ON THE DENSE TRAJECTORY OF LASOTA EQUATION


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1 UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory of dyamical system give by Lasota equatio is costructed. 1. Itroductio. The equatio u t + u x = F u) was first itroduced by McKedrick i 1926 [8] ad vo Foerster i 1959 [2]. It described the dyamics of populatio age structure. A classical system of equatios with a olocal boudary coditio has always bee the subject of iterest of the whole world of mathematics. The ext stage of research work o similar type equatios was the work of Lasota, Ważewska ad Mackey [5, 6]. They used a similar type of equatio, to be precise, equatio u t + cx) u = F x, u) x to describe blood cell populatio. Appearace of iterpretatio for this equatio ispired professor Lasota ad his parters to study chaos ad stability i dyamical systems give by this equatio. The first impulse was give by professor Lasota [3] provig the existece of ivariat measure, therefore callig this Lasota equatio would be good ad legitimate. Apart from Lasota ad oe of the authors, also Rudicki [9] ad Szarek [4] worked o ivariat measures.the Loskot [7] aalyzed them i the turbulece aspect i the Bass sese. The subject of this work is to prove the existece of a dese trajectory for Lasota equatio. I the costructio of a dese trajectory, a geeralizatio of Avez method was used. Till this time this method has bee used as a tool for ivariat measure costructio i the works of Lasota ad his studets. Costructio of this trajectory may also be called a Avez costructio, as
2 62 right iverses are the basic tool here. However, this variat of Avez method is more iterestig as it does ot require discretisatio of this system. 2. Formulatio of the theorem. I paper [1], the existece of a ivariat measure for the dyamical system geerated o the space V of all Lipschitz fuctios o [0, 1] by the equatio u 1) t + x u x = λu is proved, where λ > 1. The followig theorem esures the existece of a dese trajectory for the same system. Theorem 1. Let us cosider equatio 1) i the domai with the iitial coditio x [0; 1] t 0 u0, x) = vx). Let {T t } t 0 be the semidyamical system geerated by this problem, i.e., 2) T t v)x) = e λt vxe t ). If λ > 1, the there exists a dese trajectory of system {T t } at the space V of all Lipschitz fuctios o [0; 1] vaishig i Auxiliary elemetary lemma. To prove the theorem, the followig elemetary techical lemma is ecessary Lemma 1. Let {s }, {b }, {c } be arbitrary sequeces of positive umbers. The, there exists the sequece {a } such that for every positive iteger 3) 0 < a < c ad 4) a k s k b. k= Proof. To prove the lemma, first for ay positive iteger p we costruct the sequece {a p } satisfyig 3) for ay ad 4) for all p. The sequece {a 1 } is defied by the formula a 1 = mi { c, b s }. It is obvious that a 1 c. Moreover, a 1 k s 1 k b 1 2 k+1 = 1 2 b 1 < b 1.
3 63 Now assume the existece of sequeces {a r } for all r = 1,..., p. Assume also, that these sequeces satisfy the followig iequalities ad a r a r 1 for r = 2,..., p a r k s r < b r k= for all r. From the covergece of the last series, there follows that there exists such kp), that 5) a r k s r < 1 2 b p+1. Defie 6) a p+1 = where k=p) { { } bp+1 mi 2ϕ,p), ap a p ϕ, p) = k=+1 for p p), otherwise, a p k s. Now, havig defied the sequece {a p } for all p the sequece {a } defied by the classical diagoal formula a = a satisfies the coditio of thesis. 4. Proof of the theorem. The space V with the topology of uiform covergece is a separable metric space. Therefore, the topology has a coutable basis. Let {σ } V ad {ε } R + be such sequeces, that the set {U } =1, where 7) U = Uσ, ε ) = {v V : σ x) vx) < ε x [0; 1]} is a basis of uiform topology i V. Sice σ V, for every, there exists the optimal Lipschitz costat of σ i.e. s = sup σ x) σ y) x y. Obviously, x,y [0,1],x y sup σ x) σ y) s x y s. x,y [0;1]
4 64 Sice σ is cotiuous ad vaishes at 0, oe ca defie the sequece {κ } by the followig recurrece formula Let κ 0 = 1, κ = sup{x [0; 1] : ξ [0; x] σ x) ε }. c = κ κ 1 b = ε 1. ) 1 λ, By Lemma 1 there exists such sequece {a }, that ) λ a k κ, Defie ad By the last four formulae there is ad e λt s =k e t = a k s k ε. k= θ j = max{ λ l a j, 0} e θ k = exp =k t = θ j. ) λ e λ l a k = a k κ λθ j s k 1 exp λθ j e λθ s =k k 1 exp λθ j a s e λt k 1 ε k 1. =k
5 65 Cotiuig the proof, we have to costruct a family of right iverses of T t i.e., the family of the maps S : V V satisfyig the coditio T t Sv = v for every v V. Let σ V Defie Sσ t : V V as { Sσv)x) t e λt xxe t ) for x e t, = σx) σe t ) + e λt v1) for x > e t. From this defiitio we coclude that for every t > 0 ad for every σ V T t S t σ = id V. Let ow σ ad θ be defied as above ad let v S θ 1 σ 1... Sσ θ V ). =1 We claim, that such v exists ad is uique. The uiqueess of v follows from the cotiuity of v, v/s vaishig at zero ad a atural coditio that for every iterval [exp t ); exp t 1 )] 8) vx) = e λ θ k σ xe ) θ k = e λt σ xe t ) up to additive costat. Let v = S θ 1 σ 1... Sσ θ 0). The fuctio v is equal to every fuctio belogig to S θ 1 σ 1... Sσ θ V ) up to a additive costat o the iterval [exp t ); 1]. Moreover, v x) v +1 x) e λt +1 s +1. From the last iequality there follows that the sequece {v } coverges uiformly to some fuctio v. To complete the claim, it is sufficiet to show that v satisfies the Lipschitz coditio. From 8) there follows that o every iterval of the form [exp t ); exp t 1 )], the fuctio v satisfies the Lipschitz coditio with the costat Moreover, e λt s e t = s e λt s e t. e 1 λ)θ k 1. Whece there follows, that the fuctio v satisfies the Lipschitz coditio with costat 1, ad i cosequece v = v. To complete the proof of the theorem, it is sufficiet to prove that {T t v} t 0 is dese i V. From the defiitio of c ad
6 66 t there follows that o the iterval [0, exp =k θ )] the absolute value of fuctio v is less tha =k e λt s exp λt k 1 ) ε k 1 ad o the iterval [ ) )] exp θ ; exp θ, =k =k 1 the fuctio v is equal to the fuctio e λt k 1σ k 1 xe t k 1 ). I cosequece, T tk 1 v is less tha ε k 1 o the iterval [0, e t k] ad equal to σ k 1 up to a additive costat o the iterval [e t k, 1]. Thus T tk 1 v U k 1. Sice {U } is a basis of uiform topology i V, the last formula completes the proof. Refereces 1. Dawidowicz A.L., O the existece of a ivariat measure for the dyamical system geerated by partial differetial equatio, A. Polo. Math., XLI 1983), vo Foerster J., Some remarks o chagig populatios, i: The kietics of cell proliferatio, Stohlma F. ed.), Grue & Stratto, New York, 1959, Lasota A., Piaigiai G., Ivariat measures o topological spaces, Boll. U. Mat. Ital., 515 B) 1977), Lasota A., Szarek T., Dimesio of measures ivariat with respect to Ważewska partial differetial equatios, J. Differetial Equatios, 1962) 2004), Lasota A., WażewskaCzyżewska M., Matematycze problemy dyamiki uk ladu krwiek czerwoych, Mat. Stos., ), Lasota A., WażewskaCzyżewska M., Mackey M.C., Miimazig therapeutically iduced aemia, J. Math. Biol., ), Loskot K., Turbulet solutios of first order partial differetial equatio, J. Differetial Equatios, 581) 1985), McKedrick A.G., Applicatio of mathematics to medical problems, Ediburgh Math. Soc., ), Rudicki R., Ivariat measures for the flow of a first order partial differetial equatio, Ergodic Theory Dyam. Systems, 53) 1985), Received March 1, 2004 Jagielloia Uiversity Istitute of Mathematics ul. Reymota 4/ Kraków Polad
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