Electronic Journal of Differential Equation, Vol. 21(21, No. 7, pp. 1 5. ISSN: 172-6691. URL: http://ejde.math.wt.edu or http://ejde.math.unt.edu ftp ejde.math.wt.edu (login: ftp A THEOREM OF ROLEWICZ S TYPE FOR MEASURABLE EVOLUTION FAMILIES IN BANACH SPACES CONSTANTIN BUŞE & SEVER S. DRAGOMIR Abtract. Let ϕ be a poitive and non-decreaing function defined on the real half-line and U be a trongly meaurable, exponentially bounded evolution family of bounded linear operator acting on a Banach pace and atifing a certain meaurability condition a in Theorem 1 below. We prove that if ϕ and U atify a certain integral condition (ee the relation 1 from Theorem 1 below then U i uniformly exponentially table. For ϕ continuou and U trongly continuou and exponentially bounded, thi reult i due to Rolewicz. The proof ue the relatively recent technique involving evolution emigroup theory. Let X be a real or complex Banach pace and L (X the Banach algebra of all linear and bounded operator on X. Let T = {T (t : t } L (X be a trongly ln( T (t continuou emigroup on X and ω (T = lim t t be it growth bound. The Datko-Pazy theorem ([4], [1] tate that ω (T < if and only if for all x X the map t T (t x belong to L p (R + for ome 1 p <. Let ϕ : R + R + be a non-decreaing function uch that ϕ(t > for all t >. If for each x X, x 1 the map t ϕ( T (tx belong to L 1 (R + then T i exponentially table, i.e. ω (T i negative. Thi later reult i due to J. van Neerven [8, Theorem 3.2.2.]. The Datko-Pazy Theorem follow from thi by taking ϕ(t = t p (t. Moreover, it i eaily to ee that the above Neerven reult remain true if we replace the trongly continuity aumption about T with trongly meaurability and exponentially boundedne aumption about T. A family U = {U (t, : t } L (X i called an evolution family of bounded linear operator on X if U (t, t = I (the identity operator on X and U (t, τ U (τ, = U (t, for all t τ. Such a family i aid to be trongly continuou if for every x X, the map (t, U (t, x : {(t, : t } X. (1 are continuou, and exponentially bounded if there are ω > and K ω > uch that U (t, K ω e ω(t for all t. (2 If T = {T (t : t } L (X i a trongly continuou emigroup on X, then the family {U (t, : t } given by U (t, = T (t i a trongly continuou 2 Mathematic Subject Claification. 47A3, 93D5, 35B35, 35B4, 46A3. Key word and phrae. Evolution family of bounded linear operator, evolution operator emigroup, Rolewicz theorem, exponential tability. c 21 Southwet Texa State Univerity. Submitted September 2, 21. Publihed November 23, 21. 1
2 CONSTANTIN BUŞE & SEVER S. DRAGOMIR EJDE 21/7 and exponentially bounded evolution family on X. Converely, if U i a trongly continuou evolution family on X and U (t, = U (t, for all t then the family T = {T (t : t } i a trongly continuou emigroup on X. For more detail about the trongly continuou emigroup and other reference we refer to [1], [7]. The Datko-Pazy theorem can be alo obtained from the following reult given by S. Rolewicz ([13], [14]. Let ϕ : R + R + be a continuou and nondecreaing function uch that ϕ ( = and ϕ (t > for all t >. If U = {U (t, : t } L (X i a trongly continuou and exponentially bounded evolution family on the Banach pace X uch that up ϕ ( U (t, x dt = M ϕ <, for all x X, x 1, (3 then U i uniformly exponentially table, that i (2 hold with ome ω <. A horter proof of the Rolewicz theorem wa given by Q. Zheng [18] who removed the continuity aumption about ϕ. Other proof (the emigroup cae of Rolewicz theorem were offered by W. Littman [5] and J. van Neervan [8, pp. 81-82]. Some related reult have been obtained by K.M. Przy luki [12], G. Wei [16] and J. Zabczyk [17]. In a very recent paper R. Schnaubelt give a nice proof of a nonautonomou verion of Datko Theorem uing evolution emigroup (ee [15, Theorem 5.4]. Alo a very general Datko-Pazy type reult in the autonomou cae can be found in [9]. A family U = {U(t, : t } L(X i called trongly meaurable if for all x X the map given in (1 are meaurable. In thi note we prove the following: Theorem 1. Let ϕ : R + R + be a nondecreaing function uch that ϕ (t > for all t > and U = {U (t, : t } L (X be a trongly meaurable and exponentially bounded evolution family of operator on the Banach pace X. If U atifie the condition: (i there exit M ϕ > uch that ξ ϕ( U(t, ξx dt M ϕ <, x X, x 1, ξ, (4 (ii for all f L 1 (R +, X the map t U(, tf( : R + L 1 (R +, X (5 are meaurable, then U i uniformly exponentially table. Firtly we prove the following Lemma which i eentially contained in [1, Theorem 2.1]. Lemma 1. Let U be a trongly continuou and exponentially bounded evolution family of operator on X uch that up Then U i uniformly bounded, that i, ϕ ( U (t, x dt = M ϕ (x <, for all x X (6 up U (t, ξ = C <. t ξ
EJDE 21/7 ROLEWICZ S THEOREM 3 Proof of Lemma 1. Let x X and N (x be a poitive integer uch that M ϕ (x < N (x and let, t + N. For each τ [t N, t], we have e ωn 1 [t N,t] (u U (t, x e ω(t τ 1 [t N,t] (u U (t, τ U (τ, x (7 K ω U (u, x, for all u. Here K ω and ω are a in (2. From (6 follow that ϕ ( =. Then from (7 we obtain ( U (t, x ( 1[t N,t] (u U (t, x N (x ϕ K ω e ωn = ϕ K ω e ωn ϕ ( U (u, x du = M ϕ (x. du (8 We may aume that ϕ (1 = 1 (if not, we replace ϕ be ome multiple of itelf. Moreover, we may aume that ϕ i a trictly increaing map. Indeed if ϕ (1 = 1 and a := 1 ϕ (t dt, then the function given by t ϕ (u du, if t 1 ϕ (t = at at + 1 a, if t > 1 i trictly increaing and ϕ ϕ. Now ϕ can be replaced by ome multiple of ϕ. From (8 it follow that Now, it i eay to ee that U (t, K ω e ωn(x, for all x X. up U (t, ξ x 2K ω e ωn(x := C (x <, for all x X. (9 t ξ The aertion of Lemma 1 follow from (9 and the Uniform Boundedne Theorem. It i clear that (3 follow by (6, but in t clear if them are equivalent. In the proof of Theorem 1 we alo ue the following variant of Jenen inequality, ee e.g. [11, Theorem 3.1]. Lemma 2. Let Φ : R + R + be a convex function and w : R + R + be a locally integrable function uch that < w(tdt <. If wφ L 1 (R + and f : R + R + i uch that the map t Φ(f(t belong to L 1 (R + then ( w(tf(tdt w(tφ(f(tdt Φ. (1 w(tdt w(tdt Proof of Theorem1. Let U = {U (t, : t } be a evolution family of bounded linear operator on X. We conider the evolution emigroup aociated to U on L 1 (R +, X. Thi emigroup i defined by { U (, t f ( t, if t (T (t f ( :=, if t, (11 for all f L 1 (R +, X. Firtly we will prove that T (t act on L 1 (R +, X for each t. Indeed, if f n are imple function and f n converge punctually almot everywhere to f on R + when n, then uing the meaurability of the function
4 CONSTANTIN BUŞE & SEVER S. DRAGOMIR EJDE 21/7 given in (1 it follow that for all n N the map T(tf n are meaurable. From (4 and Lemma 1 it follow that (T(tf n ( (Tf( K f n ( t f( t a n almot everywhere for [t,, that i, the function in (11 i meaurable. On the other hand (Tf( d = t U(, tf( t dt C f L1 (R +,X <, i.e. the function T(tf belong to L 1 (R +, X. The function defined in (5 are meaurable for each f L 1 (R +, X, hence the map t T(tf : R + L 1 (R +, X (12 are alo meaurable. We do not know at thi tage if the meaurability of the function in (12 can be obtained uing only the meaurability of function in (1. We may uppoe that ϕ(1 = 1 (if not, we replae ϕ by ome multiple of itelf. The function t Φ(t := t ϕ(udu : R + R + i convex, continuou on (, and trictly increaing. Moreover Φ (t ϕ (t for all t [, 1]. Without lo of generality we may aume that up T (t 1. t Let f C c ((,, X, the pace of all continuou, X-valued function defined on R + with compact upport in (,, uch that f := up{ f(t : t (, } 1. Uing Lemma 2 (in inequality (1 we replce w( by exp 1 and the Fubini Theorem it follow that ( Φ T (t exp 1 f L1 dt (R +,X ( = Φ e ( t U (, t f ( t d dt t ( = Φ e ξ U (t + ξ, ξ f (ξ dξ dt ( e ξ Φ ( U (t + ξ, ξ f (ξ dt dξ ( e ξ ϕ ( U (t + ξ, ξ f (ξ dt dξ M ϕ e ξ f (ξ dξ M ϕ <. Let g L 1 (R +, X with g L 1 (R +,X 1 and f n C c ((,, X uch that f n 1 and exp 1 f n g in L 1 (R +, X. Let (f nk be a ubequence of (f n uch that exp 1 f nk converge at g, punctually almot everywhere for t R +, when
EJDE 21/7 ROLEWICZ S THEOREM 5 k. Uing the above etimate with f replaced by f nk, and the Dominated Convergence Theorem, it follow that Φ( T(tg L 1 (R +,X dt M ϕ <. The aertion of Theorem 1, follow now, uing a variant of Neerven reult (ee the begining of our note. We recall that U i uniformly exponentially table if and only if ω (T i negative [3, Theorem 2.2]. See alo [2], [6] and the reference therein for more detail about the evolution emigroup on half line and their connection with aymptotic behaviour of evolution familie of bounded linear operator acting on Banach pace. Acknowledgement. The author would like to thank the anonymou referee for hi/her valuable uggetion and for pointing out reference [15] and [9]. Reference [1] C. Buşe and S. S. Dragomir, A Theorem of Rolewicz type in Solid Function Space, Glagow Mathematical Journal, to appear. [2] C. Chicone and Y. Latuhkin, Evolution Semigroup in Dynamical Sytem and Differential Equation, Mathematical Survey and Monograph, Vol. 7, Amer. Math. Soc., Providence, RI, 1999. [3] S. Clark, Y. Latuhkin, S. Montgomery-Smith and T. Randolph, Stability radiu and internal veru external tability in Banach pace: An evolution emigroup approach, SIAM Journal of Control and Optim., 38(6 (2, 1757-1793. [4] R. Datko, Extending a theorem of A.M. Liapanov to Hilbert pace, J. Math. Anal. Appl., 32 (197, 61-616. [5] W. Littman, A generaliation of a theorem of Datko and Pazy, Lect. Note in Control and Inform. Sci., 13, Springer Verlag (1989, 318-323. [6] N.V. Minh, F. Räbiger and R. Schnaubelt, Exponential tability, exponential expanivene, and exponential dichotomy of evolution equation on the half-line, Integral Eqn. Oper. Theory, 3R (1998, 332-353. [7] R. Nagel (ed, One Parameter Semigroup of Poitive Operator, Lecture Note in Math., no. 1184, Springer-Verlag, Berlin, 1984. [8] J.M.A.M. van Neerven, The Aymptotic Behaviour of Semigroup of Linear Operator, Birkhäuer Verlag Bael (1996. [9] J. van Neerven, Lower emicontinuity and the theorem of Datko and Pazy, Integral Eqn. Oper. Theory, to appear. [1] A. Pazy, Semigroup of Linear Operator and Application to Partial Differential Equation, Springer Verlag, 1983. [11] J. E. PecǎriĆ, F. Prochan, & Y. L. Tong, Convex Function, Partial Ordering and Statitical Application, Academic Pre (1992. [12] K.M. Przy Luki, On a dicrete time verion of a problem of A.J. Pritchard and J. Zabczyk, Proc. Roy. Soc. Edinburgh, Sect. A, 11 (1985, 159-161. [13] S. Rolewicz, On uniform N equitability, J. Math. Anal. Appl., 115 (1886 434-441. [14] S. Rolewicz, Functional Analyi and Control Theory, D. Riedal and PWN-Polih Scientific Publiher, Dordrecht-Warzawa, 1985. [15] R. Schnaubelt, Well-poedne and aymptotic behaviour of non-autonomou linear evolution equation, preprint, http://www.mathematik.uni.hale.de/report. [16] G. Wei, Weakly l p -table linear operator are power table, Int. J. Sytem Sci.,2(1989, 2323-2328. [17] A. Zabczyk, Remark on the control of dicrete-time ditributed parameter ytem, SIAM J. Control, 12 (1974, 731-735. [18] Q. Zheng, The exponential tability and the perturbation problem of linear evolution ytem in Banach pace, J. Sichuan Univ., 25 (1988, 41-411.
6 CONSTANTIN BUŞE & SEVER S. DRAGOMIR EJDE 21/7 Contantin Buşe Department of Mathematic, Wet Univerity of Timişoara Bd. V. Parvan 4, 19 Timişoara, România. E-mail addre: bue@tim1.math.uvt.ro Sever S. Dragomir School of Communication and Informatic, Victoria Univerity of Technology PO Box 14428, Melburne City MC 81,, Victoria, Autralia. E-mail addre: ever@matilda.vu.edu.au URL: http://rgmia.vu.edu.au/ssdragomirweb.html