Uncertain Dynamic Systems on Time Scales

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1 Journl of Uncertin Sytem Vol.9, No.1, pp.17-30, 2015 Online t: Uncertin Dynmic Sytem on Time Scle Umber Abb Hhmi, Vile Lupulecu, Ghu ur Rhmn Abdu Slm School of Mthemticl Science, GCU Lhore Univerity Contntin Brncui, Str. Genev, Nr. 3, Trgu Jiu, Romni Received 28 Jnury 2013; Revied 13 Augut 2013 Abtrct In thi pper we tudy the exitence nd uniquene of the olution for the dynmic ytem on time cle with uncertin prmeter. For thi im, we introduced the notion of uncertin proce on time cle. In ddition, the liner dynmic ytem on time cle with uncertin prmeter re lo tudied. c 2015 World Acdemic Pre, UK. All right reerved. Keyword: differentil eqution, time cle, uncertin meure, uncertin proce, exitence nd uniquene, finnce, uncertin inference 1 Introduction The uncertin theory w introduced by Liu [14] n importnt tool for the tudy of ome rel world phenomen which cnnot be modeled by fuzzine. The min tge on the development of thi theory nd ome fundmentl reult cn be found in work [7, 14, 15, 16, 22]. The concept of uncertin differentil eqution w lo introduced in [14], but the exitence nd uniquene of the olution of thi kind of differentil eqution w obtined by Chen nd Liu in the work [6]. The theory of dynmic ytem on time cle imultneouly llow u to tudy continuou nd dicrete dynmic ytem. Since Hilger initil work [10] there h been ignificnt growth in the theory of dynmic ytem on time cle, covering vriety of different qulittive pect. The clculu of time cle w initited by B. Aulbch nd S. Hilger in order to crete theory tht cn unify dicrete nd continuou nlyi. We refer to the book [4, 5], nd the pper [1, 2, 21] which re more pecific with repect to our trget. Hoffcker [12], Ahlbrndt nd Morin [3] re known tht hve been worked nd demontrted the relted ide to the multivrite ce nd the tudy of prtil dynmic eqution (PDE), however, thee ide re lredy known for univrite ce of dynmic eqution [13]. The uncertin dynmic ytem re widely ued for olving different problem, nd thi technique i rther well developed. A lot of cientific work re dedicted to the development of thi method [2, 4, 5, 6, 10, 11]. Mny intereting nd importnt reult hve been obtined in thi field by different uthor. Depite of thi fct there till remin lot of unolved problem. For filling thee gp, in thi rticle we conider problem of uncertin dynmic ytem on time cle. It i well known tht the mthemticl model re necery to be ble to tudy the rel world ytem. Thee model involve different prmeter whoe vlue re determined by meurement. Even when the meurement re mde by modern technologie the end reult i tht the vlue of the prmeter re different from one meurement to nother. Depending on the nture of the vlue obtined from the meurement, the vrition of the prmeter cn be conidered to be rndom vrible, fuzzy rndom vrible or uncertin vrible. Moreover, the time vrition of the prmeter cn be continuou, dicrete, or my lternte from one time intervl to nother. In the ltter ce, the mthemticl model on time cle pper nturl one. The purpoe of thi pper i to prove the exitence nd uniquene of olution for the dynmic ytem on time cle with uncertin prmeter. The orgniztion of thi pper i follow. Section 2 preent few definition nd concept of time cle. Alo, the notion of uncertin proce on time cle i introduced. In Section 3 we prove the exitence nd uniquene of olution for the dynmic ytem on time cle with Correponding uthor. Emil: umberbb@yhoo.com (U.A. Hhmi).

2 18 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle uncertin prmeter. In the lt ection, we tudy the liner dynmic ytem on time cle with uncertin prmeter. 2 Preliminry 2.1 Time Scle By time cle T we men ny cloed ubet of R. Then T i complete metric pce with the metric defined by d(t, ) := t for t, T. Since we know tht for working into the different connected component of the time cle T, we need the concept of jump opertor. The forwrd jump opertor σ : T T i defined by σ(t) := inf T : > t}, while the bckwrd jump opertor ρ : T T i defined by ρ(t) := up T : < t}. In thi definition we put inf = up T nd up = inf T. The grinine function µ : T [0, ) i defined by µ(t) := σ(t) t. If σ(t) > t, we y t i right-cttered point, while if ρ(t) < t, we y t i left-cttered point. Point tht re right-cttered nd left-cttered t the me time will be clled iolted point. A point t T uch tht t < up T nd σ(t) = t, i clled right-dene point. A point t T uch tht t > inf T nd ρ(t) = t, i clled left-dene point. Point tht re right-dene nd left-dene t the me time will be clled dene point. The et T κ i defined to be T κ = T\m} if T h left-cttered mximum m, otherwie T κ = T. To undertnd the notion we hve to conider ome exmple for clering the btrction of the itution. Given time cle intervl [, b] T := t T : t b}, then [, b] κ T denoted the intervl [, b] T if < ρ(b) = b nd denote the intervl [, b) T if < ρ(b) < b. In fct, [, b) T = [, ρ(b)] T. Alo, for T, we define [, ) T = [, ) T. If T i bounded time cle, then T cn be identified with [inf T, up T] T. If T nd δ > 0, then we define the following neighborhood of : U T (, δ) := ( δ, + δ) T, U + T (, δ) := [, + δ) T, nd U T (, δ) := ( δ, ] T. Let R m be the pce of m-dimenionl column vector x = col(x 1, x 2,..., x m ) with norm. Definition 2. 1 ([4]) A function f : T R m i clled regulted if it right-ided limit exit (finite) t ll right-dene point in T, nd itleft-ided limit exit (finite) t ll left-dene point in T. A function f : T R m i clled rd-continuou if it i continuou t ll right-dene point in T nd it left-ided limit exit (finite) t ll left-dene point in T. Denote by C rd (T, R m ) the et of ll rd-continuou function from T into R m. Obviouly, continuou function i rd-continuou, nd rd-continuou function i regulted ([4, Theorem 1.60]). Definition 2. 2 ([4]) A function f : [, b] T R m R m i clled Hilger continuou if f i continuou t ech point (t, x) where t i right-dene, nd the limit lim f(, y) nd (,y) (t,x) both exit nd re finite t ech point (t, x) where t i left-dene. lim f(t, y) y x Definition 2. 3 ([4]) Let f : T R m nd t T κ. Let f (t) R m (provided it exit) with the property tht for every ε > 0, there exit δ > 0 uch tht f(σ(t)) f() f (t)[σ(t) ] ε σ(t) (1) for ll U T (t, δ). We cll f (t) the delt (or Hilger) derivtive ( -derivtive for hort) of f t t. Moreover, we y tht f i delt differentible ( -differentible for hort) on T κ provided f(t) exit for ll t T κ. The following reult will be very ueful. Propoition 2. 1 ([4, Theorem 1.16]) Aume tht f : T R m nd t T κ. (i) If f i -differentible t t, then f i continuou t t. (ii) If f i continuou t t nd t i right-cttered, then f i -differentible t t with f (t) = f(σ(t)) f(t). σ(t) t

3 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, (iii) If f i -differentible t t nd t i right-dene, then f f(t) f() (t) = lim. t t (iv) If f i -differentible t t, then f(σ(t)) = f(t) + µ(t)f (t). It i known [9] tht for every δ > 0 there exit t let one prtition P : = < t 1 < < t n = b of [, b) T uch tht for ech i 1, 2,..., n} either t i t i 1 δ or t i t i 1 > δ nd ρ(t i ) = t i 1. For given δ > 0 we denote by P([, b) T, δ) the et of ll prtition P : = < t 1 < < t n = b tht poe the bove property. Let f : T R m be bounded function on [, b) T, nd let P : = < t 1 < < t n = b be prtition of [, b) T. In ech intervl [t i 1, t i ) T,where 1 i n, we chooe n rbitrry point ξ i nd form the um S = n (t i t i 1 )f(ξ i ). We cll S Riemnn -um of f correponding to the prtition P. i=1 Definition 2. 4 ([8]) We y tht f i Riemnn -integrble from to b (or on [, b) T ) if there exit vector I R m with the following property: for ech ε > 0 there exit δ > 0 uch tht S I < ε for every Riemnn -um S of f correponding to prtition P P([, b) T, δ) independent of the wy in which we chooe ξ i [t i 1, t i ) T, i = 1, 2,..., n. It i eily een tht uch vector I i unique. The vector I R m i the Riemnn -integrl of f from to b, nd we will denote it by b f(t) t. Propoition 2. 2 ([8, Theorem 5.8]) A bounded function f : [, b) T R m i Riemnn -integrble on [, b) T if nd only if the et of ll right-dene point of [, b) T t which f i dicontinuou i et of meure zero. Since every regulted function on compct intervl i bounded (ee [4, Theorem 1.65]), o, we get tht every regulted function f : [, b] T R m, i Riemnn -integrble from to b. Propoition 2. 3 ([11, Theorem 5.8]) Aume tht, b T, < b nd f : T R m i rd-continuou. Then the integrl h the following propertie. (i) If T = R, then b f(t) t = b f(t)dt, where the integrl on the right-hnd ide i the Riemnn integrl. (ii) If T conit of iolted point, then b f(t) t = t [,b) T µ(t)f(t). Definition 2. 5 ([4]) A function g : T R m i clled -ntiderivtive of f : T R m if g (t) = f(t) for ll t T κ. Bohner exclimed tht ech rd-continuou function h -ntiderivtive [4, Theorem 1.74]. Propoition 2. 4 ([8, Theorem 4.1]) Let f : T R m be Riemnn -integrble function on [, b) T. If f h -ntiderivtive g : [, b] T R m, then b f(t) t = g(b) g(). In prticulr, σ(t) f() = µ(t)f(t) for t ll t [, b) T (ee [4, Theorem 1.75]). Propoition 2. 5 ([8, Theorem 4.3]) Let f : T R m be function which i Riemnn -integrble from to b. For t [, b] T, let g(t) = f(t) t. Then g i continuou on [, b] T. Further, let [, b) T nd let f be rbitrry t if i right-cttered, nd let f be continuou t if i right-dene. Then g i -differentible t nd g ( ) = f( ). Lemm 2. 1 ([21]) Let g : R R be continuou nd nondecreing function. If, t T with t, then g(τ) τ g(τ)dτ. Severl of the propertie of the Riemnn -integrl re dicued in [1, 4].

4 20 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle 2.2 Uncertin Proce on Time Scle Let Γ be n nonempty et nd let L be σ-lgebr of et of Γ. A mpping M : L [0, 1] i clled uncertin meure if it tifie the following xiom: (A1) M(Γ) = 1; (A2) M(A) M(B) for ll A, B L with A B; (A3) M(A) + M(A c ) = 1 for ll A L, where A c := Γ\A; (A4) For every countble equence A n } of element of L, we hve ( ) M A n = M (A n ). n=1 n=1 Let Γ be nonempty et, L σ-lgebr on Γ, nd M n uncertin meure. T he triplet (Γ, L, M) i clled n uncertin pce. Exmple 2. 1 Let u conider Γ = (0, 1), L the σ-lgebr of ll Borel ubet of Γ. Let λ : (0, 1) R + be defined by λ(x) = x 1 2, x (0, 1). Then the mpping M : L [0, 1] defined by M(A) = upλ(x), x A 1 up x A c λ(x), if upλ(x) < 1/2 x A if upλ(x) 1/2, x A i n uncertin meure on (0, 1). Denote by B the σ-lgebr of ll Borel ubet of R m. A function X( ) : Γ R m i clled n uncertin vrible if X i meurble function from (Γ, F) into (R m, B), tht i, X 1 (B) := γ Γ; X(γ) B} L for ll B B. A time cle uncertin proce i function X(, ) : [, b] T Γ R m uch tht X(t, ) : Γ R m i uncertin vrible for ech t T. For ech point γ Γ, the function on T given by t X(t, γ) i will be clled mple pth of the time cle uncertin proce X(, ) correponding to γ. A time cle uncertin proce X(, ) i id to be regulted (rd-continuou, continuou) if the trjectory t X(t, γ) i regulted (rd-continuou, continuou) function on [, b] T for ech γ Γ. Lemm 2. 2 Let X(, ) : [, b] T Γ R m be time cle uncertin proce. If the mple pth t X(t, γ) i Riemnn -integrble on [, b) T for every γ Γ, then the function Y (, ) : [, b] T Γ R m given by Y (t, γ) = i continuou time cle uncertin proce. X(, γ), t [, b] T Proof: From Propoition 2.5, it follow tht the function t X(, γ) i continuou for ech γ Γ. Since the Riemnn -integrl i limit of the finite um S (γ) = n i=1 (t i t i 1 )X(ξ i, γ) of meurble function, we hve tht γ X(, γ) i meurble function. Therefore, Y (, ) i continuou time cle uncertin proce.

5 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, Uncertin Initil Vlue Problem In the following, conider n initil vlue problem of the form X (t, γ) = F (t, X(t, γ), γ), t [, b] κ T X(, γ) = X 0 (γ), (2) where X 0 : Γ R m i uncertin vrible nd F : [, b] κ T Rm Γ R m tifie the following umption: (H1) F (t, x, ) : Γ R m i uncertin vrible for ll (t, x) [, b] κ T Rm, (H2) for ech γ Γ, the function F (,, γ) : [, b] κ T Rm R m i Hilger continuou function t every point (t, x) [, b] κ T Rm. By olution of (2) we men time cle uncertin proce X(, ) : [, b] κ T Γ Rm tht tifie condition in (2). Remrk 1: Conider the uncertin differentil Eqution (2) fmily (with repect to prmeter γ) of determinitic differentil eqution, nmely X (t, γ) = F (t, X(t, γ), γ), t [, b] κ T, γ Γ, (3) X(, γ) = X 0 (γ). Then i not correct to olve ech problem (3) to obtin the olution of (2). Let u give two exmple. Exmple 3. 1 Let (Γ, L, M) be n uncertin pce. Conider n initil vlue problem of the form X (t, γ) = K(γ)X 2 (t, γ), t [0, ) R, γ Γ, X(0, γ) = 1, (4) where K : Γ (0, ) i uncertin vrible. It i ey to ee tht, for ech γ Γ, X(t, γ) = 1/(1 K(γ)t) i olution of (4) on the intervl [0, 1/K(γ)]. Since for ech 0 we hve tht M(1/K(γ) > ) < 1, it follow tht not ll olution X(, γ) re well defined on ome common intervl [0, ). Exmple 3. 2 Let (Γ, L, M) be n uncertin pce nd let Γ 0 / L. It i ey to check tht, for ech γ Γ, the function X(, ) : [0, 1] R Γ R, given by 0 if γ Γ0 X(t, γ) = t 3/2 if γ Γ\Γ 0, i olution of the initil vlue problem X (t, γ) = 3 2 X(t, γ), t [0, ) R, γ Γ, X(0, γ) = 0. But X(, ) i not uncertin proce. Indeed, we hve tht tht i, γ X(1, γ) i not meurble function. γ Γ; X(1, γ) [ 1 2, 1 2 ]} = Γ 0 / L, Uing the Propoition 4 nd 5 nd [21, Lemm 2.3], it i ey to prove the following reult. Lemm 3. 3 A time cle uncertin proce X(, ) : [, b] κ T Γ Rm i the olution of the problem (2) if nd only if X(, ) i continuou time cle uncertin proce nd it tifie the following uncertin integrl eqution X(t, γ) = X 0 (γ) + F (, X(, γ), γ), t [, b] T, γ Γ. (5) The following reult i known Gronwll inequlity on time cle nd will be ued in thi pper.

6 22 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle Lemm 3. 4 ([21, Lemm 3.1]) Let rd-continuou time cle uncertin procee X(, ), Y (, ) : [, b] κ T Γ R + be uch tht X(t, γ) Y (t, γ) + where 1 + µ(t)p(t) > 0, for ll t [, b] T. Then we hve X(t, γ) Y (t, γ) + q()x(, γ), t [, b] T, γ Γ, e p (t, σ()p()y (, γ), t [, b] T, γ Γ. Theorem 3. 1 Let F : [, b] κ T Rm Γ R m tifie (H1)-(H2) nd ume tht there exit rd-continuou time cle uncertin proce L(, ) : [, b] κ T Γ R + uch tht F (t, x, γ) F (t, y, γ) L(t, γ) x y (6) for every t [, b] κ T, x, y Rm nd γ Γ. Let X 0 : Γ R m uncertin vrible uch tht F (t, X 0 (γ), γ) M, t [, b] κ T, γ Γ, (7) where M > 0 i contnt. Then the problem (2) h unique olution. Proof: To prove the theorem we pply the method of ucceive pproximtion (ee [21]). For thi, we define equence of function X n (, ) : [, b] κ T Γ Rm, n N, follow: X 0 (t, γ) = X 0 (γ) X n (t, γ) = X 0 (γ) + F (, X n 1 (, γ), γ), n 1, for every t [, b] κ T nd every γ Γ. Firt, uing (7) nd the Lemm 2.1, we oberve tht (8) X 1 (t, γ) X 0 (t, γ) F (, X 0 (γ), γ) M(t ) M(b ), t [, b] T, γ Γ. We prove by induction tht for ech integer n 2 the following etimte hold (t )n (b )n X n (t, γ) X n 1 (t, γ) M L(γ) M L(γ), t [, b] T, γ Γ, (9) n! n! where L(γ) = up L(t, γ). Suppoe tht (9) hold for n = k 2. Then, uing (6), (7) nd Lemm 2.1, we [,b] T obtin X k+1 (t, γ) X k (t, γ) L(γ) X k (, γ) X k 1 (, γ) L(γ) M k! L(γ) M k! F (, X k (, γ), γ) F (, X k 1 (, γ), γ) ( ) k ( ) k (t )k+1 (b )k+1 d = M L(γ) (k+1)! M L(γ) (k+1)!, for ll t [, b] T nd γ Γ. Thu, (9) i true for n = k + 1 nd o (9) hold for ll n 2. Further, we how tht for every n N the function X n (, γ) : [, b] T R re continuou for ech γ Γ. Let ε > 0 nd t, [, b] T be uch tht t < ε/m. We hve X 1 (t, γ) X 1 (, γ) = F (τ, X 0 (γ), γ) τ F (τ, X 0 (γ), γ) τ = t F (τ, X 0 (γ), γ) τ F (τ, X 0 (γ), γ) τ F (τ, X 0 (γ), γ) dτ M t < ε

7 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, nd o t X 1 (t, γ) i continuou on [, b] T for ech γ Γ. Since for ech n 2 X n (t, γ) X n (, γ) + + n 1 F (τ, X n 1 (τ, γ), γ) τ F (τ, X n 1 (τ, γ), γ) F (τ, X 0 (γ), γ) τ k=1 then, by induction, we obtin F (τ, X k (τ, γ), γ) F (τ, X k 1 (τ, γ), γ) τ, X n (t, γ) X n (, γ) M ( 1 + n 1 k=1 L(γ) k 1 (b ) k k! F (τ, X 0 (γ), γ) τ F (τ, X 0 (γ), γ) τ ) t 0 t. Therefore, for every n N the function X n (, γ) : [, b] T Γ R m i continuou for ech γ Γ. Now, uing Lemm 3.3 nd (8), we deduce tht the function X n (t, ) : Γ R m re meurble. Conequently, it follow tht for every n N the function X n (, ) : [, b] T Γ R i time cle uncertin proce. Further, we hll how tht the equence (X n (t, )) n N i uniformly convergent. Denote Since Y n (t, γ) = X n+1 (t, γ) X n (t, γ), n N, γ Γ. Y n (t, γ) Y n (, γ) L(γ) X n (τ, γ) X n 1 (τ, γ) τ then, reoning bove, we deduce tht the function t Y n (t, γ) re continuou on [, b] T for ech γ Γ. Now, uing (9), we obtin n 1 up X n (t, γ) X m (t, γ) t [,b] T for ll n > m > 0. Since the erie uch tht n=1 k=m up Y k (t, γ) M t [,b] T n 1 k=m L(γ) k (b ) k+1 (k + 1)! L(γ) n 1 (b ) n /n! converge, then for ech ε > 0 there exit n 0 N up X n (t, γ) X m (t, γ) ε for ll n, m n 0 nd γ Γ. (10) t [,b] T Hence, ince ([, b] T, ) i complete metric pce, it follow tht the equence (X n (t, )) n N i uniformly convergent on [, b] T. Denote X(t, γ) = lim X n(t, γ), t [, b] T, γ Γ. Obviouly, t X(t, γ) i continuou n on [, b] T for ech γ Γ. Since, by Lemm 2.2 nd (8), the function γ X n (, γ) re meurble nd X(t, γ) = lim X n(t, γ) for every t [, b] T nd γ Γ, we deduce tht γ X(t, γ) i meurble for every n t [, b] T. Therefore, X(, ) : [, b] T Γ R m i continuou time cle uncertin proce. We how tht X(, ) tifie the uncertin integrl eqution (5). For ech n N we put G n (t, γ) = F (t, X n (t, γ), γ), t [, b] T, γ Γ. Then G n (t, γ) i rd-continuou time cle uncertin proce, nd we hve tht up G n (t, γ) G m (t, γ) L(γ) up X n (t, γ) X m (t, γ), t [, b] T, γ Γ, t [,b] T t [,b] T for ll n, m n 0. Uing (10) we infer tht the equence (G n (, γ)) n N i uniformly convergent on [, b] T for ech γ Γ. If we tke m, then for ech ε > 0 there exit n 0 N uch tht for every n n 0 we hve up G n (t, γ) F (t, X(t, γ), γ) L(γ) up X n (t, γ) X(t, γ), t [, b] T, γ Γ, t [,b] T t [,b] T nd o lim G n(t, γ) F (t, X(t, γ), γ) = 0 for ll t [, b] T nd γ Γ. Alo, it ey to ee tht n up G n (, γ) F (, X(, γ), γ) t [,b] T L(γ) X n (, γ) X(, γ), γ Γ.

8 24 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle Since X(t, γ) = lim n X n(t, γ) uniformly on [, b] T, then it follow tht Now, we hve lim n G n (, γ) = F (, X(, γ), γ) for ll t [, b] nd γ Γ. up X(t, γ) X 0(γ) F (, X(, γ), γ) up X(t, γ) X n (t, γ) t [,b] T t [,b] T + up X n(t, γ) X 0 (γ) F (, X n 1 (, γ), γ) t [,b] T + up F (, X n 1 (, γ), γ) F (, X(, γ), γ). t [,b] T Uing the two previou convergence X(t, γ) = X 0 (γ) + F (, X(, γ), γ) for ll t [, b] T nd γ Γ, tht i, X(, ) tifie the uncertin integrl Eqution (5). Then, by Lemm 3.3, it follow tht X(, ) i the olution of the problem (2). Finlly, we how the uniquene of the olution. For thi, we ume tht X(, ), Y (, ) : [, b] T Γ R m re two olution of (5). Since X(t, γ) Y (t, γ) L(γ) X(, γ) Y (, γ) d, t [, b] T, γ Γ, from Lemm 3.4 it follow tht X(t, γ) Y (t, γ) 0, t [, b] T, γ Γ, nd o, the proof i complete. Let T be upper unbounded time cle. Under uitble condition we cn extend the notion of the olution of (2) from [, b] κ T to [, ) T := [, ) T, if we define F on [, ) T R m Γ nd how tht the olution exit on ech [, b] T where b (, ) T, < ρ(b). Theorem 3. 2 Aume tht F : [, ) T R m Γ R m tifie the umption of the Theorem 3.1 on ech intervl [, b] T with b (, ) T, < ρ(b). If there i contnt M > 0 uch tht F (t, x, γ) M for ll (t, x) [, b) T R m then the problem (2) h unique olution on [, ) T. Proof: Let X(, ) be the olution of (2) which exit on [, b) T with b (, ) T, < ρ(b), nd the vlue of b cnnot be increed. Firt, we oberve tht b i left-cttered point, then ρ(b) (, b) T nd the olution X(, ) exit on [, ρ(b)] T. But then the olution X(, ) exit lo on [, b] T, nmely by putting X(b, γ) = X(ρ(b), γ) + µ(b)x (ρ(b), γ) = X(ρ(b), γ) + µ(b)f (ρ(b), X(ρ(b), γ), γ). If b i left-dene point, then their neighborhood contin infinitely mny point to the left of b. Then, for ny t, (, b) T uch tht < t, we hve X(t, γ) X(, γ) F (τ, X(τ, γ), γ) τ M t. Tking limit, t b nd uing Cuchy criterion for convergence, it follow lim t b X(t, γ) exit nd i finite. Further, we define X b (γ) = lim X(t, γ) nd conider the initil vlue problem t b X (t, γ) = F (τ, X(τ, γ), γ), t [b, b 1 ] T, b 1 > σ(b), X(b, γ) = X b (γ). By Theorem 3.1, one get tht X(t, γ) cn be continued beyond b, contrdicting our umption. Hence every olution X(t, γ) of (2) exit on [, ) T nd the proof i complete.

9 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, Uncertin Liner Sytem Let : Γ R be poitively regreive uncertin vrible, tht i, 1 + µ(t)(γ) > 0 for ll γ Γ. Then, by Lemm 2.2, the function (t, γ) e (γ) (t, ) defined by ( ) log(1 + µ(τ)(γ)) e (γ) (t, ) = τ,, t T, γ Γ, µ(τ) i continuou time cle uncertin proce. For ech fixed γ Γ, the mple pth t e (γ) (t, ) i the exponentil function on time cle (ee [4]). It ey to check tht the uncertin proce (t, γ) e (γ) (t, ) i olution of the initil vlue problem (for determinitic ce, ee [4, Theorem 2.33]): X (t, γ) = (γ)x(t, γ), t [, b] κ T, γ Γ, (11) X(, γ) = 1. If : Γ R i bounded then, by the Theorem 3.1 nd 3.2, it follow tht (11) h unique olution on [, ) T. Let u denote by M m (R) the pce of ll m m mtrice. We recll tht A := up Ax ; x 1} define norm on M m (R) nd the following inequlity Ax A x hold for ll A M m (R) nd x R m. A mpping A : Γ M m (R) i clled n uncertin mtrix if ll it component ij : Γ R, i, j = 1, 2,..., m, re uncertin vrible. An uncertin mtrix A i id to be regreive if I +µ(t)a(γ) i invertible for ll t T nd γ Γ, where I i the m m identity mtrix. Moreover, the et R m = R(Γ, M m (R)) of ll regreive uncertin mtrice i group with repect to the ddition opertion define A B = A + B + µ(t)ab for ll t T. The invere element of A R m i given by A = [I + µ(t)a 1 ]A = A[I + µ(t)a] 1 for ll t T. Now conider the following homogeneou liner uncertin initil vlue problem X (t, γ) = A(γ)X(t, γ), t T, γ Γ, X(, γ) = X 0 (γ). (12) where A R m. The correponding nonhomogeneou liner uncertin initil vlue problem i X (t, γ) = A(γ)X(t, γ) + H(t, γ), t T, γ Γ, X(, γ) = X 0 (γ). (13) where H : T Γ R m i n uncertin proce. Theorem 4. 1 Suppoe tht A : Γ M m (R) i regreive nd bounded uncertin mtrix, X 0 : Γ R m i bounded uncertin vrible, nd H(, ) : [, ) T Γ R m i rd-continuou time cle uncertin proce. If there i contnt ν > 0 uch tht H(t, γ) ν for ll t [, b) T with b (, ) T, < ρ(b), then the initil vlue problem (13) h unique olution on [, ) T. Proof: Firt, we oberve tht we put F (t, x, γ) := A(γ)x + H(t, γ), then F tifie the condition (H 1 ) nd (H 2 ). Moreover, F (t, x, γ) F (t, y, γ) A(γ) x y for every t [, ) T, x, y R m nd γ Γ. Therefore, by the Theorem 3.1, it follow tht (13) h unique olution on [, b] κ T. Further, let X(t, ) be the olution of (13) which exit on [, b) T with b (, ) T, < ρ(b). Alo, let N > 0 be uch tht A(γ) N. Then we hve X(t, γ) X(, γ) ν(t ) + N A(γ)X(, γ) + X(, γ). H(, γ)

10 26 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle Then, by the Corollry 6.8 in [4], it follow tht X(t, γ) (1 + ν N )e N(t, ) ν N (1 + ν N )e N(b, ). Hence F (t, X(t, γ), γ) M := ν +(1+ ν N )e N (b, ). Proceeding in the proof of the Theorem 3.2 it follow tht the unique olution of (13) exit on [, ) T. A mpping Ψ : T Γ M m (R) i clled n uncertin mtrix proce if ll it component ψ ij : T Γ R, i, j = 1, 2,..., m, re uncertin proce. An uncertin mtrix proce Ψ i id to be regreive if Ψ(t, ) R m for ll t T. In the following, uppoe tht A : Γ M m (R) i regreive nd bounded uncertin mtrix. An uncertin mtrix proce Ψ A i id to be n uncertin mtrix olution of the the following homogeneou liner uncertin differentil eqution X (t, γ) = A(γ)X(t, γ), t T, γ Γ, (14) if ech column of Ψ A tifie (14). An uncertin fundmentl mtrix of (14) i n uncertin mtrix olution Ψ A of (14) uch tht det Ψ A (t, γ) 0 for ll t T nd γ Γ. An uncertin trnition mtrix of (14) t initil time T i n uncertin fundmentl mtrix Ψ A uch tht Ψ A (, γ) = I for ll γ Γ. The uncertin trnition mtrix of (14) t initil time T will be denoted by U A (t, ). Therefore, the uncertin trnition mtrix of (14) t initil time T i the unique olution of the following uncertin mtrix initil vlue problem Φ (t, γ) = A(γ)Φ(t, γ), Φ(, γ) = I, (15) nd X(t, γ) = U A (t, )X(, γ), t, i the unique olution of the following uncertin initil vlue problem X (t, γ) = A(γ)X(t, γ), t T, γ Γ, X(, γ) = X 0 (γ). The exitence nd uniquene of the olution of (15) follow from the Theorem 3.1. The uncertin trnition mtrix of (14) t initil time T i lo clled the uncertin mtrix exponentil function (t ), nd it i denoted by e A(γ) (t, ) or e A (t, ). In the following theorem we give ome propertie of the uncertin trnition mtrix. The proof of the theorem i the me tht in [4, Theorem 5.21]. Theorem 4. 2 If A : Γ M m (R) i regreive nd bounded uncertin mtrix, then (1) U A (t, t) = I; (2) U A (σ(t), ) = [I + µ(t)a(γ)]u A (t, ); (3) U 1 A (t, ) = U T A (t, ); T (4) U A (t, ) = U 1 A (, t) = U T A (, t); T (5) U A (t, )U A (, r) = U A (t, r), for ll t,, r T with t > > r nd ll γ Γ. Theorem 4. 3 (Vrition of Contnt). Suppoe tht the umption of the Theorem 4.1 hold. Then the unique olution X(, ) : [, ) T Γ R m of the uncertin initil vlue problem (13) i given by X(t, γ) = U A (t, )X 0 (γ) + U A (t, σ())h(, γ), t [, ) T, γ Γ. (16) Proof: Indeed, we cn rewritten (16) [ ] X(t, γ) = U A (t, ) X 0 (γ) + U A (, σ())h(, γ).

11 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, Uing the product rule to differentite X(t, ), we infer [ ] X (t, γ) = A(γ)U A (t, ) X 0 (γ) + U A (, σ())h(, γ) +U A (σ(t), )U A (, σ(t))h(t, γ) = A(γ)X(t, γ) + H(t, γ) Obviouly, X(, γ) = X 0 (γ). Therefore, X(t, γ) i the olution of (16). Corollry 4. 1 Let X 0 : Γ R m be bounded uncertin vrible. If A : Γ M m (R) i regreive nd bounded uncertin mtrix, then the unique olution of the uncertin initil vlue problem (12) i given by X(t, γ) = U A (t, )X 0 (γ), t [, ) T, γ Γ. Theorem 4. 4 (Vrition of Contnt). Suppoe tht the umption of the Theorem 4.1 hold. Then the unique olution X(, ) : [, ) T Γ R m of the following uncertin initil vlue problem X (t, γ) = A T (γ)x σ (t, γ) + H(t, γ), t [, ) T, γ Γ, (17) X(, γ) = X 0 (γ), on [, ) T given by X(t, γ) = Ψ A T (γ)(t, )X 0 (γ) + Proof: Indeed, we cn rewrite (17) tht i, Ψ A T (γ)(t, )H(, γ), t [, ) T, γ Γ. (18) X (t, γ) = A T (γ)[x(t, γ) + µ(t)x (t, γ)] + H(t, γ) = A T (γ)x(t, γ) µ(t)a T (γ)x (t, γ) + H(t, γ), [I + µ(t)a T (γ)]x (t, γ) = A T (γ)x(t, γ) + H(t, γ). Since the mtrix A(γ) i regreive, then A T (γ) i lo regreive, nd hence we infer tht tht i, X (t, γ) = [I + µ(t)a T (γ)] 1 A T (γ)x(t, γ) + [I + µ(t)a T (γ)] 1 H(t, γ) = A T (γ)x(t, γ) + [I + µ(t)a T (γ)] 1 H(t, γ), X (t, γ) = A T (γ)x(t, γ) + [I + µ(t)a T (γ)] 1 H(t, γ). Now, uing the Theorem 4.3 nd the propertie of the uncertin trnition mtrix, we obtin tht tht i (18). X(t, γ) = U A T (t, )X 0 (γ) + = U A T (t, )X 0 (γ) + = U A T (t, )X 0 (γ) + = U A T (t, )X 0 (γ) + U A T (t, σ())[i + µ(t)a T (γ)] 1 H(, γ) U T A (t, σ())[i + µ(t)a T (γ)] 1 H(, γ) [I + µ(t)a(γ)] 1 U A (σ(), t)} T H(, γ) U A (, t)h(, γ), Corollry 4. 2 Let X 0 : Γ R be bounded uncertin vrible. If A : Γ M m (R) i regreive nd bounded uncertin mtrix, then the unique olution of the following uncertin initil vlue problem X (t, γ) = A(γ)X σ (t, γ), t [, ) T, γ Γ, i given by X(, γ) = X 0 (γ), X(t, γ) = U A T (t, )X 0 (γ), t [, ) T, γ Γ.

12 28 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle Exmple 4. 1 Let u conider Γ = (0, 1), L the σ-lgebr of ll Borel ubet of Γ, M the uncertin meure on Γ defined in Exmple 2.1, nd the following initil vlue problem X (t, γ) = γx(t, γ) + e γ (t, 0), t [0, ) T, γ Γ, (19) X(0, γ) = γ. Then, by the Theorem 4.1 nd 4.3, the initil vlue problem (19) h unique olution on [0, ) T, given by tht i, Next, conider two prticulr ce. X(t, γ) = γe γ (t, 0) + 0 e γ (t, σ())e γ (, 0), ( ) 1 X(t, γ) = e γ (t, 0) γ µ()γ, t [0, ) T. If T = R, then µ(t) = 0 for ll t N, nd e γ (t, 0) = e γt. Moreover, in thi ce we hve µ()γ = 0 d = t. It follow tht the initil vlue problem X (t, γ) = γx(t, γ) + e γt, t [0, ) X(0, γ) = γ, h the olution X(t, γ) = (γ + t)e γt, t [0, ). If T = N, then µ(n) = 1 for ll n N, nd e γ (n, 0) = (1 + γ) n. Moreover, in thi ce we hve µ()γ = [0,n) γ = n 1 + γ. It follow tht the difference initil vlue problem Xn+1 (γ) = (1 + γ)x n (γ) + (1 + γ) n, n N X 0 (γ) = γ, h the olution X n (γ) = (γ + n 1 + γ )(1 + γ)n, n N. Exmple 4. 2 Let u conider Γ = (0, 1),L the σ-lgebr of ll Borel ubet of Γ, M the uncertin meure on Γ defined in Exmple 2.1, nd the following initil vlue problem X (t, γ) = γx σ (t, γ) + e γ (t, ), t [0, ) T, γ Γ, (20) X(0, γ) = γ. The initil vlue problem (20) h unique olution on [, ) T, given by tht i, X(t, γ) = γe γ (t, ) + 0 e γ (t, )e γ (, 0), X(t, γ) = (γ + t) e γ (t, 0), t [0, ) T, γ (0, 1). If T = R, then µ(t) = 0 for ll t R, nd e γ (t, 0) = e γt. It follow tht X(t, γ) = (γ + t) = e γt, t [0, ) T, γ (0, 1).

13 Journl of Uncertin Sytem, Vol.9, No.1, pp.17-30, If T = hn with h > 0, then µ(t) = h for ll t hn, nd e γ (t, 0) = (1 + γh) t/h. It follow tht the h-difference initil vlue problem Xt+h (γ) = 1 1+γh X t(γ) + h(1 + γh) t/h 1, t hn X 0 (γ) = γ, h the unique olution X t (γ) = (γ + t) (1 + γh) t/h, t hn. If T = 2 N, then µ(t) = t for ll t 2 N, nd e γ (t, 0) = initil vlue problem Xt (γ) = (1 + γt)x 2t (γ) t X 1 (γ) = γ, h the unique olution X t (γ) = (γ + t) [1,t) [1,t) [0,t) (1 + γ) 1, t 2 N. (1 + γ) 1. It follow tht the 2-difference (1 + γ) 1, t 2 N Exmple 4. 3 Let u conider Γ = (0, 1), L the σ lgebr of ll Borel ubet of Γ, M the uncertin meure on Γ defined in Exmple 2.1, nd the following initil vlue problem X (t, γ) = [ 1 γ 0 1 ] [ γ X(t, γ), X(0, γ) = 1 ], t [0, ) T, (21) [ ] 1 γ where 1 µ(t) 0 for t [0, ) T. The mtrix A = h the eigenvlue λ = 1, λ 2 = 1 with the [ ] [ ] 1 1 correponding eigenvector v 1 =, v 0 1 =, repectively. Then Ψ A (t, γ) = γ 2 [ e 1 (t, 0) e 1 (t, 0) 0 2 γ e 1(t, 0) ] [, Ψ 1 A (, γ) = e1 (, 0) 0 nd therefore, the uncertin trnition mtrix for (21) i given by [ γ e 1 (t, ) U A (t, ) = 2 e 1(t, ) γ 2 e 1(t, ) 0 e 1 (t, ) It follow tht the olution of the initil vlue problem (21) i given by X(t, γ) = U A (t, 0)X(0, γ) = [ 3γ 2 e 1(t, 0) γ 2 e 1(t, 0) e 1 (t, 0) ]. γ 2 e ] 1(, 0) γ 2 e 1(, 0) ] t [0, ) T, γ (0, 1). 5 Concluion The theory of dynmicl ytem with uncertin prmeter cn be uccefully pplied to modeling mny rel-world phenomen uch in biology to tudy the popultion growth, in epecilly of microorgnim uch bcteri, in phyicl ytem where heterogeneou micro-cle tructure re preent, in the propgtion of electromgnetic wve through dielectric mteril with vribility in the relxtion time nd o on. Some of thee ppliction will be the ubject of future work. Acknowledgment Thi work w upported by Higher Eduction Commiion of Pkitn.

14 30 U.A. Hhmi et l.: Uncertin Dynmic Sytem on Time Scle Reference [1] Agrwl, R.P., nd M. Bohner, Bic clculu on time cle nd ome of it ppliction, Reult Mth, vol.35, pp.3 22, [2] Agrwl, R.P., Bohner., M., O Regn, D., nd A. Peteron, Dynmic eqution on time cle: urvey, Journl of Computtionl nd Applied Mthemtic, vol.141, no.1-2, pp.1 26, [3] Ahlbrndt, C.D., nd C. Morin, Prtil differentil eqution on time cle, Journl of Computtionl nd Applied Mthemtic, vol.141, pp.35 55, [4] Bohner, M., nd A. Peteron, Dynmic Eqution on Time Scle: An Introduction with Appliction, Birkhuer, Boton, [5] Bohner, M., nd A. Peteron, Advnce in Dynmic Eqution on Time Scle, Birkhuer, Boton, [6] Chen, X., nd B. Liu, Exitence nd uniquene theorem for uncertin differentil eqution, Fuzzy Optimiztion nd Deciion Mking, vol.9, no.1, pp.69 81, [7] Go, X., Some propertie of continuou uncertin meure, Interntionl Journl of Uncertinty, Fuzzine nd Knowledge-Bed Sytem, vol.17, no.3, pp , [8] Gueinov, G.S., Integrtion on time cle, Journl of Mthemticl Anlyi nd Appliction, vol.285, pp , [9] Gueinov, G.S., nd B. Kymkln, Bic of Riemnn delt nd nbl integrtion on time cle, Journl of Differentil Eqution nd Appliction, vol.8, pp , [10] Hilger, S., Ein Mkettenklkl mit Anwendung uf Zentrummnnigfltigkeiten, Ph.D. Thei, Univerität Würzburg, [11] Hilger, S., Anlyi on meure chin- unified pproch to continuou nd dicrete clculu, Reult Mth, vol.18, pp.18 56, [12] Hoffcker, J., Bic prtil dynmic eqution on time cle, Journl of Difference Eqution nd Appliction, vol.8, no.4, pp , [13] Jckon, B., Prtil dynmic eqution on time cle, Journl of Computtionl nd Applied Mthemtic, vol.186, pp , [14] Liu, B., Uncertinty Theory, 2nd Edition, Springer-Verlg, Berlin, [15] Liu, B., Fuzzy proce, hybrid proce nd uncertin proce, Journl of Uncertin Sytem, vol.2, no.1, pp.3 16, [16] Liu, B., Some reerch problem in uncertin theory, Journl of Uncertin Sytem, vol.2, no.4, pp.3 10, [17] Lungn, C., nd V. Lupulecu, Rndom dynmicl ytem on time cle, Electronic Journl of Differentil Eqution, vol.2012, no.86, pp.1 14, [18] Mlinowki, M.T., On rndom fuzzy differentil eqution, Fuzzy Set nd Sytem, vol.160, pp , [19] Mlinowki, M.T., Exitence theorem for olution to rndom fuzzy differentil eqution, Nonliner Anlyi, vol.73, pp , [20] Sngl, S., Stochtic Differentil Eqution, Ph.D. Dierttion, Miouri Univerity of Science nd Technology, [21] Tidell, C.C., nd A.H. Zidi, Succeive pproximtion to olution of dynmic eqution on time cle, Communiction on Applied Nonliner Anlyi, vol.16, no.1, pp.61 87, [22] You, C., Some convergence theorem of uncertin equence, Mthemticl nd Computer Modelling, vol.49, no.3-4, pp , 2009.