NONLINEAR PERTURBATIONS FOR LINEAR NONAUTONOMOUS IMPULSIVE DIFFERENTIAL EQUATIONS AND NONUNIFORM (<i>H,K,µ,ν</i>)-DICHOTOMY

Jimin Zhang; Liu Yang; Meng Fan; Ming Chen

doi:10.11948/2018.1085

2018 Volume 8 Issue 4

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Jimin Zhang, Liu Yang, Meng Fan, Ming Chen. NONLINEAR PERTURBATIONS FOR LINEAR NONAUTONOMOUS IMPULSIVE DIFFERENTIAL EQUATIONS AND NONUNIFORM (H,K,µ,ν)-DICHOTOMY[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1085-1107. doi: 10.11948/2018.1085

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Jimin Zhang, Liu Yang, Meng Fan, Ming Chen. NONLINEAR PERTURBATIONS FOR LINEAR NONAUTONOMOUS IMPULSIVE DIFFERENTIAL EQUATIONS AND NONUNIFORM (H,K,µ,ν)-DICHOTOMY[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1085-1107. doi: 10.11948/2018.1085

NONLINEAR PERTURBATIONS FOR LINEAR NONAUTONOMOUS IMPULSIVE DIFFERENTIAL EQUATIONS AND NONUNIFORM (H,K,µ,ν)-DICHOTOMY

1 School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang, 150080, China;
2 College of Automation, Harbin Engineering University, Harbin, Heilongjiang, 150001, China;
3 School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, China;
4 Department of Mathematics, Dalian Maritime University, Dalian, Liaoning 116026, China;
5 Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin, Heilongjiang, 150080, China

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Abstract

Abstract

We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation x'=A(t)x,t≠τ_i,∆x|_t=τi=B_ix(τ_i),i ∈ Z in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general dichotomy on R called the nonuniform (h,k,µ,ν)-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlinear perturbations, we establish a new version of the GrobmanHartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive differential equations x'=A(t)x+f(t,x), t≠τ_i,∆x|_t=τi=B_ix(τ_i) + g_i(x(τ_i)),i ∈ Z with the help of nonuniform (h,k,µ,ν)-dichotomies.
- Nonautonomous impulsive differential equations /
- topological equivalence /
- nonuniform(h,k,µ,ν)-dichotomy /
- invariant manifolds
MSC: 34D09;37D10;34A37

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References

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