Abstract
Abstract In this paper, a nonlinear differential equation x ′ = A ( t , x ) + f ( t ) is considered. Some new sufficient conditions for the existence of a bounded solution and an asymptotically almost periodic solution, which generalize and improve the previously known results, are established by using a dissipative-type condition for A ( t , x ) . Finally, an example is presented to illustrate the feasibility and effectiveness of the new results.
Highlights
1 Introduction In recent years, almost periodic solutions and their various generalizations have attracted the attention of many researchers
The existence of a bounded solution and an asymptotically almost periodic solution are two important properties which have a close relation to the applications of neural networks, epidemiology, etc., so they have been widely studied
Medvedev [ ] gave a sufficient condition to guarantee the existence of a bounded solution of the following equation: x = A(t, x) + f (t), ( . )
Summary
Almost periodic solutions and their various generalizations have attracted the attention of many researchers (see [ – ] and the references therein). Thanh and Nguyen Truong [ ] considered the following difference equation: x(n + ) = Ax(n) + f (n), n ∈ N, where N is a natural number and A is a bounded linear operator on a Banach space, the sequences {x(n)}n∈N are totally ergodic, σ := σ (A) ∩ is countable and the sequence {f (n)}n∈N is asymptotically almost periodic, the sequence {x(n)}n∈N is asymptotically almost periodic. F (t) ∈ C(R+, Rn) is an asymptotically almost periodic function if and only if for any > , there exist positive numbers L( ) and T( ) such that any interval of length L( ) contains an ω such that when t ≥ T( ), f (t + ω) – f (t)
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