The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses

Haide Gou () and Tianxiang Wang ()
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Haide Gou: Northwest Normal University
Tianxiang Wang: Northwest Normal University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 499-523

Abstract: Abstract 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 $$C_0$$ C 0 -semigroup $$W(\cdot )$$ W ( · ) generated by the operator $$-A$$ - 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 $$C_0$$ C 0 -semigroup $$W(\cdot )$$ W ( · ) and non-instantaneous impulsive function $$\gamma _k$$ γ k are compact, $$W(\cdot )$$ W ( · ) is not compact and $$\gamma _k$$ γ k is compact, $$W(\cdot )$$ W ( · ) and $$\gamma _k$$ γ k are not compact, respectively. At last, two examples are given to illustrate the abstract results.

Keywords: Hilfer evolution equations; Non-instantaneous impulses; Lower and upper solutions; Monotone iterative method; Mild solution; 26A33; 34G20; 35R12; 47D06 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00271-4

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