EXISTENCE OF MILD SOLUTION FOR -HILFER FRACTIONAL CAUCHY VALUE PROBLEM OF SOBOLEV TYPE

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.