For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, we use Perron ’ s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in [1] , to show that if the associative operator is semi - Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both T <sup>+</sup> and T<sup>-</sup>. Moreover, we also give the converse result that the linear systems ha ve an exponential dichotomy on both T <sup>+</sup> and T<sup>-</sup> then the associative operator is Fredholm on T .
Zhang, J., Fan, M. and Zhu, H. (2010) Necessary and Sufficient Criteria for the Existence of Exponential Dichotomy on Time Scales. Computers & Mathematics with Applications, 600, 2387-2398. https://doi.org/10.1016/j.camwa.2010.08.034
Coppel, W.A. (1965) Stability and Asymptotic Behavior of Differential Equations. Heath Mathematical Monographs. Heath & Co., Boston.
Coppel, W.A. (1978) Dichotomies in Stability Theory. Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin. https://doi.org/10.1007/BFb0067780
Palmer, K.J. (1988) Exponential Dichotomies and Fredholm. Proceedings of the American Mathematical Society, 104, 149-156. https://doi.org/10.1090/S0002-9939-1988-0958058-1
Palmer, K.J. (1984) Exponential Dichotomies and Transversal Homoclinic Points. Journal of Differential Equations, 55, 225-256. ps://doi.org/10.1016/0022-0396(84)90082-2
Palmer, K.J. (1987) A Perturbation Theorem for Exponential Dichotomies. Proceedings of the Royal Society of Edinburgh Section A, 106, 25-37. https://doi.org/10.1017/S0308210500018175
Palmer, K.J. (1987) Exponential Dichotomies for Almost Periodic Equations. Proceedings of the American Mathematical Society, 101, 293-298. https://doi.org/10.1090/S0002-9939-1987-0902544-6
Palmer, K.J. (2000) Shadowing in Dynamical Systems: Theory and Applications, Mathematics and Its Applications, Vol. 501, Kluwer Academic Publishers, Dordrecht, Boston.
Sacker, R.J. and Sell, G.R. (1978) A Spectral Theory for Linear Differential Systems. Journal of Differential Equations, 27, 320-358. https://doi.org/10.1016/0022-0396(78)90057-8
Sacker, R.J. and Sell, G.R. (1994) Dichotomies for Linear Evolutionary Equations in Banach Spaces. Journal of Differential Equations, 113, 17-67. https://doi.org/10.1006/jdeq.1994.1113
Fenichel, N. (1979) Geometric Singular Perturbation Theory for Ordinary Differential Equations. Journal of Differential Equations, 31, 53-98. https://doi.org/10.1016/0022-0396(79)90152-9
Fenichel, N. (1971) Persistence and Smoothness of Invariant Manifolds for Flows. Indiana University Mathematics Journal, 21, 193-226. https://doi.org/10.1512/iumj.1972.21.21017
Beyn, W. (1994) On Well-Posed Problems for Connecting Orbits in Dynamical Systems. Contemporary Mathematics, 172, 131-168. https://doi.org/10.1090/conm/172/01802
Beyn, W.J. (1990) The Numerical Computation of Connecting Orbits in Dynamical Systems. IMA Journal of Numerical Analysis, 10, 379-405. https://doi.org/10.1093/imanum/10.3.379
Beyn, W.J. and Lorenz, J. (1999) Stability of Traveling Waves: Dichotomies and Eigenvalue Conditions on Finite Intervals. Numerical Functional Analysis and Optimization, 20, 201-244. https://doi.org/10.1080/01630569908816889
Ascher, U., Mattheij, R.M. and Russell, R.D. (1988) Solution of Boundary Value Problems for ODEs. Prentice Hall, Englewood Cliffs.
Broer, H.W., Osinga, H.M. and Vegter, G. (1997) Algorithms for Computing Normally Hyperbolic Invariant Manifolds. Zeitschrift für Angewandte Mathematik und Physik, 48, 480-524. https://doi.org/10.1007/s000330050044
Dieci, L. and Lorenz, J. (1995) Computation of Invariant Tori by the Method of Characteristics. SIAM Journal on Numerical Analysis, 32, 1436-1474. https://doi.org/10.1137/0732066
Barreira, L., Dragicevic, D. and Valls, C. (2016) Fredholm Operators and Nonuniform Exponential Dichotomies. Chaos, Solitons and Fractals, 85, 120-127. https://doi.org/10.1016/j.chaos.2016.01.021
Barreira, L., Dragicevic, D. and Valls, C. (2017) Nonuniform Exponential Dichotomies and Fredholm Operators for Flows. Aequationes Mathematicae, 91, 301-316.
Hilger, S. (1990) Analysis on Measure Chains—A Unified Approach to Continuous and Discrete Calculus. Results in Mathematics, 18, 18-56. https://doi.org/10.1007/BF03323153
Zhang, J., Fan, M. and Zhu, H. (2009) Existence and Roughness of Exponential Dichotomies of Linear Dynamic Equations on Time Scales. Computers & Mathematics with Applications, 59, 2658-2675. https://doi.org/10.1016/j.camwa.2010.01.035
Zhang, J., Song, Y. and Zhao, Z. (2013) General Exponential Dichotomies on Time Scales and Parameter Dependence of Roughness. Advances in Difference Equations, 2013, 339.
Yang, L., Zhang, J., Chang, X. and Liu, Z. (2015) Exponential Dichotomy on Time Scales and Admissibility of the Pair . Advances in Difference Equations, 2015, 69.
Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser Boston Inc., Boston. https://doi.org/10.1007/978-1-4612-0201-1
Bohner, M. and Peterson, A. (2003) Advances in Dynamic Equations on Time Scales. Birkhauser Boston Inc., Boston.
Potzsche, C. (2004) Exponential Dichotomies of Linear Dynamic Equations on Measure Chains under Slowly Varying Coefficients. Journal of Mathematical Analysis and Applications, 289, 317-335. https://doi.org/10.1016/j.jmaa.2003.09.063
Taylor, A.E. (1958) Introduction to Functional Analysis. Wiley, New York.