Abstract
By constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, some existence theorems of mild -periodic ( -quasi) solutions for abstract impulsive evolution equations are obtained in ordered Banach spaces. These results partially generalize and extend the relevant results in ordinary differential equations and partial differential equations.
Highlights
Introduction and Main ResultImpulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants
In view of multiple applications of the impulsive differential equations, it is necessary to develop the methods for their solvability
The monotone iterative technique of Lakshmikantham et al see 1–3 is such a method which can be applied in practice
Summary
Impulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants. I T t t ≥ 0 is a compact semigroup, ii K is regular in X and T t is continuous in operator norm for t > 0, they built a mixed monotone iterative method for the PBVP 1.5 , and they proved that, if the PBVP 1.5 has coupled lower and upper quasisolutions i.e., L ≡ 0 and without impulse in 1.2 and 1.3 v0 and w0 with v0 ≤ w0, nonlinear term f satisfies one of the following conditions: F1 f : J × X × X → X is mixed monotone, F2 There exists a constant M1 > 0 such that f t, u2, w − f t, u1, w ≥ −M1 u2 − u1 , ∀t ∈ J, v0 t ≤ u1 ≤ u2 ≤ w0 t , v0 t ≤ w ≤ w0 t , 1.6 and f t, u, v is nonincreasing on v.
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