Abstract
We consider the problem of existence and structure of solutions bounded on the entire real axis of nonhomogeneous linear impulsive differential systems. Under assumption that the corresponding homogeneous system is exponentially dichotomous on the semiaxes and and by using the theory of pseudoinverse matrices, we establish necessary and sufficient conditions for the indicated problem.
Highlights
We consider the problem of existence and structure of solutions bounded on the entire real axis of nonhomogeneous linear impulsive differential systems
The ideas proposed in these works were developed and generalized in numerous other publications 5. The aim of this contribution is, using the theory of impulsive differential equations, using the well-known results on the splitting index by Sacker 6 and by Palmer 7 on the Fredholm property of the problem of bounded solutions and using the theory of pseudoinverse matrices 5, 8, to investigate, in a relevant space, the existence of solutions bounded on the entire real axis of linear differential systems with impulsive action
We consider the problem of existence and construction of solutions bounded on the entire real axis of linear systems of ordinary differential equations with impulsive action at fixed points of time x Atxft, t / τi, 1
Summary
I − P X−1 τi γi s ds f s ds , t ≤ 0. Assume that the homogeneous system 2 is e-dichotomous on R and R− with projectors P and Q, respectively, and such that P Q QP Q In this case, the system 2 has r-parameter set of solutions bounded on R in the form 14. The nonhomogeneous impulsive system 1 has for arbitrary f t ∈ BC R \ {τi}I and γi ∈ Rn an r-parameter set of solutions bounded on R in the form x t, cr Xr t cr f γi t is the generalized Green operator 16 of the problem of finding bounded solutions of the impulsive system 1 with the property. If condition 11 is satisfied, the nonhomogeneous impulsive system 1 possesses a unique solution bounded on R in the form xt f γi t, where f γi t is the generalized Green operator 16 of the problem of finding bounded solutions of the impulsive system 1. The nonhomogeneous impulsive system 1 possesses a unique solution bounded on R for all f t ∈ BC R \ {τi}I and γi ∈ Rn
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