Research Article The Equivalence of Datko and Lyapunov Properties for (h, k)-trichotomic Linear Discrete-Time Systems

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1 Discrete Dynaics in Nature and Society Volue 206, Article ID , 8 pages Research Article The Equivalence of Datko and Lyapunov Properties for (h, k)-trichotoic Linear Discrete-Tie Systes Claudia-LuiniuaMihiu, Mihail Megan,,2 and Traian CeauGu Departent of Matheatics, Faculty of Matheatics and Coputer Science, West University of Tiişoara, Vasile Pârvan Boulevard No. 4, Tiişoara, Roania 2 Acadey of Roanian Scientists, Independenţei 54, Bucharest, Roania Correspondence should be addressed to Mihail Megan; egan@ath.uvt.ro Received 22 July 205; Accepted 26 January 206 Acadeic Editor: Allan C. Peterson Copyright 206 Claudia-Luiniţa Mihiţ et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. The ai of this paper is to characterize a general property of (h, k)-trichotoy through soe Lyapunov functions for linear discrete-tie systes in infinite diensional spaces. Also, we apply the results to illustrate necessary and sufficient conditions for nonunifor exponential trichotoy and nonunifor polynoial trichotoy.. Introduction In the last few years an iportant developent has been ade in the field of the asyptotic behaviors of dynaical systes. Aong the ost iportant asyptotic behaviors studied, we ention the properties of stability, dichotoy, and trichotoy (see [ 4] and the references therein). A rearkable characterization for the stability property of continuous dynaical systes was proved by Datko in 972 (see [8]) and later, Przyłuski and Rolewicz obtain in [5] a siilar result for discrete-tie systes. This was a starting point for the developent of the area and the results were extended to the dichotoy case in [6, 7]. An iportant generalization of the dichotoy concept (approached in various anners in [2, 3, 6, 8]) is the notion of trichotoy, the ost coplex asyptotic property of dynaical systes. The trichotoy supposes the splitting of the state space, at any oent, into three subspaces: the stable subspace, the unstable subspace, and the central subspace. The concept of (exponential) trichotoy was introduced by Elaydi and Hajek (see [9, 0]) for nonlinear differential equations and later, the case of difference equations is treated by Elaydi and Janglajew in []. Also, iportant contributions on the line of trichotoy in discrete-tie are due to Cuevas and Vidal [7], López-Fenner and Pinto [3], Megan and Stoica [9, 20], Papaschinopoulos [2], and Popa et al. [22]. In[23],A.L.SasuandB.Sasuproposeaninterestingtechnique for exponential trichotoy of difference equations, the adissibility technique, and in [24] the authors obtain for the first tie nonlinear conditions for the exponential trichotoy in infinite diensional spaces. The Lyapunov functions represent an iportant tool in the study of the asyptotic properties of dynaical systes (see, e.g., [4, 5, 25, 26]). The objective of this paper is to approach the general concept of (h, k)-trichotoy, where h and k are growth rates, for linear discrete-tie systes in Banach spaces and as particular cases we deduce the results for (nonunifor) exponential trichotoy and (nonunifor) polynoial trichotoy. Also, we obtain necessary and sufficient conditions for a general concept of (h, k)-trichotoy (called (h, k)- trichotoy of Datko type) and the ain result is the characterization of this concept of trichotoy in ters of Lyapunov functions.

2 2 Discrete Dynaics in Nature and Society The results are applied to illustrate criteria through the Lyapunov functions for nonunifor exponential trichotoy and nonunifor polynoial trichotoy. 2. Growth Rates Definition. An increasing sequence h : N [,), h(n) is called a growth rate if li n. Definition 2. One says that the growth rate ( ) satisfies hypothesis (H) if there exist a growth rate ( ) and M (, ) such that (H ) (H 2 ) M; () M h2 f 2, (2) for all (, n) N N, >n. Now, we present soe exaples of growth rates which satisfy hypothesis (H). Exaple 3. Let e nα, α>0,and e nβ ; β (0, α) is a growth rate with (H ) (H 2 ) e n(β α) e α M; e α (3) eβ e j(α β) e(α β) e n(α β) e α β e α for all (, n) N 2, >n. e α e β e2(α β) M h2, Exaple 4. If (n+) α with α>,then (n+) β with β (, α) is a growth rate which satisfies the following: (H ) (H 2 ) (n+) β α M (, ) ; f 2 (n+) α β+ (j + ) α β+ ( n)(+) α β+ (+) 2(α β+) h2 f 2 for all (, n) N 2, >n. M h2 f 2, (4) (5) (6) Exaple 5. Let (n+) α e nγ with α>and γ>0.then (n+) β e δn with β (, α), δ (0, γ) is a growth rate with the following properties: (H ) e n(δ γ) (n+) α β+ (H 2 ) for all (, n) N 2, >n. 3. (h, k)-trichotoy e n(δ γ) (j + ) α β+ e j(γ δ) ( n)(+) α β+ e (γ δ) (+) 2(α β+) e 2(γ δ) h2 f 2 e γ M; e γ (7) eδ M h2 f 2, Let X be a real or coplex Banach space and B(X) the Banach algebra of all bounded linear operators on X. I represents the identity operator on X and the nors on X and on B(X) will be denoted by.also, (8) Δ(, n) N 2 : n}, (9) where N is the set oonnegative integers. We consider the linear discrete-tie syste (A) with A:N B(X), A(n) A n. Every solution of (A) is given by for all (, n) Δ,where x n+ A n x n, n N, (0) x A n x n, () A n fl A,...,A n, I, Reark 6. We observe that for all (, n), (n, p) Δ. if >n if n. (2) A n Ap n Ap, (3) Definition 7. AsequenceP:N B(X), P(n) P n is called a projections sequence on X if for all n N. P 2 n P n, (4)

3 Discrete Dynaics in Nature and Society 3 Definition 8. One says that P P n,p2 n,p3 n } is a faily of (i) suppleentary projections sequences if (s ) (s 2 ) P i n Pj n P n +P2 n +P3 n I, n N; (5) 0, ij, i,j, 2, 3} ; (6) (ii) invariant projections sequences for (A) if P i n+ A n A n P i n, n N, i, 2, 3}. (7) In what follows, we consider h, k : N [,) two growth rates and a pair (A, P), wherep P n,p2 n,p3 n } is a faily of suppleentary and invariant projections sequences for (A). Definition 9. The pair (A, P) is called (h, k)-trichotoic if there exists a nondecreasing sequence (s n ), s n,suchthat (ht ) (ht 2 ) (kt 3 ) (kt 4 ) h An P n x s n P n x ; (8) h P2 n x s An P2 n x ; (9) k n An P3 n x k s n P3 n x ; (20) k n P3 n x k s An P3 n x, (2) for all (, n, x) Δ X. In the particular case when (s n ) is a constant sequence, (A, P) is called uniforly (h, k)-trichotoic. As particular cases of (h, k)-trichotoy we reark the following: (i) if e nα, k n e nβ with α, β > 0 we obtain the concept of (nonunifor) exponential trichotoy and if (s n ) is constant it results in the property of unifor exponential trichotoy; (ii) if (n+) α, k n (n+) β with α, β > we recover the concept of (nonunifor) polynoial trichotoy and if (s n ) is constant it results in the property of unifor polynoial trichotoy; (iii) if P 3 n 0for all n Nit results in the notion of h- dichotoy, nonunifor exponential dichotoy (for e nα, α > 0), unifor exponential dichotoy (for e nα, α>0,and(s n ) constant), nonunifor polynoial dichotoy (for (n+) α, α>), and unifor polynoial dichotoy (for (n + ) α, α>,and(s n ) constant). We give a general exaple of a pair (A, P) which is (h, k)- trichotoic. Exaple 0. Let ( ) and (k n ) be two growth rates and (s n ) a nondecreasing sequence of positive real nubers, s n. Let P P n,p2 n,p3 n } be a faily of suppleentary projections sequences with P i n+ Pi n Pi n, n N, i,2,3. (22) Linear discrete-tie syste (A), defined by A n s n P n + s + +s n+ P 2 n n+ s + k n+s n P 3 n n k n s, (23) n+ verifies the relation Then P i n+ A n A n P i n, i,2,3. (24) A n s n h s P n + h s s n P 2 n + k s n k n s P 3 n, (25) for all (, n) Δ. For all (, n, x) Δ X the following properties hold: (i) (ii) (iii) (iv) s n h An P n x s P n x s n P n x ; (26) h P2 n x h s 2 s n P2 n x s An P2 n x ; (27) s n k n An P3 n x k s P3 n x k s n P3 n x ; (28) k n P3 n x k2 k n s n P3 n x k s An P3 n x and we deduce that the pair (A, P) is (h, k)-trichotoic. (29) Reark. It is obvious that if the pair (A, P) is uniforly (h, k)-trichotoic then it is also (h, k)-trichotoic. In the following exaple we show that the converse iplication is not valid. Exaple 2. We consider P P n,p2 n,p3 n } afailyof suppleentary projections sequences with the property P i n+ Pi n Pi n, n N, i,2,3. (30) Linear discrete-tie syste (A) is given by u n A n P n u + u n+ P 2 n + V n P 3 n n+ V, (3) n+ where u n e 5n/(2+w n) and V n e 4n/(4+w n) and w n 6, 7, 8, 6, 7, 8,...} is a periodic sequence. u n

4 4 Discrete Dynaics in Nature and Society We have that P is invariant for (A) and A n u n P n u + u P 2 n u + V n P 3 n n V, (32) for all (, n) Δ. A siple coputation shows that for e nα, k n e nβ with α (0, /2), β (0, /3),ands n e n/2 the pair (A, P) is (h, k)-trichotoic. If we suppose that (A, P) is uniforly (h, k)-trichotoic, then, for e nα, 4k+, n4k, w n 6,andw 8, we obtain which is a contradiction. e α e k+ M, (33) 4. (h, k)-trichotoy of Datko Type Let ( ) be a growth rate which satisfies hypothesis (H) and let ( ) be a growth rate given by Definition 2. We consider a pair (A, P), wherep P n,p2 n,p3 n } is a faily of suppleentary and invariant projections sequences for (A). The following result ephasizes a necessary condition for (h, k)-trichotoy. Theore 3. If (A, P) is (h, k)-trichotoic and ( ) satisfies hypothesis (H) then there exist a growth rate (g n ) and a nondecreasing sequence (d n ), d n,suchthat (fd ) An j P n x d n P n x, (n, x) N X; (34) + (fd 2 ) (gd 3 ) An j P2 n x d f An P2 n x, + (gd 4 ) (, n, x) Δ X, >n; (35) g j An j P3 n x d n g n P3 n x, (n, x) N X; (36) g j An j P3 n x g d An P3 n x, Proof. It is easy to see that, for (, n, x) Δ X, >n. d n M s n, g n k n, (37) (38) where M (,), ( ) are given by Definition 2, and (s n ), (k n ) aregivenbydefinition9,relations(fd ), (fd 2 ), (gd 3 ), and (gd 4 ) are satisfied. A necessary condition for polynoial trichotoy is represented by the following. Corollary 4. If the pair (A, P) is polynoially trichotoic, then there are a nondecreasing sequence (d n ), d n,andtwo constants a, b > 0 such that (pd ) (j + ) a An j P n x d n (n+) a P n x, + (pd 2 ) (n, x) N X; (j + ) a An j P2 n x d (+) a An P2 n x, (pd 3 ) (, n, x) Δ X, >n; (j + ) b An j P3 n x d n (n+) b P3 n x, + (pd 4 ) (n, x) N X; (j + ) b An j P3 n x d (+) b An P3 n x, Proof. It results fro Theore 3. (, n, x) Δ X, >n. (39) (40) (4) (42) Definition 5. One says that the pair (A, P) adits a (h, k)- trichotoy of Datko type if there exists a nondecreasing sequence (d n ) with d n such that (hd ) An j P n x d n P n x, (n, x) N X; (43) + (hd 2 ) (kd 3 ) An j P2 n x d h An P2 n x, + (, n, x) Δ X, >n; (44) An j P3 n x d n k n P3 n x, (n, x) N X; (45)

5 Discrete Dynaics in Nature and Society 5 (kd 4 ) An j P3 n x k d An P3 n x, (, n, x) Δ X, >n. As particular cases, we ention the following: (46) (i) if e nα with α>0and k n e nβ with β>0we obtain the notion of exponential trichotoy of Datko type; (ii) if (n+) α with α>and k n (n+) β with β> we recover the concept of polynoial trichotoy of Datko type. Reark 6. Theore 3 ephasizes that if ( ), (k n ) are two growth rates, ( ) satisfies hypothesis (H), andthepair (A, P) is (h, k)-trichotoic, then (A, P) adits a (f, g)- trichotoy of Datko type. Theore 7. If the pair (A, P) adits a (h, k)-trichotoy of Datko type, then (A, P) is (h, k)-trichotoic. Proof. (ht ) Using condition (hd ) we obtain h An P n x d n P n x, (, n, x) Δ X. (47) (ht 2 ) Siilarly, by (hd 2 ) for we have h P2 n x d An P2 n x, (, n, x) Δ X, >n. Obviously, the relation is valid for n. (kt 3 ) Inequality (kd 3 ) iplies that (48) k n An P3 n x d nk P3 n x, (, n, x) Δ X. (49) Corollary 9. The pair (A, P) is exponentially trichotoic if and only if there exist the constants a, b > 0 and a nondecreasing sequence (d n ), d n,withthefollowing properties: (ed ) e ja An j P n x d ne na P n x, (n, x) N X; (5) + (ed 2 ) (ed 3 ) e ja An j P2 n x d e a An P2 n x, e bj An j P3 n x d ne nb P3 n x, + (ed 4 ) Proof. e jb An j P3 n x d e b An P3 n x, (, n, x) Δ X, >n; (n, x) N X; (, n, x) Δ X, >n. Necessity. It is a particular case of Theore 3 for e nα, e na with a (0, α), (52) (53) (54) (kt 4 ) By (kd 4 ) (for )wededuce (55) k n P3 n x k d An P3 n x, (, n, x) Δ X, >n, (50) k n e nβ g n e nb, with β>0, bβ+α a. andtheinequalityisverifiedforn. So, thepair (A, P) is (h, k)-trichotoic. Corollary 8. If there exist two constants a, b > and a nondecreasing sequence (d n ), d n, such that conditions (pd ) and (pd 3 ) ((pd 2 ) and (pd 4 ), resp.) fro Corollary 4 are fulfilled for all (n, x) N X(for all (, n, x) Δ X, resp.) with >nthen (A, P) is polynoially trichotoic. Proof. It results fro Theore 7 for (n+) a with a> and k n (n+) b,withb>. The following result represents a characterization for the exponential trichotoy. Sufficiency. Using Theore 7, for e ja and e jb we obtain that (A, P) is exponentially trichotoic. Reark 20. The previous result shows that the exponential trichotoy and the exponential trichotoy of Datko type are equivalent. 5. Lyapunov Functions for (h, k)-trichotoy Throughout this section, ( ), (k n ) represent two growth rates, (A) is a linear discrete-tie syste, and P P n,p2 n,p3 n } is a faily of suppleentary and invariant projections sequences for (A).

6 6 Discrete Dynaics in Nature and Society Definition 2. Two appings L,L 2 :Δ X R + are called (h, k)-lyapunov functions for the pair (A, P) if there exists a nondecreasing sequence (t n ), t n,suchthat (hl ) L (, p, P p x) + (kl 3 ) (hl 2 2 ) + Ap j P p x L (n, p, P p x) t p P p x ; (56) + k p Ap j P3 p x L (n,p,p 3 p x) t p P3 p x ; (57) L 2 (n, p, P 2 p x) + h Ap j P2 p x L 2 (, p, P 2 p x) (58) t Ap P2 p x ; (kl 3 2 ) k Ap j P3 p x L 2 (, p, P 3 p x) t Ap P3 p x, (59) for all (, n, p) N 3 with >n pand for all x X. In particular, if (i) e nα,withα > 0, k n e nβ,withβ > 0 then the (h, k)-lyapunov functions are called exponential Lyapunov functions; (ii) (n+) α,withα>0, k n (n+) β,withβ>0then the (h, k)-lyapunov functions are called polynoial Lyapunov functions. Exaple 22. On Xl (N, R),theBanachspaceofbounded real-valued sequences, endowed with the nor x sup x n, x (x n) n N X, (60) n N we consider P P n,p2 n,p3 n },with P n xχ 3Nx, P 2 n xχ 3N+x, P 3 n xχ 3N+2x, where χ A represents the characteristic function of set A. Also, linear discrete-tie syste (A) is defined by (6) For the growth rates e 2n and k n e 4n we define the exponential Lyapunov functions: L (, n, x) L 2 (, n, x) j+ e 2j 2n An j P n x + e 4n 4j An j P3 n x, j+ e 2 2j An j P2 n x + e 4j 4 An j P3 n x, if >n 0, if n. (64) After soe coputations, we obtain that for t n e 4(n+) /(e 4 ) the appings L and L 2 are exponential Lyapunov functions for the pair (A, P). In the following, we give a characterization for the (h, k)-trichotoy of Datko type in ters of (h, k)-lyapunov functions. Theore 23. The pair (A, P) adits a (h, k)-trichotoy of Datko type if and only if there exist two (h, k)-lyapunov functions for (A, P). Proof. Necessity.LetL,L 2 :Δ X R +, defined by L (, n, x) L 2 (, n, x) j+ An j P n x + k n An j P3 n x, (65) j+ h An j P2 n x + k An j P3 n x, if >n 0, if n, respectively. Then (hl ) L (, p, P p x) + j+ + + h p Ap j P p x + Ap j P p x + Ap j P p x h p Ap j P p x L (n, p, P p x) d p P p x ; (66) (67) A n e 3n 3(n+) P n +e3(n+) 3n P 2 n +P3 (62) n andwehavethat A n e3n 3 P n +e3 3n P 2 n +P3 n, (63) for all (, n) Δ. (kl 3 ) + k p + k p Ap j P3 p x Ap j P3 p x L (n,p,p 3 p x) d p P3 p x ; (68)

7 Discrete Dynaics in Nature and Society 7 (hl 2 2 ) (kl 3 2 ) L 2 (n, p, P 2 p x) + h Ap j P2 p x n h n Ap j P2 p x + h Ap j P2 p x jp h jp Ap j P2 p x L 2 (, p, P 2 p x) d Ap P2 p x ; k Ap j P3 p x k Ap j P3 p x L 2 (, p, P 3 p x) (69) jp (70) d Ap P3 p x, for all (, n, p) N 3 with >n pand for all x X. Sufficiency. (hd ) Fro (hl ),forpnwe obtain + An j P n x t n P n x, (7) and for it results in condition (hd ). (hd 2 ) For pnin (hl 2 2 ) we deduce h An j P2 n x t An P2 n x, (72) for all (, n) Δ, > n, x X. (kd 3 ) Putting pnin (kl 3 ) we obtain k n + and for it results in (kd 3 ). (kd 4 ) By condition (kl 3 2 ) it follows An j P3 n x t n P3 n x, (73) k An j P3 n x t An P3 n x, (74) for all (, n) Δ, > n, x X. Thus, it results in that (A, P) adits a (h, k)-trichotoy of Datko type. A sufficient condition for (h, k)-trichotoy given through the Lyapunov functions is as follows. Corollary 24. If there exist two (h, k)-lyapunov functions L,L 2 for the pair (A, P),then(A, P) is (h, k)-trichotoic. Proof. It is obtained fro Theores 7 and 23. Corollary 25. Let ( ), (k n ) be two growth rates such that ( ) satisfies hypothesis (H). If(A, P) is (h, k)-trichotoic then there exist two (f, g)-lyapunov functions for (A, P), where ( ), (g n ) are the growth rates given by Definition 2 (Theore 3, resp.). Proof. It is iediate by Reark 6 and Theore 23. An iportant characterization for the exponential trichotoy in ters of Lyapunov functions is represented by the following. Corollary 26. The pair (A, P) is exponentially trichotoic if and only if there exist L, L 2 two exponential Lyapunov functions for (A, P). Proof. It results by Reark 20 and Theore 23. Corollary 27. If (A, P) is polynoially trichotoic, then there exist L, L 2 polynoial Lyapunov functions for (A, P). Proof. It is a consequence of Corollary 25. Corollary 28. If there exist two (h, k)-lyapunov functions for the pair (A, P),where (n+) α, with α >, k n (n+) β, with β >, then (A, P) is polynoially trichotoic. Proof. It results fro Theores 7 and 23 for (n+) α, k n (n+) β, Conflict of Interests with α>, with β>. (75) (76) The authors declare that there is no conflict of interests regarding the publication of this paper. References [] A. I. Alonso, J. Hong, and R. Obaya, Exponential dichotoy and trichotoy for difference equations, Coputers & Matheatics with Applications,vol.38,no.,pp.4 49,999. [2] M.-G. Babuţia, M. I. Kovács, M. Lăpădat, and M. Megan, Discrete (h, k)-dichotoy and rearks on the boundedness of the projections, Operators, vol. 204, Article ID 96345, 6 pages, 204. [3]M.G.Babuţia, M. Megan, and I. L. Popa, On (h, k)-dichotoies for nonautonoous linear difference equations in Banach spaces, International Differential Equations, vol. 203, Article ID 76680, 7 pages, 203. [4] L. Barreira and C. Valls, Lyapunov functions for trichotoies with growth rates, Differential Equations, vol.248, no., pp. 5 83, 200.

8 8 Discrete Dynaics in Nature and Society [5] L. Barreira and C. Valls, Lyapunov sequences for exponential trichotoies, Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no., pp , 200. [6] A. J. G. Bento and C. M. Silva, Nonunifor (μ, ])-dichotoies and local dynaics of difference equations, Nonlinear Analysis: Theory, Methods & Applications,vol.75,no.,pp.78 90,202. [7] C. Cuevas and C. Vidal, Weighted exponential trichotoy of linear difference equations, Dynaics of Continuous, Discrete & Ipulsive Systes. Series A. Matheatical Analysis, vol.5, no. 3, pp , [8] R. Datko, Unifor asyptotic stability of evolutionary processes in a Banach space, SIAM Journal on Matheatical Analysis,vol.3,no.3,pp ,972. [9] S. Elaydi and O. Hajek, Exponential trichotoy of differential systes, Matheatical Analysis and Applications,vol. 29, no. 2, pp , 988. [0] S. Elaydi and O. Hajek, Exponential dichotoy and trichotoy oonlinear differential equations, Differential Integral Equations,vol.3,no.6,pp ,990. [] S. Elaydi and K. Janglajew, Dichotoy and trichotoy of difference equations, Difference Equations and Applications,vol.3,no.5-6,pp ,998. [2] Z.-G. Li, X.-Q. Song, and X.-L. Yang, On nonunifor polynoial trichotoy of linear discrete-tie systes in Banach spaces, Applied Matheatics, vol. 204, ArticleID , 6 pages, 204. [3] J. López-Fenner and M. Pinto, (h, k)-trichotoies and asyptotics oonautonoous difference systes, Coputers & Matheatics with Applications,vol.33,no.0,pp.05 24,997. [4] M. Megan and L. Buliga, Nonunifor exponential trichotoy for difference equations in Banach spaces, Hot Topics in Operator Theory, Theta Series in Advanced Matheatics, vol.9, pp. 5 58, [5] K.M.PrzyłuskiandS.Rolewicz, Onstabilityoflineartievarying infinite-diensional discrete-tie systes, Systes & Control Letters,vol.4,no.5,pp ,984. [6] I.-L. Popa, M. Megan, and T. Ceauşu, Exponential dichotoies for linear discrete-tie systes in Banach spaces, Applicable Analysis and Discrete Matheatics, vol.6,no.,pp.40 55, 202. [7] P. Preda and M. Megan, Exponential dichotoy of evolutionary processes in Banach spaces, Czechoslovak Matheatical Journal, vol. 35, no. 0, pp , 985. [8] B. Sasu, On exponential dichotoy of variational difference equations, Discrete Dynaics in Nature and Society, vol.2009, Article ID , 8 pages, [9] M. Megan and C. Stoica, On unifor exponential trichotoy of evolution operators in Banach spaces, Integral Equations and Operator Theory, vol. 60, no. 4, pp , [20] M. Megan and C. Stoica, Trichotoy for discrete skewevolution seiflows in Banach spaces, Annals of the Tiberiu Popoviciu Seinar of Functional Equations, Approxiation and Convexity,vol.5,pp.79 85,2007. [2] G. Papaschinopoulos, On exponential trichotoy of linear difference equations, Applicable Analysis, vol.40,no.2-3,pp , 99. [22] I.-L. Popa, M. Megan, and T. Ceauşu, On h-trichotoy of linear discrete-tie systes in Banach spaces, Acta Universitatis Apulensis,vol.39,pp ,204. [23] A. L. Sasu and B. Sasu, Input-output adissibility and exponential trichotoy of difference equations, Matheatical Analysis and Applications, vol. 380, no., pp. 7 32, 20. [24] B. Sasu and A. L. Sasu, Nonlinear criteria for the existence of the exponential trichotoy in infinite diensional spaces, Nonlinear Analysis: Theory, Methods & Applications,vol.74,no. 5, pp , 20. [25] G. Papaschinopoulos, A characterization of exponential trichotoy via Lyapunov functions for difference equations, Matheatica Japonica,vol.37,no.3,pp ,992. [26] I.-L. Popa, T. Ceauşu, and M. Megan, Nonunifor power instability and Lyapunov sequences, Applied Matheatics and Coputation,vol.247,pp ,204.

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