On The Structure Of The Set Of Bounded Solutions For A Type Of Non-autonomous Differential Equations

In this paper, we give some precise descriptions of the structure and behavior of bounded solutions for non-autonomous second-order equation:with compact base space Hand driving systemθin case f(θ_th,x)is strictly decreasing in x on some interval (a, b). The stability of the bounded solutions with respect to perturbations in f.In the second part of the paper, we will try to establish some analogous new results for (1.1) by developing some elementary techniques in [9],[10],[13] and [15],combining with the basic theory of pullback attractors for co-cycle dynamical systems .We also establish some stability results concerning the structure of bounded solutions.Throughout the paper, we will always assume the equation satisfies the following structure conditions:·(F1):g(x)≥δ> 0,and g(x) is Lipschitz continuous.·(F2):f(θ_th,x)is continious and strictly decreasing with respect to x.·(F3):(H,d) is a compact metric space.H is minimal,moreover,θ_tH=H,i.e (?)h,{θ_th|ι∈R}=H.In the third section we will describe an example of the above theorem and conclusions .At last, we summarize the results of the whole paper.