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))& tin [0,T] u(0)=u_0, D^{alpha}u(0)=u_1end{cases}end{eqnarray*} begin{eqnarray*} begin{cases} u^{(alpha)}(t)=Au(t)+f(t,u(t))& tin [0,T] u(0)=u_0, D^{alpha}u(0)=u_1 end{cases} label{pb1} end{eqnarray*} Where $alphain (1,2]$, 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.