Characterizations of the (h, k, µ, ν) Trichotomy for Linear Time-Varying Systems

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1 Characterizations of the h, k, µ, ν) Trichotoy for Linear Tie-Varying Systes arxiv: v1 [ath.ds] 6 Dec 2015 Ioan-Lucian Popa, Traian Ceauşu, Mihail Megan Astract The present paper considers a concept of h, k, µ, ν) trichotoy for noninvertile linear tie-varying systes in Hilert spaces. This work provides a characterization for linear tievarying systes that adits a h, k, µ, ν) trichotoy in ters of two coupled systes having a h, µ, ν) dichotoy. 1 Introduction In the qualitative theory of difference equations the notions of dichotoy and trichotoy play a vital role. This fact is very well analysed in [1] and [6] where it is proved that the property of exponential trichotoy is necessary in the presence of ergodic solutions of linear differential and difference equations with ergodic perturations. Besides the classical concepts of unifor and nonunifor exponential trichotoy see [5] and reference therein for ore details), in [10] J. Lopez-Fener and M. Pinto extend the previous concept to the so called h, k) trichotoy using two sequences with positive ters. Recently, a new nonunifor concept of µ, ν) dichotoy is proposed in [3] with increasing sequences which go to infinity. These sequences, called growth rates, have een used to extend the classical concepts of exponential and polynoial dichotoy and trichotoy, oth for unifor and nonunifor approaches. We enhance the work developed in [2], [4], [13], [14], [15] and [16] and all included references for a detailed discussion considering this approach. One prole that lies within the ain interest in the asyptotic ehavior of the solution of difference equation is the characterization of the trichotoy in ters of dichotoies. In [5] S. Elaydi and K. Janglajew considered invertile) difference equations on the entire axis Z and 1

2 they proved that if the difference equation has an E-H) trichotoy that also has exponential dichotoy on Z and Z +. This result was extended y S. Matucci [11] to the concept of l p trichotoy. Siilarly, in [5] is proved that l p trichotoy on Z iplies l p dichotoy on Z + and on Z and l p dichotoy on Z iplies a trivial l p trichotoy. In [7] is proved that for alost periodic difference equations the notion of exponential trichotoy on Z iplies exponential dichotoy on Z. Tacking into consideration the previous three results, there naturally arises the question aout what happens when we consider noninvertile) difference equations only defined on Z +. This represents the ain goal of the present paper, to provide a characterizations for linear tie varying systes i.e. difference equations) that are defined only on Z + and not assued to e invertile. Thus, we consider an generalization fro h, k, µ, ν) dichotoy to h, k, µ, ν) trichotoy in discrete tie fro [16] and we prove that linear tie-varying LTV) systes that adit such a trichotoy can e characterized in ters of two coupled systes that adit a h,µ,ν) dichotoy in the sense of J. Lopez-Fener and M. Pinto [10], i.e. a concept that uses only sequences with positive ters. 2 Preliinaries In this section we introduce our notations and present soe definitions and useful details. We denoted y Z the set of real integers, Z + is the set of all n Z, n 0, and Z is the set of all n Z, n 0. We also denote y the set of all pairs of real integers,,n) with n 0. The nor on the Hilert space H and on BH) the Banach algera of H will e denoted y. The identity operator on H is denoted y I. We will e considering the discrete-tie linear tie-varying syste x n+1 = A n x n, n Z + A) where A n ) n Z+ BH) is a given sequence. If for every n Z + the sequence A n ) n Z+ is invertile, then the LTV syste A) is called reversile. The state transition atrix for the LTV syste A) is defined as A n A 1 A n if > n := I if = n. 1) It is ovious that the transition atrix satisfies the propagator property, i.e. A n Ap n = A p, for all,n),n,p) and every solution of the LTV syste A) satisfies x = A n x n, for all,n). 2

3 AsequenceP n ) n Z+ BH)iscalledaprojections sequenceifp n ) 2 = P n, for all n Z +. A projections sequence P n ) n Z+ with the property A n P n = P n+1 A n for every n Z + is called invariant for the syste A). As a consequence of the invariance property we get the following relation that A n P n = P A n, for all,n). An increasing sequence µ : Z + [1, ) is a growth rate sequence if µ0) = 1 and li µn) = +. n + Definition 1. The LTV syste A) adits a h, k, µ, ν) trichotoy if there exist invariant projections P i n ) n Z + BH), i {1,2,3} satisfying P 1 n +P 2 n +P 3 n = I, P i np j n = 0 2) for all n Z + and i,j {1,2,3}, i j and there exist growth rate sequences h n,, µ n, ν n and soe positive constants K > 0, a > 0, 0 and ε 0 such that ) a A n Pnx 1 hn K µ ε n Pnx, 1 3) h ) Pnx 2 K ν k A ε n Pnx, 2 4) ) a A n P3 n x K h µ ε n P3 nx, 5) P 3 nx K k h n ) A n P 3 nx, 6) for all,n), x H and the restriction of A n KerP i n : KerPi n KerP i is an isoorphis for all,n) and all i {2,3}. Moreover, the LTV syste A) adits a h,k,µ,ν) dichotoy if adits a h,k,µ,ν)-trichotoy with P 3 n = 0, for all n Z +. The notion of h, k, µ, ν) trichotoy is a natural generalization of the classical concepts of unifor nonunifor) concepts of exponential and polynoial trichotoy [9], [17]). The constants a and play the role of Lyapunov exponents while ε easures the nonunifority of trichotoies. In [16] is pointed out that aε < 0 can siplify the previous expressions i.e. 3)-6)), ut in this way we can see etter how the Lyapunov exponents are involved in this characterization. One can see that not all the definitions the sequences considered are assued to e growth rate sequences. For exaple in [10], J. López-Fenner and M. Pinto siply consider sequences with positive ters. In the sae line of reasoning, if we consider in previous definition h n,, µ n, ν n four sequences with positive ters we can denote this notion as FP h,k,µ,ν)-trichotoy and for P 3 n = 0 the notion of FP h,k,µ,ν)-dichotoy. 3

4 Also we can ention a particular case of previous notion that will e used in this paper, i.e. FP h,µ,ν) dichotoy otained for h n =. Reark 1. See, [15]) One can see that relation 2), the orthogonality property fro Definition 1 can e rewritten as follows: a) in ters of two projection sequences, i.e. there exists Q i n ) n Z + BH), i {1,2} such that Q 1 n Q2 n = Q2 n Q1 n = 0, for all n Z + 7) where Q 1 n = P 1 n and Q 2 n = P 2 n +P 3 n. ) in ters of four projection sequences, i.e. there exists four projection sequences R i n ) n Z + BH), i {1,2,3,4} such that R 1 n +R4 n = R2 n +R3 n = I, R1 n R2 n = R2 n R1 n = 0, R 3 nr 4 n = R 4 nr 3 n, for all n Z + 8) where R 1 n = P1 n, R2 n = P2 n, R3 n = P1 n +P3 n and R4 n = P2 n +P3 n. Reark 2. a) If we consider P 1 n = Q 1 n, P 2 n = Q 2 n, P 3 n = I Q 1 n Q 2 n, then P i n), i {1,2,3} are orthogonal projection sequences if and only if Q i n), i {1,2} are orthogonal projection sequences. ) If Pn i), i {1,2,3} are orthogonal projection sequences then Ri n ), i {1,2,3,4} defined y R 1 n = P 1 n, R 2 n = P 2 n, R 3 n = P 1 n+p 3 n and R 4 n = P 2 n+p 3 n are also orthogonal. Conversely, we have that P 1 n = R 1 n, P 2 n = R 2 n and P 3 n = R 3 nr 4 n, n Z +. 3 Main Results We associate to the LTV syste A) the syste x n+1 = B n x n, n Z + B) with B n = hn+1 h n ) a/2 +1 ) /2An. Following 1) we have that the state transition atrix associated to B) satisfies B n = h h n ) a/2 k ) /2 A n, for all,n). 9) 4

5 In addition, using the orthogonality property fro Definition 1, we have that P 1 n and P 2 n +P 3 n are also projections sequences which are invariant for the LTV syste B), i.e. and for all,n). B n P 1 n = P 1 nb n B n P 2 n +P 3 n) = P 2 +P 3 )B n Theore 1. If the LTV syste A) is h,k,µ,ν) trichotoic then the LTV syste B) is FP h,µ,ν) dichotoic, where h n = ha n. Proof. Take P 1 n = S 1 n. By 3) we have A n S 1 nx K hn h ) a µ ε n S 1 nx, for all,n) and x H. Using Reark 2 a) if follows that the orthogonality property allow us to consider the following Pythagoras equality for all n Z + and x H. Thus, we otain ) 2 Pn 2 +Pn)x 3 2 K 2 ν 2ε P 2 n +P3 n )x 2 = P 2 n x 2 + P 3 n x 2, k K 2 k Hence, for Pn 2 +P3 n = S2 n we have that PA 2 n x 2 +K 2 k ) 2 ν A 2ε n Pn 2 +Pn)x 3 2 S 2 n x K k ) An S2 n x, ) 2 ν 2ε P 3 A n x 2 for all,n) and all x H. We consider the sequence h n : Z + 0, ) defined y h n = ha n. Taking into account relation 9) we have that hn B n S1 n x K µ h ε n S1 n x S 2 nx K hn h B n S 2 nx for all,n) and x H. Further, one can easily see that B n KerS 2 n : KerS2 n KerS 2 is an isoorphis for all,n) and so it follows that LTV syste B) is FP h,µ,ν) dichotoic with projections P 1 n and P 2 n +P 3 n. This copletes the proof of the theore. 5

6 Now we associate to A) the LTV syste x n+1 = C n x n, n Z + C) ) a/2 ) /2An with C n = hn h n Ovious, the state transition atrix C n associated to the LTV syste C) checks ) a/2 ) /2 C n = hn A n, for all,n). 10) h k Also, one can see that P 2 n and P 1 n +P 3 n are projections sequences which are invariant for the syste C). Moreover, we have that LTV systes B) and C) are couplet together, i.e. ) a ) C n = hn B n, for all,n). h k Theore 2. If the LTV syste A) is h,k,µ,ν) trichotoic then the LTV syste C) is FP h,µ,ν) dichotoic, where h n = 1 h n. Proof. Proceeding as in the proof of Theore 1, we consider the projection sequences T 1 n = P 1 n +P 3 n and T 2 n = P 2 n. It follows fro Reark 2 a) and Pythagoras equality, i.e. P 1 n +P3 n )x 2 = P 1 n x 2 + P 3 n x 2 that for all,n) and x H we have A n T 1 nx K T 2 nx K k h h n ) a µ ε n T 1 nx ) A n T 2 nx. We consider the sequence h n : Z + 0, ) defined y h n = 1. Fro equation 10) if follows hn that hn CT n nx 1 K µ h ε n Tnx 1 and T 2 nx K hn h C n T 2 nx for all,n) and x H. In addition, C n KerT 2 n : KerT2 n KerT 2 is an isoorphis for all,n). This allows us to show that LTV syste C) adits a FP h,µ,ν) dichotoy with projection sequences P 2 n and P 1 n +P 3 n. 6

7 Reark 3. Using h, k, µ, ν) trichotoy we have that there exists the projection sequences S i n ) n Z + BH), T i n ) n Z + BH), i {1,2} satisfying the following properties S 1 n +S 2 n = I, T 1 n +T 2 n = I; S 1 ns 2 n = S 2 ns 1 n = 0, T 1 nt 2 n = T 2 nt 1 n = 0; S 1 nt 1 n = T 1 ns 1 n = S 1 n, S 2 nt 1 n = T 1 ns 2 n = S 2 n T 2 n = T 1 n S 1 n, T 2 ns 2 n = S 2 nt 2 n = T 2 n, T 2 ns 1 n = S 1 nt 2 n = 0. for all n Z +. It is ovious that if we consider Q 1 n = S 1 n and Q 2 n = T 2 n then we have that Q 1 n and Q 2 n are orthogonal. Reark 4. Using h, k, µ, ν) trichotoy property and Theores 1 and 2 we conclude that syste A) is reducile to the coupled systes B) and C) such that B n S 1 nx K hn h µ ε n S 1 nx 11) S 2 nx K hn h B n S 2 nx 12) hn C n T1 n x K µ h ε n T1 nx 13) hn Tn 2 x K ν h ε Cn T2 nx 14) and B n KerS 2 n : KerS2 n KerS 2, C n KerT 2 n : KerT2 n KerT 2 are isoorphiss for all,n) and x H. Theore 3. If LTV coupled systes B) and C) adits FP h,µ,ν) dichotoy respectively FP h,µ,ν) dichotoy then the LTV syste A) adits a h,k,µ,ν) trichotoy. Proof. Take P 1 n = S 1 n. By 11) we have that ) h A n a 1/2 P1 n x = n k n h a h k B n S1 n x h a n k n h a h a k K n k h a µ ε n Snx 1 ) a hn = K µ ε n P1 n x 7

8 for all,n) and x H. Fro 14), considering Pn 2 = Tn, 2 we otain ) k Pn 2 1/2 ) x K n h a 1/2 Cn T2 n x h a n k k = K n h a = K h a n k k ) A n P 2 nx h ν ε a n k n h a k A n Pnx 2 for all,n) and x H. Now, we consider P 3 n = T 1 ns 2 n. By Reark 3, 12) and 13) we otain ) h A n a 1/2 ) P3 n x = An T1 n S2 n x = k 1/2 C n T1 n S2 n x and, for Pn 3 = S2 n T1 n, we have h n h a n ) h a 1/2 ) k 1/2 ) k 1/2 ) h a n K n h a 1/2 h a n k µ ε n T1 n S2 n x ) a h = K µ ε n Pnx 3 ) h Pn 3 a 1/2 x = S2 n T1 n x K n k h a = K n k = K k h a k n ) An P3 n x h a k n Bn S2 n T1 n x h a k h a n k n A n P 3 nx for all,n) and x H. Moreover, A n KerP i n : KerPi n KerP i is an isoorphis for all,n) and all i {2,3}. This iplies that LTV syste A) is h,k,µ,ν) trichotoic, which concludes our proof. Coining Theores 1, 2 and 3 we have the following Theore 4. The LTV syste A) is h,k,µ,ν) trichotoic with projection sequences P i n), i {1,2,3} if and only if a) B) if FP h,µ,ν)-dichotoic with projection sequences Sn i ), i {1,2}; ) C) if FP h,µ,ν)-dichotoic with projection sequences T i n), i {1,2}. Reark 5. The theores fro this article include and generalize the case considered in [8]. For the continuous case of evolution operators we can refer to [12]. 8

9 4 Conclusion In this paper, we have considered the prole of h, kµ, ν) trichotoy for noninvertile) linear tie-varying systes in Hilert spaces. It is proved that noninvertile LTV systes defined on Z + that adit a h,k,µ,ν)-trichotoy can e decoposed into two coupled systes having h, µ, ν)-dichotoy. References [1] A.I. Alonso, J. Hong, R. Oaya, Exponential dichotoy and trichotoy for difference equations, Cop. Math. Appl., ), [2] M. G. Bauţia, M. Megan, I.-L. Popa, On h,k)-dichotoies for nonautonoous linear difference equations in Banach spaces, Int. J. Differ. Equ., Volue 2013, Article ID , 7 pages [3] A. J. G. Bento, C. M. Silva, Nonunifor µ, ν) dichotoies and local dynaics of difference equations, Nonlinear Anal., ), [4] X.Y. Chang, J.M. Zhang, J.H. Qin, Roustness of nonunifor µ, ν) dichotoies in Banach spaces, J. Math. Anal. Appl ) [5] S. Elaydi, K. Janglajew, Dichotoy and trichotoy of difference equations, J. Difference Equ. Appl ), no. 5-6, [6] J. Hong, R. Oaya, A. Sanz, Existence of a class of ergodic solutions iplies exponential trichotoy, Appl. Math. Lett ), [7] J. Hong, C. Nunez, The Alost Periodic Type Difference Equations, Mathl. Coput. Modelling Vol. 28, No ), pp [8] M. Lăpădat, On a,,c)-trichotoy and d,)-dichotoy of linear discrete-tie systes in Banach spaces, Int. J. Pure Appl. Math., Vol. 194, 32015), [9] Z. Li, X. Song, X. Yang. On Nonunifor Polynoial Trichotoy of Linear Discrete-Tie Systes in Banach Spaces, J. Appl. Math., vol. 2014, Article ID , 6 pages [10] J. López-Fenner, M. Pinto, h,k)-trichotoies and Asyptotics of Nonautonoous Difference Systes, Coputers Math. Applic., Vol. 33, No. 10, , [11] S. Matucci, The l p trichotoy for difference systes and applications, Arch. Math., Vol ), No. 5,

10 [12] M.I. Kovacs, M. Megan, C.L. Mihit, On h, k)-dichotoy and h, k)-trichotoy of Noninvertile Evolution Operators in Banach Spaces,An. Univ. Vest Tiiş. Ser. Mat.-Infor., 22014), [13] I. L. Popa, M. Megan, T.Ceauşu, Exponential dichotoies for linear discrete-tie systes in Banach spaces, Appl. Anal. Discrete Math., vol. 6, no ), pp [14] I.-L. Popa, M. Megan, T. Ceauşu,Nonunifor Exponential Dichotoies in Ters of Lyapunov Functions for Noninvertile Linear Discrete-Tie Systes, Sci. World J., Volue 2013, Article ID , 7 pages [15] I.-L. Popa, M. Megan, T. Ceauşu, On h-trichotoy of Linear Discrete-Tie Systes in Banach Spaces, Acta Univ. Aplulensis Ser Mat. Inf., 39) 2014, [16] J. Zhang, M. Fan, L. Yang, Nonunifor h, k, µ, ν)-dichotoy with Applications to Nonautonoous Dynaical Systes, arxiv: v1 [17] X. Song, T. Yue, D. Li, Nonunifor Exponential Trichotoy for Linear Discrete-Tie Systes in Banach Spaces, J. Funct. Spaces Appl., vol. 2013, Article ID , 6 pages. Ioan-Lucian Popa Departent of Matheatics 1 Decerie 1918 University of Ala Iulia Ala Iulia, Roania eail: lucian.popa@ua.ro Traian Ceauşu Departent of Matheatics West University of Tiişoara, Tiişoara, eail: ceausu@ath.uvt.ro Mihail Megan Acadey of Roanian Scientists Bucharest, Roania eail: egan@ath.uvt.ro 10