12] O. Kaleva, O noes on fuzzy differenial equaions, Nonlinear Analysis, 64 (2006),

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