Dichotomy Of Poincare Maps And Boundedness Of Some Cauchy Sequences

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1 Applied Mathematics E-Notes, 2(202), 4-22 c ISSN Available free at mirror sites of Dichotomy Of Poincare Maps And Boundedness Of Some Cauchy Sequences Abar Zada y, Sadia Arshad z, Gul Rahmat x, Rohul Amin { Received 4 March 20 Abstract Let fu(p; q)g pq0 be the N-periodic discrete evolution family of m m matrices having complex scalers as entries generated by L(C m )-valued, N-periodic sequence of m m matrices (A n) where N 2 is a natural number. We proved that the Poincare map U(N; 0) is dichotomic if and only if the matrix P V = N U(N; )e i is invertible and there exists a projection P which commutes with the map U(N; 0) and the matrix V, such that for each 2 R and each vector b 2 C m the solutions of the discrete Cauchy sequences x n+ = A nx n + e in P b; x 0 = 0 and y n+ = An y n + e in (I P )b; y 0 = 0 are bounded. Introduction It is well-nown, see [2], that a matrix A is dichotomic, i.e. its spectrum does not intersect the unit circle if and only if there exists a projector, i.e. an m m matrix P satisfying P 2 = P, which commutes with A and has the property that for each real number and each vector b 2 C m ; the following two discrete Cauchy problems xn+ = Ax n + e in P b; n 2 Z + () x 0 = 0 and yn+ = A y n + e in (I P )b; n 2 Z + y 0 = 0 have bounded solutions. In particular, the spectrum of A belongs to the interior of the unit circle if and only if for each real number and each m-vector b; the solution of the Cauchy problem () is bounded. Continuous version of the above result is given in [4]. On the other hand, in [3], it is shown that an N-periodic evolution family U = fu(p; q)g pq0 of bounded linear operators acting on a complex space ; is uniformly Mathematics Subject Classi cations: 35B35 y Department of Mathematics, Abdul Wali Khan University, Mardan, Paistan z Abdus Salam School of Mathematical Sciences (ASSMS), GCU, Lahore, Paistan x Abdus Salam School of Mathematical Sciences (ASSMS), GCU, Lahore, Paistan { Department of Mathematics, University of Peshawar, Peshawar, Paistan (2) 4

2 Zada et al. 5 exponentially stable, i.e. the spectral radius of the Poincare map U(N; 0) is less than one, if and only if for each real number and each N-periodic sequence (z n ) decaying to n = 0; we have n sup e i U(n; )z = M(; b) < : n Recently in [], it is proved that the spectral radius of the matrix U(N; 0) is less than one, if for each real and each m-vector b; the operator V := P N ei U(N; ) is invertible and N sup e i(j ) U(N; j)b n < : j This note is a continuation of the latter quoted paper. In fact, we prove that the matrix U(N; 0) is dichotomic if and only if for each real and each m-vector b; the operator V := P N ei U(N; ) is invertible and solutions of the two discrete Cauchy sequences lie (A; P b; x 0 ; 0) are bounded. 2 Preliminary Results Consider the following Cauchy Problem zn+ = Az n ; z n 2 C m ; n 2 Z + z n (0) = z 0 : (3) where A is an m m matrix. It is easy to chec that the solution of (3) is A n z 0. Consider the following lemma which is used in Theorem. LEMMA. Let N be a natural number. If q n is a polynomial of degree n and N q n = 0 for all n = 0; ; 2 : : : where z n = z n+ z n then q is a C m -valued polynomial of degree less than or equal to N. For proof see [2]. Let p A be the characteristic polynomial associated with the matrix A and let (A) = f ; 2 ; : : : ; g, m be its spectrum. There exist integer numbers m ; m 2 ; : : : ; m such that p A () = ( ) m ( 2 ) m2 : : : ( ) m ; m + m m = m: Then in [2] we have the following theorem. THEOREM. For each z 2 C m there exists w j 2 W j := er(a j I) mj, (j 2 f; 2; : : : ; g) such that A n z = A n w + A n w A n w : Moreover, if w j (n) := A n w j then w j (n) 2 W j for all n 2 Z + and there exist a C m - valued polynomials q j (n) with deg (q j ) m j such that w j (n) = n j q j (n); n 2 Z + ; j 2 f; 2; : : : ; g:

3 6 Dichotomy of Poincare Maps FROOF. Indeed from the Cayley-Hamilton theorem and using the well nown fact that er[pq(a)] = er[p(a)] er[q(a)] whenever the complex valued polynomials p and q are relatively prime, it follows that C m = W W 2 W : (4) Let z 2 C m. For each j 2 f; 2; : : : ; g there exists a unique w j 2 W j such that z = w + w w and then A n z = A n w + A n w A n w ; n 2 Z + : Let q j (n) = n j w j (n). Successively one has Again taing ; 2 q j (n) = [q j (n)] q j (n) = ( n j w j (n)) = ( n j A n w j ) = (n+) j A n+ w j n j A n w j = (n+) j (A j I)A n w j : = [ (n+) j (A j I)A n w j ] = (n+2) j (A j I)A (n+) w j (n+) j (A j I)A n w j = (n+2) j (A j I) 2 A n w j : Continuing up to m j we get mj q j (n) = (n+mj) j (A j I) mj A n w j : But w j (n) belongs to W j for each n 2 Z +. Thus mj q j (n) = 0. Using Lemma, we can say that the degree of polynomial q j (n) is less than or equal to m j. 3 Dichotomy and Boundedness A family U = fu(p; q) : (p; q) 2 Z + Z + g of an m m complex valued matrices is called discrete periodic evolution family if it satis es the following properties.. U(p; q)u(q; r) = U(p; r) for all p q r 0; 2. U(p; p) = I for all p 0 and 3. there exists a xed N 2 such that U(p + N; q + N) = U(p; q) for all p; q 2 Z + ; p q:

4 Zada et al. 7 Let us consider the following discrete Cauchy problem: ( zn+ = A n z n + e in b; n 2 Z + z 0 = 0; where the sequence (A n ) is N-periodic, i.e. A n+n = A n for all n 2 Z + and a xed N 2: Let An U(n; j) = A n 2 A j if j n ; I if j = n; then, the family fu(n; j)g nj0 is a discrete N-periodic evolution family and the solution (z n ) of the Cauchy problem (A n ; ; b) 0 is given by: n z n = U(n; j)e i(j j ) b Let us denote by C = fz 2 C : jzj = g, C + = fz 2 C : jzj > g and C = fz 2 C : jzj < g. Clearly C = C [ C + [ C. Then with the help of above partition of C for matrix A we give the following de nition: DEFINITION. The matrix A is called: (i) stable if (A) is the subset of C or, equivalently, if there exist two positive constants N and such that A n Ne n for all n = 0; ; 2 : : :, (ii) expansive if (A) is the subset of C + and (iii) dichotomic if (A) have empty intersection with set C. It is clear that any expansive matrix A whose spectrum consists of ; 2 ; : : : ; is an invertible one and its inverse is stable, because (A ) = f ; 2 ; : : : ; g C : P Let L := U(N; 0), V = N U(N; )e i and A i A j = A j A i for any i; j 2 f; 2; : : : ; ng. We recall that a linear map P acting on C m is called projection if P 2 = P. THEOREM 2. Let N 2 be a xed integer number. The matrix L is dichotomic if and only if the matrix V is invertible and there exists a projection P having the property P L = LP and P V = V P such that for each 2 R and each vector b 2 C m the solutions of the following discrete Cauchy problems ( xn+ = A n x n + e in P b; n 2 Z + (5) x 0 = 0 and ( yn+ = An y n + e in (I P )b; n 2 Z + y 0 = 0 : (6)

5 8 Dichotomy of Poincare Maps are bounded. PROOF. Necessity: Woring under the assumption that L is a dichotomic matrix we may suppose that there exists 2 f; 2; : : : ; g such that j j j 2 j j j < < j + j j j: Having in mind the decomposition of C m given by (4) let us consider = W W 2 W ; 2 = W + W +2 W : Then C m = 2 : De ne P : C m! C m by P x = x, where x = x + x 2, x 2 and x It is clear that P is a projection. Moreover for all x 2 C m and all n 2 Z +, this yields P L x = P (L (x + x 2 )) = P (L (x ) + L (x 2 )) = L (x ) = L P x; where the fact that is an L invariant subspace, was used. Then P L = L P. Similarly by using the fact that and 2 are V invariant subspaces we can prove that P V = V P. We now that the solution of the Cauchy problem (5) is: x n = n U(n; j)e i(j ) P b: j Put n = N + r, where r = 0; ; 2; : : : ; N : Then x N+r = N+r j U(N + r; j)e i(j ) P b: Let and Then Thus A = f; + N; : : : ; + ( )Ng; where 2 f; 2; : : : ; Ng R = fn + ; N + 2; : : : ; N + rg: R [ ([ N A ) = f; 2; : : : ; ng: N x N+r = e i U(N + r; j)e ij P b + e i U(N + r; j)e ij P b j2a j2r N = e i U(N + r; + sn)e i(+sn) P b + e i s=0 U(N + r; N + )e i(n+) P b

6 Zada et al. 9 N = e i U(r; 0)U(N; 0) ( s ) U(N; )e i(+sn) P b + e i s=0 U(r; )e i(n+) P b: Let z = e in, also we now that L = U(N; 0), thus x N+r = e i U(r; 0) L ( s ) z s e i z s=0 U(r; )e i P b N U(N; )e i P b + = e i U(r; 0) L z 0 + L 2 z + + L 0 z +e i z U(r; )e i P b: N U(N; )e i P b We now that P N U(N; )ei = V thus x N+r = e i U(r; 0) L z 0 + L 2 z + + L 0 z V P b + e i z U(r; )e i P b: By our assumption we now that L is dichotomic and jz j = thus z is contained in the resolvent set of L therefore the matrix (z I L) is an invertible matrix. Thus x N+r = e i U(r; 0)(z I L) (z I L )V P b + e i z = e i U(r; 0)(z I L) (z I L )P V b + e i z U(r; )e i P b U(r; )e i P b: We now that V is a surjective map, so there exists b 0 such that V b = b 0 then x N+r = e i U(r; 0)(z I L) (z I L )P b 0 + e i z Taing norm of both sides x N+r = e i U(r; 0)(z I L) (z I L )P b 0 + e i z U(r; )e i P b: U(r; )e i P b

7 20 Dichotomy of Poincare Maps x N+r U(r; 0)(z I L) zp b 0 + U(r; 0)(z I L) P L b 0 + U(r; )P b = U(r; 0)(z I L) P b 0 + U(r; 0)(z I L) P L b 0 + U(r; )P b: Using THEOREM, We have Thus L b 0 = p () + 2p 2 () + + p (); P L b 0 = p () + 2p 2 () + + p (); where each p i () are C m -valued polynomials with degree at most (m i ) for any i 2 f; 2; : : : ; g. From hypothesis we now that j i j < for each i 2 f; 2; : : : ; g. Thus P L b 0! 0 when! and so x N+r is bounded for any r = 0; ; 2; : : : ; N. Thus x n is bounded. For the second Cauchy problem: We have y n = n U (n; j)e i(j ) (I P )b: j where U (n; j) = A n A n 2 A j if j n ; I if j = n: It is easy to chec that U (n; j) is also a discrete evaluation family. n = N + r, where r = 0; ; 2; : : : ; N : Then By putting y N+r = N+r j U (N + r; j)e i(j ) (I P )b: As A i A j = A j A i for all i; j 2 f; 2; : : : ; ng thus L = U (N; 0). By similar procedure as above we obtained that y N+r = U (r; 0)(z I L ) (I P )V (b) + U (r; 0)(z I L ) L (I P )V (b) + U (r; )(I P )b: Since (I P )V b 2 2 the assertion would follow. But 2 = W + W +2 W : Each vector from 2 can be represented as a sum of vectors w + ; w +2 ; : : : ; w : It would be su cient to prove that L w j! 0; for any j 2 f + ; : : : ; g: Let W 2

8 Zada et al. 2 fw + ; W +2 ; : : : ; W g; say W = er(l I), where is an integer number and jj >. Consider r 2 W n f0g such that (L I)r = 0 and let r 2 ; r 3 ; : : : ; r given by (L I)r j = r j ; j = 2; 3; : : : ; : Then B := fr ; r 2 ; : : : ; r g is a basis in Y: It is then su cient to prove that L r j! 0; for any j = ; 2; : : : ; : For j = we have that L r = r! 0: For j = 2; 3; : : : ;, denote = L r j : Then (L I) = 0 i.e. C + C C = 0; for all (7) where = : Passing for instance at the components, it follows that there exists a C m -valued polynomial P having degree at most and verifying (7) such that = P (): Thus! 0 as! ; i.e. L r j! 0 for any j 2 f; 2; : : : ; g: Thus (y n ) is bounded. Su ciency: Suppose to the contrary that the matrix L is not dichotomic. Then (L) \ 6=. Let! 2 (L) \ : Then there exists a nonzero y 2 C m such that Ly =!y. It is easy to see that L y = w y: Choose 0 2 R such that e i 0 N =!. We now that x N+r ( 0 ; b) = e i 0 U(r; 0) L z L 2 z L 0 z 0 P V0 b + e i 0 z U(r; )e i 0 P b : But V 0 is surjective, thus there exists b 0 2 C m such that V 0 b 0 = y, so x N+r ( 0 ; b 0 ) = e i 0 U(r; 0) L z L 2 z L 0 z 0 P y + e i 0 z U(r; )e i 0 P b 0 Clearly = e i 0 U(r; 0) P L yz P L 2 yz P L 0 yz 0 + e i 0 z U(r; )e i 0 P b = e i 0 U(r; 0)P L yz L 2 yz L 0 yz 0 + e i 0 z U(r; )e i 0 P b = e i 0 U(r; 0)P [e i 0z 0 ] + e i 0 z x N ( 0 ; b 0 )! when! : U(r; )e i 0 P b Thus a contradiction arises. In [] an example, in terms of stability is given which shows that the assumption on invertibility of V ; for each real number ; cannot be removed.

9 22 Dichotomy of Poincare Maps References [] S. Arshad, C. Buse, A. Nosheen and A. Zada, Connections between the stability of a Poincare map and boundedness of certain associate sequences, Electronic Journal of Qualitative Theory of Di erential Equations, 6(20), 2. [2] C. Buse and A. Zada, Dichotomy and bounded-ness of solutions for some discrete Cauchy problems, Proceedings of IWOTA 2008, Operator Theory, Advances and Applications, (OT) Series Birhäuser Verlag, Eds: J. A. Ball, V. Bolotniov, W. Helton, L. Rodman and T. Spitovsy, 203(200), [3] C. Buse, P. Cerone, S. S. Dragomir and A. Sofo, Uniform stability of periodic discrete system in Banach spaces, J. Di erence Equ. Appl., 2()(2005), [4] A. Zada, A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems, Electronic Journal of Di erential Equations, 94(2008), 5.