Existence of extremal mild solutions for the initial value problem of evolution equations with non-instantaneous impulses. (English) Zbl 1386.34112
Summary: In this article, we are concerned with the initial value problem of a class of evolution equations with non-instantaneous impulses in an ordered Banach spaces \(E\). By introducing a new concept of lower and upper mild solutions, we construct a new monotone iterative method for the initial value problem of evolution equations with non-instantaneous impulses and obtain the existence of extremal mild solutions between lower and upper mild solutions for the problem under the situation that the associated semigroup is compact and equicontinuous, respectively. An example is also given to illustrate the feasibility of our abstract results.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34K30 | Functional-differential equations in abstract spaces |
| 35R12 | Impulsive partial differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
evolution equation; non-instantaneous impulses; lower and upper mild solutions; monotone iterative technique; mild solutionReferences:
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