The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses. (English) Zbl 1514.34109
Summary: In this article, we study the existence of mild solutions for a class of Hilfer fractional evolution equations with non-instantaneous impulses in ordered Banach spaces. The definition of mild solutions for our problem was given based on a \(C_0\)-semigroup \(W(\cdot)\) generated by the operator \(-A\) and probability density function. By means of monotone iterative technique and the method of lower and upper, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained under the situation that the corresponding \(C_0\)-semigroup \(W(\cdot)\) and non-instantaneous impulsive function \(\gamma_k\) are compact, \(W(\cdot)\) is not compact and \(\gamma_k\) is compact, \(W(\cdot)\) and \(\gamma_k\) are not compact, respectively. At last, two examples are given to illustrate the abstract results.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 26A33 | Fractional derivatives and integrals |
| 34A08 | Fractional ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 35R12 | Impulsive partial differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
Hilfer evolution equations; non-instantaneous impulses; lower and upper solutions; monotone iterative method; mild solutionReferences:
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