Haihua Wang, Jie Zhao, Junhua Ku , Yanqiong Liu

EXISTENCE OF MILD SOLUTION FOR $(k, \Psi)$-HILFER FRACTIONAL CAUCHY VALUE PROBLEM OF SOBOLEV TYPE

Introduction

Existence of mild solution for $(k, \psi)$-hilfer fractional cauchy value problem of sobolev type. Investigate mild solutions for (k,Ψ)-Hilfer fractional Cauchy problems (Sobolev type). This study generalizes the (α,β,k)-resolvent family, deriving new existence theorems.

Abstract

In the context of solving $(k, \Psi)$-Hilfer fractional differential equations with Sobolev type, we initially explore a more generalized version of the $(\alpha, \beta, k)$-resolvent family. Subsequently, we present various properties associated with this resolvent family. Specific instances of this resolvent family, such as the $C_0$ semigroup, sine family, cosine family and others, have been previously discussed in other academic papers. By combining this resolvent family with the $(k, \Psi)$-Hilfer fractional derivative, we examine the existence of mild solutions to $(k, \Psi)$-Hilfer Sobolev type fractional evolution equations, without requiring the existence of the inverse of $E$. Ultimately, two existence theorems are derived. Received: June 2, 2024Accepted: July 26, 2024

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