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

Haihua Wang; Jie Zhao; Junhua Ku; Yanqiong Liu

Haihua Wang Department of Mathematics, Qiongtai Normal University, Haikou, Hainan 571100, P.R. China
Jie Zhao Department of Mathematics, Qiongtai Normal University, Haikou, Hainan 571100, P.R. China
Junhua Ku Department of Mathematics, Qiongtai Normal University, Haikou, Hainan 571100, P.R. China
Yanqiong Liu Department of Mathematics, Qiongtai Normal University, Haikou, Hainan 571100, P.R. China

Article ID: 2447

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Keywords: $(\alpha, \beta, k)$-resolvent family, $(k, \Psi)$-Hilfer Sobolev type fractional differential equations, Mild solution, existence

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.

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