In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value problem for impulsive parabolic equations is illustrated to our theory results.
In this paper we consider the nonlinear difference equation with several delays (axn+1 + bxn) k − (cxn) k + ∑m i=1 pi(n)x k n−σi = 0 where a, b, c ∈ (0, ∞), k = q/r, q, r are positive odd integers, m, σi are positive integers, {pi(n)}, i = 1, 2, . . . , m, is a real sequence with pi(n) ≥ 0 for all large n, and lim inf n→∞ pi(n) = pi < ∞, i = 1, 2, . . . , m. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.