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Novel method for generalized stability analysis of nonlinear impulsive evolution equations. (English) Zbl 1272.34083
The authors basically deal with Cauchy problems for nonlinear impulsive evolution equations \left\{\begin{aligned} & u'(t)+(A+B)u(t)=0,\quad t>0,\;t\neq\tau_k,\\ &u(0)=u_0,\\ &\Delta u(t)=I_k(u(t)),\quad t=\tau_k, \end{aligned}\right. where $$A: D(A)\subset X\to X$$ is a linear unbounded operator in a Banach space $$X$$, $$B: X \to X$$ is a nonlinear operator and $$u_0\in X$$. Moreover, the impulsive time sequence $$\{\tau_k\}_{k=0}^{\infty}$$ satisfies $$0=\tau_0<\tau_1<\dotsb<\tau_k<\dotsb$$, $$\lim\limits_{k\to\infty}\tau_k=+\infty$$; $$\Delta u(\tau_k)=u(\tau_k^+)-u(\tau_k^-)$$, $$I_k:X\to X$$, $$u(\tau_k^+)=\lim\limits_{h\to 0^+}u(\tau_k+h)$$ and $$u(\tau_k^-)=\lim\limits_{h\to 0^-}u(\tau_k+h)$$.
Using Banach’s contraction principle, the authors establish the existence and uniqueness of a generalized solution $$u:[0,+\infty)\to X$$ to the above system which tends to zero at an appropriate decay rate $$w(t)^{-1}$$ as $$t\to\infty$$. Some stability results of the above system are also proved by using Schauder’s fixed point theorem by means of a growth restriction for the nonlinear perturbation and the impulsive term. Moreover, stable manifolds for the associated singular perturbation problems with impulses are considered.
The reviewer has two comments on that paper. Firstly, $$PL^{\infty}(0,\infty;X)$$ seems not to be a Banach space as it is stated on p. 1213. Secondly, the statement of the compactness criterion on p. 1218 is not clear.
##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34D20 Stability of solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses 34E15 Singular perturbations for ordinary differential equations
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