Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 4(2), 68-71, February (2015) Res.J.Recent Sci. International Science Congress Association 68 On Uniform Exponential Stability of Self adjoint Evolution Family: By Weak Rolewicz Approach Akbar Zada, IhsanUllah Khan, Tahir Hussain and Nisar Ahmad Department of Mathematics University of Peshawar, Peshawar, PAKISTANAvailable online at: www.isca.in,www.isca.me Received 2nd January 2014, revised 20th March 2014, accepted 11th May 2014Abstract In this article we prove that ifP{P(,)}stst³³is a self adjoint and strongly q-periodic continuous evolution family of bounded linear operators acting on a complex or real Hilbert space then is uniformly exponentially stable if for each unit vectorxHthe integral(P(,0),)sxxdsis bounded, wheree)::0,RR++=¥®is a non-decreasing function such that(0) = 0 and(s) �0 for all()Î¥. Keywords: EvolutionFamily, uniform exponential stability, strong rolewicz condition. IntroductionIn 2012, Constantin Buse and Gul Rahmatproved a result for positive evolution family by Weak Rolewicz type approach. In fact they proved that if is a non-decreasing function andP{P(,)}stst³³is a positive and strongly q-periodic continuous evolution family of bounded linear operators acting on a complex or real Hilbert spacesatisfying(|P(,0),|)sxxds¥then the family is uniformly exponentially stable. They asked a question whether the above result is true for self adjoint q-periodic evolution families or not. We worked on that problem and conclude that the result holds true for self adjoint q-periodic evolution families. It will be better to describe the historical background of the study before writing our result. Datko brought forth one of the important result in the stability of strongly continuous semigroup which argues that a strongly continuous semigroupS{S()}of bounded linear operator sacting on complex or real Banach space is uniformly exponentially stable if and only if ||S()||sds¥. Pazy had a research on the results of Datko and furtherimproved his attempt by stating that a strongly continuous semigroup of bounded linear operators acting on real or complex Banach space is uniformly exponentially stable if and only if||S()||,sds¥for any. Working with similar problem Rolewicz generalizes the Pazy theorem a step ahead. He stated that if ||S()||,sds¥then the semigroupis uniformly exponentially stable. Later on special cases were proved by Zabcyzyk and Przyluski. Zhengand Littmanobtained the new proofs of Rolewicz from which they discard the condition of continuity on f . Letbe a Banach space andbe its dual space thenS{S()}is called weak stable for, if |S(),|.sxxds¥ (1)It is important to note that weak stability of a semigroup does not imply its uniform exponential stability, counter examples can be found in9-11. For further results on this topic we recommend12-13. Recently Constantin Buse and Gul Rahmattried to extend the result of (1) to evolution family by Weak Rolewicz type approach. Our aim is to improve the resultfor self adjoint q-periodic evolution families. In the first section of this article we will give some preliminaries and in second section we will present our main result. Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(2), 68-71, February (2015) Res.J.Recent Sci International Science Congress Association 69 Preliminaries We denote by RCandN the sets of real numbers, complex numbers and the non-negative integers respectively. () A s denotes the spectral radius of A and L ( X ) the space of all bounded linear operators acting on X . As usual, .,.  denotes the scalar product on a Hilbert space H . The norms in X , H , () LX , () LH will be denoted by the same symbol ||.|| . A family satisfying the following properties is called q-periodic(for some 1 q ³ ) strongly continuous evolution family. i. P(,)1 ss = , ii. P(,)(,)P(,) stPtrsr = , iii P(,)P(,) sqtqst ++= , iv The map (,)P(,):{(,):,} ststxststRwherestH ®Î³®is continuous for all st ³ . A family P is said to be exponentially bounded if their exists 0 vRandM γ such that () ||P(,)||. vst stMeforallst - £³ (2.1)The growth bound of exponentially bounded evolution family P is defined by () (P):inf{:0||P(,)||}. vstvwvRthereisMsuchthatstMe=γ£The family P is uniformly exponentially stable if (P)0 w .An evolution family P is called selfadjoint if each operator P(,) st with st ³ , is self adjoint. Here we will recall few lemmas from the paper of Constant in Buse and Gul Rahmat, without proofs so that the paper will be self-contained. Lemma 2.1 : If the spectral radius of () KLX Î is greater or equal to 1 then for all 0 e 1and any sequence ()0() nnbwithbasn ®®¥ and ||()|| nb ¥ 1 £ there exists aunit vector vX Î such that(1).|||||| bKvforallnN e -£Î , where X is a complex Banach space. Throughout this article ( n p ) will denote a sequence of non-negative real numbers such that nn qppforeverynN a+ ££-£Î and some positive real number a . Lemma 2.2 Let 0 P{P(,)} st st ³³ is a q-periodic ( 1 q ³ ) strongly continuous evolution family of bounded linear operators acting on a Banach space X. Suppose that ( n p ) is a sequence as defined before. If 0 P{P(,)} st st ³³ is not uniformly exponentially stable then there exists a positive constant C with the properties that for every sequence ( ) ()0||()||1 nnnkwithkasnandk ®®¥£ there exists a unit vector vX Î such that 10 ||||P(,0)||. nn CkpvforallnN £Î(3)An evolution family 0 P{P(,)} st st ³³ is said to satisfy the strong discrete Rolewicz condition if (||P(,0)||). px ¥ (4)Lemma2.3 Let 0 P{P(,)} st st ³³ is a strongly continuous q-periodic ( 1 q ³ ) evolution family acting on X . If the family P satisfies(||P(,0)||)px ¥ then it is uniformly exponentially stable, where f is an R-function. Results and Discussion Let 0 P{P(,)} st st ³³ is a strongly continuous evolution family of bounded linear operators acting on complex Hilbert space H . When P is self adjoint (.P(,)P(,)) ieststforeveryst * =³ then 2 P(2,2),P(,)P(,),||P(,)||. stxxststxxstx The following property of self adjoint operators is very important for our results. Let U and V are two self adjoint operators, then 222 |UV,|U,V,,. xyyyxxforallxyH (5)Before proving our result we recall Theorem 3.3 1 , without proof. Theorem 3.1Let f is an R-function and 0 P{P(,)} st st ³³ is an evolution family which is positive, strongly continuous and q-periodic ( 1 q ³ )acting on a complex Hilbert space . H If J(x)= (P(,),)||||1, soxxdsforallxHwithx ¥Î= then P is uniformly exponentially stable. The problem that was left open by Constantin Buse and Gul Rahmat was “whether the result holds true for self adjoint q-periodic evolution families”. We tried over that problem and after simple arrangement we got the result. The outcome which has proved simply can be seen in the following lines. Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(2), 68-71, February (2015) Res.J.Recent Sci International Science Congress Association 70 Theorem3.2 Let f is an R-function and 0 P{P(,)} ts ts ³³ is an evolution family which is self adjoint, strongly continuous and q-periodic ( 1 q ³ ) acting on complex Hilbert space . H If (P(,),)||||1 soxxdsforallxHwithx ¥Î= , then P is uniformly exponentially stable. Proof: As f can be considered a continuous function so applying the mean value theorem to the function (P(2,,) ssoxx f  on the interval [nq, (n+1)q], we find () n px in the same interval such that (6) 111 2 222222224(22) |P(,0),||P(,2())P(2(),0),| P(,2()),)P(2(),0),P(2(),0),,|P(,0),|P(2(),0),.nnnnnnnqwqwnnSetsnq sxyspxpxxy spxyypxxxMepxxxhencesxyMepxxx=+  Thus for any unit vector x in , H we have 122241 |P(,0),|P(2(),0),. nnqw sxxpxxx Me  Since f is a non-decreasing function, so we get that 2241 (|P((1),0),|)(P(2(),0),). qw nqxxpxxx Me ff  Taking summation on both sides 2 40201 (|P((1),0),|) (P(2(),0),).qwnqxxMepxxx +  (7)From inequality (3.2) we can write (P(2(),0),).pxxx¥Then (3.3) implies that 2401 (|P((1),0),|). qwnqxxMe ¥ (8)Let n+1=2mthen (3.4) can be written as (||P(,0)||).qwmqxMe¥Hence using Lemma 2.3 we conclude that P is uniformly exponentially stable. Conclusion The main objectives of this article were to give a positive answer to the question putted recently by Constant in Buse and Gul Rahmat. In fact they proved a result about the exponential stability and positive evolution family by weak Rolewicz approach, then they stated that” is it possible to prove the same result for self ad joint evolution family”. In this article we conclude that the same result is still true for self ad joint evolution family. References 1.Buse C. and Rahmat G., Weak Rolewicz theorem in Hilbert spaces, Electronic journal of Differential Equations, 218, 1-10 (2012)2.Datko R., Extending a theorem of A. M. Liapunov to Hilbert space, Journal of Math. Anal, Appl., 32, 610-616 (1970)3.Pazy A., On the Applicability ofLyapunov's Theorem in Hilbert Space, SIAM J. Math. Anal, , 291-294 (1972)4.Rolewicz S., On uniform N-equistability, J. Math. Anal. Appl., 115, 434-441 (1986)5.Zabczyk J., Remarks on the control of discrete time distributed parameter systems, SIAM, J. Control, 12,731-735 (1974)6.Przyluski K.M., On a discrete time version of a problem of A. J. Pritchard and J. Zabcyzyk, Proc. Roy. Soc. Edinburgh, Sect. A, 101, 159-161 (1985)7.Zheng Q., The exponential stability and the perturbation problem of linear evolution systems in Banach spaces, J. Sichuan Univ., 25,401-411 (1988) (in Chinese) 00(P(,),)(P(2,),)(P(2(),),)(P(2(),),), (P(2(),),)(P(,),). soxxdssoxxdsqpxoxxpxoxxhence pxoxxsoxxds ffff¥¥ Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(2), 68-71, February (2015) Res.J.Recent Sci International Science Congress Association 71 8.Littman W., A generalization of a theorem of Datko and Pazy, Lecture Notes in control and Inform. Sci., 130, Springer-Verlag, Berlin, 318-323 (1989)9.Greiner G., Voight J. and Wol M., On the spectral bound of the generator of semi groups of positive operators, J. Operator Theory, 5(2), 245-256 (1981)10.Huang F., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Di. Eq, , 43-56 (1983)11.Jan van Neerven, Straub B., and Weis L., On the asymptotic behaviour of a semi group of linear operators, Indag. Math. (N.S.), 4(6), 453-476 (1995)12.Buse C. and Dragomir S.S., A Theorem of Rolewicz's type on Solid Function Spaces, Glasgow Mathematical Journal, 44, 125-135 (2002)13.Buse C. and Dragomir S.S., A Rolewicz's type Theorem. An evolution semi group approach, Electronic Journal Differential Equations, 45, 1-5 (2001)14.Storozhuk K.V., On the Rolewicz theorem for evolution operators, Proc. Amer. Math. Soc., 135, 6, 1861-1863 (2007)