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Báo cáo hóa học: "EXPONENTIAL DICHOTOMY OF DIFFERENCE EQUATIONS IN l p -PHASE SPACES ON THE HALF-LINE"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXPONENTIAL DICHOTOMY OF DIFFERENCE EQUATIONS IN l p -PHASE SPACES ON THE HALF-LINE | EXPONENTIAL DICHOTOMY OF DIFFERENCE EQUATIONS IN Zp-PHASE SPACES ON THE HALF-LINE NGUYEN THIEU HUY AND VU THI NGOC HA Received 14 November 2005 Revised 12 March 2006 Accepted 17 May 2006 For a sequence of bounded linear operators Anio 0 on a Banach space X we investigate the characterization of exponential dichotomy of the difference equations vn 1 Anvn. We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn 1 Anvn fn in Ip spaces 1 p oo . Then we apply the results to study the robustness of exponential dichotomy of difference equations. Copyright 2006 N. T. Huy and V. T. N. Ha. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction and preliminaries We consider the difference equation Xn 1 Anxn n E N 1.1 where An n 0 1 2 . is a sequence of bounded linear operators on a given Banach space X xn E X for n E N. One of the central interests in the asymptotic behavior of solutions to 1.1 is to find conditions for solutions to 1.1 to be stable unstable and especially to have an exponential dichotomy see e.g. 1 5 7 12 16-20 and the references therein for more details on the history of this problem . One can also use the results on exponential dichotomy of difference equations to obtain characterization of exponential dichotomy of evolution equations through the discretizing processes see e.g. 4 7 9 18 . One can easily see that if An A for all n E N then the asymptotic behavior of solutions to 1.1 can be determined by the spectra of the operator A. However the situation becomes more complicated if An nEN is not a constant sequence because in this case the spectra of each operator An cannot determine the asymptotic behavior of the solutions to 1.1 . Therefore in order to find the conditions for 1.1 to have an exponential dichotomy one .