# Study of fractional evolution equations involving Hilfer fractional derivative of order $1<\gamma<2$ and type $0 \leq \delta \leq 1$

## TL;DR

This paper explores fractional evolution equations with Hilfer derivatives of order between 1 and 2, introducing new fractional cosine operator functions to define mild solutions in Banach spaces.

## Contribution

It introduces a new family of fractional cosine operator functions for Hilfer derivatives and establishes a framework for mild solutions of semilinear evolution equations.

## Key findings

- Defined a family of fractional cosine operator functions.
- Provided properties and applications of these functions.
- Presented an example illustrating the theory.

## Abstract

In this paper we investigate fractional differential equations with Hilfer fractional derivative of order $1<\gamma<2$ and type $\delta \in [0,1]$ in a Banach space. We introduce a family of general fractional cosine operator functions of order $1<\gamma<2$ and type $\delta \in [0,1]$ and discuss their properties to give a suitable definition of mild solution of a semilinear evolution equation. In last section an example is presented.