Conformable Fractional Cosine Families of Operators

Abstract

In this paper we are concerned with the problem \begin{eqnarray*}\begin{cases} u^{(\alpha)}(t)=Au(t)+f(t,u(t))& t\in [0,T]\\ u(0)=u_0, D^{\alpha}u(0)=u_1\end{cases}\end{eqnarray*} \begin{eqnarray*} \begin{cases} u^{(\alpha)}(t)=Au(t)+f(t,u(t))& t\in [0,T]\\ u(0)=u_0, D^{\alpha}u(0)=u_1 \end{cases} \label{pb1} \end{eqnarray*} Where $\alpha\in (1,2]$, and we use the conformable derivative. We give the notion of $\alpha$-Cosine families and proveded the existence and uniqueness of the problem 0.1.

Keywords

• $\alpha$-cosine families
• conformable derivative
• mild solution