Perron s theorem for q delay difference equations

Transcription

1 Applied Mthemtic & Informtion Science 5(1) (2011), An Interntionl Journl c 2011 NSP Perron theorem for q dely difference eqution J. O. Alzbut 1 nd T. Abdeljwd 2 1 Deprtment of Mthemtic nd Phyicl Science, Prince Sultn Univerity P.O.Box Riydh 11586, Sudi Arbi Emil Addre: jlzbut@pu.edu. 2 Deprtment of Mthemtic nd Computer Science, Çnky Univerity Ankr, Turkey Emil Addre: thbet@cnky.edu.tr Received Oct. 27, 2009; Revied Feb. 15, 2010; Accepted Mr. 4, 2010 In thi pper, we prove tht if liner q dely difference eqution tifie Perron condition then it trivil olution i uniformly ymptoticlly tble. Keyword: Perron, q clculu, q difference eqution, uniform ymptotic tbility. 1 Introduction Studie on liner q difference eqution hve trted t the beginning of the lt century in intenive work by Jckon [1], Crmichel [2], Mon [3], Adm [4], Trjitzinky [5] nd ome other uthor uch Poincre, Picrd nd Rmnunjn. However, from the 1930 up to the beginning of the 1980, the theory of liner q difference eqution h lgged noticebly behind the iter theorie of liner difference nd differentil eqution. Since the 1980, n extenive nd omewht urpriing interet in the ubject reppered in mny re of mthemtic, phyic nd ppliction including minly new difference clculu nd orthogonl polynomil, q combintoric, q rithmetic, integrble ytem nd vritionl q clculu, ee the recent pper [6 11]. The ymptotic propertie of q difference eqution hve been rrely conidered in the literture. There re few recent reult deling with ubject like pectrl nlyi, ocilltion behvior of olution, fctoriztion method nd ymmetrie of thee type of eqution, we nme the pper [12 18]. The tbility behvior of q dely difference eqution, in prticulr, i our min concern in thi pper. We trt by mentioning few bckground detil tht erve to motivte the reult of thi pper.

2 It i well known in the theory of ordinry differentil eqution (ee eg. [19, pge 120]) tht if for every continuou function f(t) bounded on [0, ), the olution of the eqution x (t) = A(t)x(t) + f(t), t 0 tifying x(0) = 0 i bounded on [0, ), then the trivil olution of the correponding homogeneou eqution x (t) = A(t)x(t), t 0 i uniformly ymptoticlly tble. Lter on, thi reult, which i known Perron Theorem [20], w extended in [19, pge 371] to dely differentil eqution. Indeed, it w hown tht if for every continuou function f(t) bounded on [0, ), the olution of the eqution x (t) = A(t)x(t) + B(t)x(t τ) + f(t), t 0, τ > 0 tifying x(t) = 0 for t [ τ, 0], i bounded on [0, ), then the trivil olution of the eqution x (t) = A(t)x(t) + B(t)x(t τ), t 0 (1.1) i uniformly ymptoticlly tble. Perron theorem for impulive dely differentil eqution h been conidered in the pper [21]. A dicrete nlogue of the bove reult h been publihed in [22]. In prticulr, Perron theorem h been proved for eqution of form x(n) = A(n)x(n) + B(n + 1)x(n j + 1), n 0, j {2, 3, 4,...}, (1.2) where x(t) denote the forwrd difference x(t + 1) x(t). Recently, we proved thi reult for type of impulive dely difference eqution [23]. For more relted reult, ee the pper [24 26]. To the bet of uthor knowledge, however, there re few reult concerning tbility of q dely difference eqution [27, 28]. Motivted by thi, we contribute to the theory of q difference eqution by proving Perron theorem for type of q dely difference eqution of the form D q x(t) = A(t)x(t) + qb(qt)x(q α 0+1 t), t q Z, where q Z = {q i : i Z} with q > 1 nd α 0 N Adjoint eqution nd olution repreenttion We introduce ome preliminry nottion tht would help in undertnding lter nlyi. For the function f : q Z R, the expreion D q f(t) = f(qt) f(t) (q 1)t (2.1) i clled the q derivtive (or Jckon derivtive [29]) of function f. Together with the definition of q derivtive, rie nturlly tht of the q integrl of given function. In view of Definition 1.71 in [30, pge 26], one cn deignte the indefinite q-integrl by f(t) q t = F (t) + C, (2.2)

3 76 where F i pre ntiderivtive of f nd C i n rbitrry contnt. The definite q integrl turn out to be defined follow b β 1 q 1 b f(t) q t = (q 1) q i f(q i ) = (q 1) tf(t), (2.3) i=α where = q α nd b = q β. We note tht (2.3) i the q nlogue of formul (ii) of Theorem 1.79 in [30, pge 29]. We re now in poition to define the q nlogue of ome well known rule of clculu. The q derivtive of the compoition of function f nd g i given by t= D q (f g)(t) = (D q f)(g(t))d q g(t), (2.4) where g(t) = ct, c R. The q derivtive of the product of function f nd g i interpreted D q (fg)(t) = f(qt)d q g(t) + D q f(t)g(t). (2.5) The fundmentl theorem of clculu for q difference opertor turn out to be defined follow b D q f(t) q t = f(b) f(), (2.6) where, b q Z. By men of (2.4), one cn write the Newton Libniz formul for q difference opertor in thi form h(t) D q f(x) q x = f(h(t))d q h(t) f(k(t))d q k(t), (2.7) k(t) where h(t) = q α 0 t nd k(t) = qt. We hll prove Perron theorem for the q dely difference eqution D q x(t) = A(t)x(t) + qb(qt)x(q α0+1 t), t q Z, α 0 N, (2.8) where it i umed tht A, B : q Z R m m re bounded mtrice. Eqution (2.8) i being deignted to be the q verion of eqution (1.1) nd (1.2). By olution of (2.8), we men function x which i defined for ll t [q α0+1 t 0, ) q Z nd tifie (2.8) for t [t 0, ) q Z = {q i : i β}, β Z. It i ey to ee tht for ny given t 0 = q α1 nd initil condition of the form x(t) = ϕ(t), t [q α 0+1 t 0, t 0 ] q Z (2.9) (2.8) h unique olution x(t) which i defined for t [q α0+1 t 0, ) q Z nd tifie the initil condition (2.9).

4 We hll trt by contructing the djoint eqution of (2.8) with repect to function reemble the one obtined in [19, pge 359]. It turn out tht the q nlogue of thi function h the form < y(t), x(t) >= y T (t)x(t) + where T denote the trnpoition. Define the eqution q α 0 t qt 77 y T (α)b(α)x(q α 0 α) q α, (2.10) D q y(t) = A T (t)y(qt) q α 0 B T (q α 0 t)y(q α 0 t). (2.11) Lemm 2.1. Let x(t) be ny olution of (2.8) nd y(t) be ny olution of (2.11), then < y(t), x(t) >= c = contnt. (2.12) Proof. Clerly, it uffice to how tht D q < y(t), x(t) >= 0. Then [ α q 0 t D q < y(t), x(t) >= D q (y T (t)x(t)) + D q By virtue of reltion (2.5) nd (2.7), we hve qt ] y T (α)b(α)x(q α 0 α) q α. D q < y(t), x(t) > = y T (qt)d q x(t) + D q y T (t)x(t) + y T (q α 0 t)b(q α 0 t)x(t)d q q α 0 t y T (qt)b(qt)x(q α0+1 t)d q qt. In view of eqution (2.8) nd (2.11), we obtin D q < y(t), x(t) > = y T (qt) [ A(t)x(t) + qb(qt)x(q α0+1 t) ] [ y T (qt)a(t) + q α0 y T (q α0 t)b(q α0 t) ] x(t) + q α 0 y T (q α 0 t)b(q α 0 t)x(t) qy T (qt)b(qt)x(q α0+1 t) = 0. Thu, < y(t), x(t) >= c = contnt. The proof i complete. By virtue of Lemm 2.1, we my y tht eqution (2.11) i n djoint of (2.8). It i ey to verify lo tht the djoint of (2.11) i (2.8), tht i, they re mutully djoint of ech other. Definition 2.1. A mtrix olution X(t, ) of (2.8) tifying X(t, t) = I nd X(t, ) = 0 for t < i clled fundmentl mtrix of (2.8). Definition 2.2. A mtrix olution Y (t, ) of (2.11) tifying Y (t, t) = I nd Y (t, ) = 0 for t > i clled fundmentl mtrix of (2.11).

5 78 It i to be noted tht the contruction of function (2.10) i of pecil interet in itelf. We hll ue function (2.10) to derive the olution repreenttion of eqution (2.8) nd (2.11). In view of reltion (2.12), we oberve tht < y(t), x(t) >=< y(t 0 ), x(t 0 ) >. (2.13) Looking t function (2.10), if we replce x() by X(, t 0 ) nd y() by Y (, t) in (2.13) nd ue the propertie of the fundmentl mtrice, we get the identity X(t, t 0 ) = Y T (t 0, t). (2.14) Furthermore, replcing y() by Y (, t) in (2.13) nd uing identity (2.14) nd the propertie of the fundmentl mtrix Y (t, ), we hve the following reult. Lemm 2.2. Let X(t, ) be fundmentl mtrix of (2.8) nd t 0 = q β = q α 1 (1 α 1 β). If x(t) i olution of (2.8), then x(t) = X(t, t 0 )x(t 0 ) + q α 0 t0 qt 0 X(t, α)b(α)x(q α0 α) q α. (2.15) One cn lo obtin the olution repreenttion of eqution (2.11) in like mnner. Indeed, upon replcing x() by X(, t) in reltion (2.13), we cn derive the olution repreenttion of the djoint eqution (2.11). Nmely, Lemm 2.3. Let Y (t, ) i fundmentl mtrix of (2.11) nd t 0 = q β = q α 1 (1 α 1 β). If y(t) i olution of (2.11), then Conider the eqution y(t) = Y (t, t 0 )y(t 0 ) + q α 0 t0 qt 0 Y (t, q α 0 α)b T (α)y(α) q α. (2.16) D q x(t) = A(t)x(t) + qb(qt)x(q α 0+1 t) + f(t), t q Z, (2.17) where f : q Z R m. Then the olution repreenttion of (2.17) i given by the following reult. Lemm 2.4. Let X(t, ) be fundmentl mtrix of (2.8) nd t 0 = q β = q α1 (1 α 1 β). If x(t) i olution of (2.17), then x(t) = X(t, t 0 )x(t 0 )+ q α 0 t0 qt 0 X(t, α)b(α)x(q α0 α) q α+ t 0 X(t, qα)f(α) q α. (2.18) The proof of the bove ttement i trightforwrd nd cn be chieved by direct ubtitution nd by uing the reltion D q f(t, τ) q τ = D t qf(t, τ) q τ + f(qt, t).

6 79 3 Perron theorem Perron condition for eqution (2.8) i formulted follow. Definition 3.1. Eqution (2.8) i id to verify Perron condition if for every bounded function f(t) on [, ) q Z, the olution of (2.17) with x(t) = 0 for t [q α 0+1, ] i bounded on [, ) q Z. Lemm 3.1. If eqution (2.8) verifie Perron condition, then there exit contnt C uch tht where denote ny convenient mtrix norm. X(t, qα) q α < C for t, t q Z, (3.1) Proof. By virtue of Lemm 2.4, the olution of (2.17) tifying (2.9) with ϕ(t) = 0 h the form x(t) = X(t, qα)f(α) q α. Let B denote the et of ll bounded function f on [, ) q Z upplied by the norm f = up t [, )q f(t). Clerly, B i Bnch pce. Z For ech t [, ) q Z, define equence of liner opertor U t : B R m by U t (f) = X(t, qα)f(α) q α. By uing the etimte U t (f) X(t, qα) qα f, it follow tht the opertor U t re bounded. By virtue of Perron condition, we deduce tht for ech f B we cn find c f > 0 uch tht up U t [, )q Z t(f) c f. Hence, by uing the Bnch Steinhu Theorem, there exit contnt L > 0 uch tht up U t (f) L f, for ll f B. (3.2) t For fixed t [, ) q Z, let x rk (1 r, k m) be the element of the mtrix X(t, qα) where α < t, α q Z. Let e p denote the cnonicl bi hving the unity t the p-th plce nd zero otherwie. Let fα r be the element of B with it α component the vector V r of R m nd zero otherwie, where V r = m k=1 ignx rke k. The vector Xfα(α) r will hve it r th component equl to m k=1 x rk. From (3.2), we cn write where M 2 = L up r V r. Hence X(t, qα)f r α(α) q α M 2, m x rk (t, qα) q α M 2. r=1

7 80 Since the bove reltion i true for every r, we tke the ummtion m k=1 of both ide to deduce tht there exit C uch tht (3.1) hold. The proof i finihed. Lemm 3.2. If eqution (2.8) verifie Perron condition, then there exit contnt M > 0 uch tht X(t, ) < M for t. Proof. Hving tken into ccount tht Y T (r, t) tifie eqution (2.11), we integrte both ide with repect to r from to t ( ) to get Y T (t, t) Y T (, t) = It follow tht Y T (, t) = I + Y T (qr, t)a(r) q r Y T (qr, t)a(r) q r + q α 0 Chnging the vrible qu = q α 0 r, we obtin Y T (, t) = I + Y T (qr, t)a(r) q r + q q α 0 1 t q α 0 Y T (q α 0 r, t)b(q α 0 r) q r. q α 0 1 Y T (q α 0 r, t)b(q α 0 r) q r. Y T (qr, t)b(qr) q r. Uing the reltion Y T (, t) = X(t, ) nd tht Y (r, t) = 0 for r > t, we hve X(t, ) = I + X(t, qr)a(r) q r + q X(t, qr)b(qr) q r. q α 0 1 Tking the norm of both ide yield X(t, ) 1 + γ(1 + q) X(t, qr) q r, where γ = mx{up t A(t), up t B(t) }. Employing inequlity (3.1) reult in the deired concluion. Lemm 3.3. If eqution (2.8) verifie Perron condition, then it zero olution i uniformly tble. Proof. Let x(t; t 0, ϕ) be the olution of (2.8) tifying (2.9). In view of Lemm 2.2, the olution h the form x(t; t 0, ϕ) = X(t, t 0 )x(t 0 ) + Chnging the vrible u = q α 0, we get x(t; t 0, ϕ) = X(t, t 0 )x(t 0 ) + q α 0 τ 1 (t 0) σ(t 0 ) 0 X(t, )B()x(q α ) q, t t 0. q α+1 t 0 X(t, q α 0 )B(q α 0 )ϕ() q, t t 0.

8 81 By virtue of Lemm 3.2, we obtin x(t; t 0, ϕ) M 1 ϕ 0, where M 1 = M(1 + γα 0 q α 0 ) nd ϕ 0 = up t [q α+1 t 0,t 0 ] x(t). Thu, the trivil olution i uniformly tble. Theorem 3.1. If eqution (2.8) verifie Perron condition, then it zero olution i uniformly ymptoticlly tble. Proof. In view of Lemm 3.3, one cn deduce tht it remin to prove tht uniformly with repect to t 0 nd ϕ. For our purpoe, let λ t 0, then the olution h the form x(t; t 0, ϕ) = X(t, λ)x(λ; t 0, ϕ) + lim x(t; t 0, ϕ) = 0 (3.3) t λ q α+1 λ Integrting both ide with repect to λ from t 0 to t, we hve or t 0 x(t; t 0, ϕ) q λ = (n n 0 1)x(t; t 0, ϕ) = + + t 0 X(t, λ)x(λ; t 0, ϕ) q λ λ t 0 n 1 q α+1 λ X(t, q α0 )B(q α0 )x(; t 0, ϕ) q. X(t, q α0 )B(q α0 )x(; t 0, ϕ) q q λ m 0=n 0 X(q n, q m0 )x(q m0 ; q n0, ϕ) n 1 m 0 1 m 0=n 0 k=m 0 α 0 +1 X(q n, q k+α0 )B(q k+α0 )x(q k ; q n0, ϕ), where t = q n, t 0 = q n0, λ = q m0 nd = q k. Interchnging the order of ummtion to get (n n 0 1)x(q n ; q n 0, ϕ) = n 1 m 0=n 0 X(q n, q m0 )x(q m0 ; q n0, ϕ) n 0 1 k=n 0 α 0 +1 n α 0 k+α 0 1 k=n 0 m 0 =k+1 n 2 k+α 0 1 k=n α 0+1 m 0=k+1 m 0 =n 0 X(q n, q k+α0 )B(q k+α0 )x(q k ; q n0, ϕ) X(q n, q k+α0 )B(q k+α0 )x(q k ; q n0, ϕ) n 1 X(q n, q k+α 0 )B(q k+α 0 )x(q k ; q n 0, ϕ).

9 82 Uing tht X(, t) = 0 for < t, the lt term of the bove eqution vnihe. Tking the norm for both ide nd uing Lemm 3.1, Lemm 3.2 nd Lemm 3.3, we obtin (n n 0 1) x(q n ; q n0, ϕ) M 1 C ϕ 0 + γ ϕ 0 Mα 2 0M 1 q α0 n 1 + γ ϕ 0 M 1 (α 0 1)q α0 k=n 0 +α 0 1 X(q n, q k+α0 ). It follow tht where (n n 0 1) x(q n ; q n 0, ϕ) M 2 ϕ 0, M 2 = M 1 C + γmm 1 α 2 0q α0 + γm 1 (α 0 1)Cq α0. Thu, x(q n ; q n 0, ϕ) M 2 (n n 0 1) ϕ 0. Letting t which tke plce n, we get the deired concluion (3.3). The olution i uniformly ymptoticlly tble. Reference [1] H. F. Jckon, q-difference eqution, Am. J. Mth. 32 (1910), [2] R. D. Crmichel, The generl theory of liner q difference eqution, Am. J. Mth. 34 (1912), [3] T. E. Mon, On propertie of the olution of liner q difference eqution with entire function coefficient, Am. J. Mth. 37 (1915), [4] C. R. Adm, On the liner ordinry q difference eqution, Am. Mth. Ser. II, 30 (1929), [5] W. J. Trjitzinky, Anlytic theory of liner q difference eqution, Act Mthemtic, (1933). [6] G. Bngerezko,The fctoriztion method for the Akey-Wilon polynomil, J. Comput. Appl. Mth. 107 (2) (1999), [7] G. Bngerezko, Vritionl q clculu, J. Mth. Anl. Appl. 289 (2) (2004), [8] G. Bngerezko, Vritionl clculu on q nonuniform lttice, J. Mth. Anl. Appl. 306 (1) (2005), [9] A. Fitouhi nd N. Bettibi, Appliction of the Mellin trnform in quntum clculu, J. Mth. Anl. Appl. 328 (1) (2007), [10] A. Fitouhi nd L. Dhoudi, On q Pley-Wiener theorem, J. Mth. Anl. Appl. 294 (1) (2004), [11] M. E. H. Imil nd M. Rhmn, Invere opertor, q frctionl Integrl nd q Bernoulli polynomil, J. Approx. Theory 114 (2) (2002),

10 [12] M. Bohner nd M. Adivr, Spectrum nd principl vector of q difference eqution, Indin J. Mth. 48 (1) (2006), [13] M. Bohner nd M. Adivr, Spectrl nlyi of q difference eqution with pectrl ingulritie, Mth. Comput. Modelling 43 (7 8) (2006), [14] M. Bohner nd M. Unl, Kneer theorem in q clculu, J. Phy. A: Mth. Gen. 38 (2005), [15] G. Bngerezko, The fctoriztion method for the generl econd order q difference eqution nd the Lguerre Hn polynomil on the generl q lttice, J. Phy. A: Mth. Gen. 36 (2003), [16] J. P. Bézivin, Sur le éqution functionnelle ux q différence, Aequtone Mth. 43 (1993), [17] D. Levi, J. Negro nd M. A. del Olmo, Dicrete q derivtive nd ymmetrie of q difference eqution, J. Phy. A:Mth. Gen. 37 (2004), [18] K. Victor nd C. Pokmn, Quntum clculu. Univeritext. Springer-Verlg, New York, [19] A. Hlny, Differentil Eqution: Stbility, Ocilltion, Time Lg, Acdemic Pre Inc., [20] O. Perron, Die Stbilittfrge bei Differentilgleichungen, Mth. Z. 32 (1930), [21] M. U. Akhmet, J. Alzbut nd A. Zfer, Perron theorem for liner impulive differentil eqution with ditributed dely, J. Comput. Appl. Mth. 193 (1) (2006), [22] J. O. Alzbut nd T. Abdeljwd, Perron Type Criterion for Liner Difference Eqution with Ditributed Dely, Dicrete Dyn. Nt. Soc., doi:1155/ [23] J. O. Alzbut nd T. Abdeljwd, Aymptotic behvior for cl of liner impulive difference eqution: Perron like theorem, Dyn. Syt. Appl., ubmitted. [24] A. Anokhin, L. Bereznky nd E. Brvermn, Exponentil tbility of liner dely impulive differentil eqution, J. Mth. Anl. Appl. 193 (1995), [25] V. A. Tyhkevich, A perturbtion-ccumultion problem for liner differentil eqution with time lg, Differ. Equ. 14 (1978), [26] N. V. Azbelev, L. M. Bereznkii, P. M. Simonov nd A. V. Chitykov, Stbility of liner ytem with time lg, Differ. Equ. 23 (1987), [27] A. Murt nd R. Youef, Stbility nd periodicity in dynmic dely eqution, Comput. Mth. Appl. 58 (2) (2009), [28] D. R. Anderon, R. J. Krueger nd A. C. Peteron, Dely dynmic eqution with tbility, Adv. Difference Equ. Volume 2006, Art. ID 94051, [29] G. Bngerezko, An Introduction to q difference eqution: http// [30] M. Bohner nd A. Peteron, Dynmic Eqution on Time Scle: An Introduction with Appliction, Boton MA, Birkhuer,

11 84 Jehd i Pletinin. He w born in 1972 in Jeddh, Sudi Arbi. He h received hi Ph.D. degree in 2004 from Middle Et Technicl Univerity in Ankr, Turkey. He begn hi creer prt time teching itnt in Deprtment of Mthemtic t Middle Et Technicl Univerity. He erved lecturer nd n itnt profeor of pplied mthemtic in Çnky Univerity during the period Currently, he i n ocite profeor of pplied mthemtic t Prince Sultn Univerity in Riydh, Sudi Arbi. Dr. Jehd reerch interet re focued on tudying the qulittive propertie of olution (ocilltion, periodicity, tbility nd controllbility) of differentil eqution, difference eqution nd eqution involving impule, dely nd frctionl derivtive. Biologicl model which re governed by uch type of eqution re lo mong hi field. Thbet Abdeljwd i Pletinin. He w born in 1971 in Wet Bnk where he h grown up until he finihed hi high chool in He h received hi Ph.D. degree in 2000 from Middle Et Technicl Univerity in Ankr, Turkey. He h been working n itnt profeor in Deprtment of Mthemtic nd Computer Science t Çnky ince Dr. Thbet h different reerch field of interet: functionl nlyi; topology nd fixed point theory; frctionl dynmicl ytem; vritionl clculu with dely nd frctionl vritionl clculu nd tbility of dynmicl ytem.