Null Solutions of Difference Systems Under Vanishing Perturbation

References

• Agarwal , R. 1992 . Difference Equations and Inequalities New York : Marcel Dekker .
   Google Scholar
• Aulbach , B. 1984 . Continuous and Discrete Dynamics near Manifolds of Equilibria , Lecture Notes in Mathematics 1058 Berlin : Springer .
   Google Scholar
• Benzaid , Z. and Lutz , D. 1987 . Asymptotic representation of solutions of perturbed systems of linear difference equations . Studies Appl. Math. , 77 : 195 – 221 .
   Web of Science ®Google Scholar
• Bohner , M. and Peterson , A. 2001 . Dynamic Equations on Time Scales: An Introduction with Applications Boston : Birkhäuser .
   Google Scholar
• Castillo , S. and Pinto , M. 2002 . “ Levinson's Theorem in Functional Dynamics Equations in Time Scale ” . In Proceedings of 4th International Conference of Differential Equations and Applications 2001 Reading : Gordon and Breach .
   Google Scholar
• Castillo , S. and Pinto , M. 1998 . Asymptotic summation for second order finite difference systems . J. Comp. Math. , 35 ( 4 ) : 117 – 129 .
   Google Scholar
• Castillo , S. and Pinto , M. 2000 . “ Improvements for asymptotic results in functional difference equations ” . In Proc. 5th ICDEA
   Google Scholar
• Castillo , S. and Pinto , M. 1997 . “ Asymptotic formulae for solutions of delay-difference systems ” . In Advances in Difference Equations, Proceedings of the Second International Conference on Difference Equations Veszprém 1995 Edited by: Elaydi , S. , Györi , I. and Ladas , G. 107 – 117 . Amsterdam : Gordon and Breach Publishers .
   Google Scholar
• Castillo , S. and Pinto , M. 2001 . Asymptotic formulae for nonlinear functional difference equations . Comp. Math. Appl. , 42 : 551 – 559 .
   Web of Science ®Google Scholar
• Coffman , C. V. 1964 . Asymptotic behavior of solutions of ordinary difference equations . Trans. Amer. Math. Soc. , 110 : 22 – 51 .
   Google Scholar
• Coffman , C. V. and Schäffer , J. J. 1967 . “ Dichotomies for linear difference equations ” . In Math. Ann. 139 – 167 .
   Google Scholar
• Coppel , W. A. 1978 . Dichotomies in Stability Theory , Lecture Notes in Math. 629 Berlin : Springer .
   Google Scholar
• Cuevas , C. and Pinto , M. 2000 . Asymptotic behavior of solutions of Volterra difference systems . J. Comp. Appl. Math. , 113 : 217 – 225 .
   Web of Science ®Google Scholar
• Cuevas , C. and Pinto , M. 2001 . Asymptotic properties of solutions of nonautonomous Volterra difference systems with infinite delay . Comp. Appl. Math. , 42 : 671 – 685 .
   Google Scholar
• Eastham , M. S. P. 1989 . The Asymptotic Solution of Linear Differential Systems (Applications of the Levinson's Theorem Oxford : Clarendon Press .
   Google Scholar
• Elaydi , S. 1999 . An Introduction to Difference Equations, , 2nd Ed. New York : Springer .
   Google Scholar
• Elaydi , S. , Papaschinopoulos , G. and Shinas , J. 1997 . “ Asymptotic theory for non invertible systems ” . In Proceedings of 2nd International Conference of Differential Systems and Applications , Veszprém 1995 107 – 117 . Gorden and Breach .
   Google Scholar
• Kelley , W. and Peterson , A. 1991 . Difference Equations: An Introduction with Applications New York : Academic Press .
   Google Scholar
• Levinson , N. 1948 . The asymptotic nature of solutions of linear differential equations . Duke Math. J. , 15 : 111 – 126 .
   Web of Science ®Google Scholar
• López , J. and Pinto , M. 1997 . (h,k) Trichotomies and asymptotics of nonautonomous difference systems . J. Comp. Math. , 33 ( 10 ) : 105 – 124 .
   Google Scholar
• López , J. and Pinto , M. 1999 . “ Existence and uniqueness of bounded solutions for a class of difference systems with dichotomies of summable type ” . In New Developments in Difference Equations and Applications (Taipei 1997 183 – 188 . Amsterdam : Gorden and Breach .
   Google Scholar
• Massera , J. L. and Schäffer , J. J. 1966 . Linear Differential Equations and Functional Spaces New York : Academic Press .
   Google Scholar
• Mate , A. and Nevai , P. 1990 . A generalization of Poincaré's theorem for recurrence equations . J. Approx. Theory , 63 : 92 – 97 .
   Web of Science ®Google Scholar
• Medina , R. and Pinto , M. 1992 . Asymptotic representation of solutions of linear second order difference equations . J. Math. Anal. Appl. , 165 ( 2 ) : 505 – 515 .
   Web of Science ®Google Scholar
• Medina , R. and Pinto , M. 1996 . Asymptotic constancy of solutions of systems of difference equations . Int. J. Math. Math. Sci. , 19 ( 2 ) : 343 – 350 .
   Google Scholar
• Medina , R. and Pinto , M. 1997 . Variationally stable difference equations. Proc. 2nd. World Cong. Nonlinear Anal., Athens 1996 . Nonlinear Anal. , 30 ( 2 ) : 1141 – 1152 .
   Google Scholar
• Medina , R. and Pinto , M. 1996 . Asymptotic representation of solutions of ordinary difference systems . Appl. Anal. , 61 ( 1–2 ) : 31 – 44 .
   Google Scholar
• Naulún , R. and Pinto , M. 1995 . Roughness of (h,k) dichotomies . J. Diff. Eqs , 118 ( 1 ) : 20 – 35 .
   Google Scholar
• Naulún , R. and Pinto , M. Asymptotic behavior of second order Poincaré difference equations in preparation
   Google Scholar
• Naulún , R. and Pinto , M. 1996 . “ Reduction principle for difference systems ” . In Proceedings of International Confererence on Difference Equations San Antonio, USA 379 – 387 . Amsterdam : Gordon and Breach Publishers .
   Google Scholar
• Naulún , R. and Pinto , M. 1997 . Stability of discrete dichotomies for linear difference systems . J. Diff. Eqs Appl. , 3 ( 2 ) : 101 – 123 .
   Google Scholar
• Palmer , K. 1988 . Exponential dichotomies, the shadowing lemma and transversal homoclinic points . Dyn. Rep. , 1 : 265 – 306 .
   Google Scholar
• Perron , O. 1909 . Über einen Satz des Herrn Poincaré . J. Reine Angew. Math. , 136 : 17 – 37 .
   Google Scholar
• Pinto , M. 1994 . Discrete dichotomies, advances in difference equation I . Comput. Math. Appl. , 28 ( 1–3 ) : 259 – 270 .
   Web of Science ®Google Scholar
• Pinto , M. 1998 . Weighted-bounded and convergent solutions . Adv. Diff. Eqn II, Comput. Math. Appl. , 36 : 391 – 400 .
   Google Scholar
• Pinto , M. 1988 . Asymptotic integration of systems resulting from the perturbations of an h-system . J. Math. Anal. Appl. , 131 : 194 – 216 .
   Google Scholar
• Pinto , M. 1990 . Variationally stable differential systems . J. Math. Anal. Appl. , 151 : 254 – 260 .
   Google Scholar
• Pinto , M. 1992, 1993 . Proceedings of the First World Congress of Nonlinear Analysts Tampa Edited by: Lakshmikantham , V. and de Gruyter , W. Dichotomies for differential systems with impulse effect, In: eds
   Google Scholar
• Pinto , M. 1994 . About the conditional stability of the piecewise continuous solutions of differential equations . Diff. Eqs Dyn. Syst. , 2 ( 3 ) : 217 – 230 .
   Google Scholar
• Pinto , M. 2001 . Asymptotic behavior of systems of Poincaré type Fac Ciencias, U. Chile, UNAP, U. Búo-Búo, UFRO .
   Google Scholar
• Pinto , M. 1995 . Asymptotic equivalence of difference systems . J. Diff. Eqs Appl. , 1 : 249 – 262 .
   Google Scholar
• Pinto , M. 1996 . Asymptotic equivalence of differential systems with impulse effect . Dyn. Cont., Discrete Impulsive Syst. , 2 ( 2 ) : 205 – 218 .
   Google Scholar
• Pinto , M. A generalization to Poincaré's theorem for recurrent systems, to appear
   Google Scholar
• H. , Poincaré . 1885 . Sur les equations linéaires aux différentielles et aux différences finites . Am. J. Math. , 7 : 203 – 258 .
   Google Scholar
• Schinas , J. 1974 . Stability and conditional stability of time-difference equations in Banach spaces . J. Inst. Math. Appl. , 14 : 335 – 346 .
   Google Scholar