ABSTRACT
Proof of Proposition 1 . Part "=>". Let (X, U) be the principal solution of (H) at a, i .e. Xa = 0, Ua = I. Suppose that t E (a, b] is right-dense. Then Xt is nonsingular and
Bt = XtX;! AtBt = Dt � O. Let (x, u) be a solution of (H) with Xa = O. Then U � ) t = ( � )t d on I, where d = ua• because of the unique solvability of the initial value problems
Xt = Xtd = IltDtc and xi Bl (I - IltAt)xr = dT xl' Bl (I - IltAt)Xf d = ll�cT Dtc � 0 ,
Part "{:". First let t E (a , bj be dense. We show that Xt is nonsingular. Suppose in contrary that there exists 0 f-d E IRn such that Xtd = O. Then for ( ;. ) := ( i5 ) d we have Xa = Xad = P and Xt = Xtd = 0, which contradicts the assumption. Let now t E II<. We show Ker Xf � Ker Xt . This is trivial if t is right-dense, so that we suppose for a moment that t is right-scattered. Take any d E Ker Xf . Putting ( ;. ) : = ( i5 ) d on I we obtain Xa = Xad = 0 and xf = Xf d = 0 E 1m IltAtBt . Therefore, if Xt f-0 then (5) implies xi Bl (/ - IltAt )xf > 0, which contradicts xf = O. Thus, d E Ker Xt and Ker Xf � Ker Xt follows. Finally, let t E II< and we show Dt � O. If t is right-dense then Dt = Bt � 0, so we may concentrate only on right-scattered t . To this end, take any c E IRn and set d := I1-t (Xf) t AtEtc and ( ;. ) := ( i5 ) d on I as before. Then again Xa = 0 and by Lemma I (ii),
xr = Xfd = IltXf (XntAtBtc = IltAtEtc E Im lltAtBt . Hence, by Lemma 1 (iii), Xt = Xtd = IltDtc and
xi Bl (I - I1-tAt)xr = dT xl' Bl (/ - I1-tAt)Xf d = ll�cT Dtc. If Xt f-0 then the assumption (5) guarantees cT Dtc > O. Whereas if Xt = 0 then Dtc = :t Xt = 0, i .e. cT Dtc = O. Altogether, Dt � 0 and the proof is complete. •
Sophia R.-J . Jang
Department of Mathematics and Statistics
Texas Tech University
Abstract. Let A > 0 be an irreducible and primitive k x k matrix. We investigate the asymptotic properties of a system of difference equations of the form y(t + 1) = [A + B(t)]y(t ) , where B(t) is an arbitrary k x k matrix. We characterize conditions on A and B(t) such that the normalized solution is asymptotically stabilized in the positive cone.
