In the context of solving -Hilfer fractional differential equations with Sobolev type, we initially explore a more generalized version of the -resolvent family. Subsequently, we present various properties associated with this resolvent family. Specific instances of this resolvent family, such as the C₀ semigroup, sine family, cosine family and others, have been previously discussed in other academic papers. By combining this resolvent family with the -Hilfer fractional derivative, we examine the existence of mild solutions to -Hilfer Sobolev type fractional evolution equations, without requiring the existence of the inverse of E. Ultimately, two existence theorems are derived.

-resolvent family, -Hilfer Sobolev type fractional differential equations, Mild solution, existence.

Received: June 2, 2024; Accepted: July 26, 2024; Published: August 9, 2024

How to cite this article: Haihua Wang, Jie Zhao, Junhua Ku and Yanqiong Liu, Existence of mild solution for -Hilfer fractional Cauchy value problem of Sobolev type, Advances in Differential Equations and Control Processes 31(4) (2024), 439-472. https://doi.org/10.17654/0974324324024

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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