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

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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 \(C_0\)-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.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (12061062).

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Haide Gou & Tianxiang Wang

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  1. Haide Gou
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  2. Tianxiang Wang
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Correspondence to Haide Gou.

Additional information

Communicated by Rahul Roy.

Supported by the National Natural Science Foundation of China (Grant No. 12061062).

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Gou, H., Wang, T. The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses. Indian J Pure Appl Math 54, 499–523 (2023). https://doi.org/10.1007/s13226-022-00271-4

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