Existence Results for Nonautonomous Impulsive Fractional Evolution Equations

Abstract

\noindent {\bf ABSTRACT}
\end{center}
\par In this paper, we investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations
\begin{align*}
\begin{cases}
\frac{d^q u(t)}{dt^q} &\quad =~~ -A(t)u(t)+f(t,(K_1 u)(t),(K_2 u)(t),\dots,(K_n u)(t),t \in I=[0,T] \\
\Delta y|_{t=t_k} &\quad =~~ I_k(y(t_k^-)),t = t_k, k = 1,2,\dots,m, \\
u(0) &\quad =~~ A^{-1}(0)g(u)+u_0
\end{cases}
\end{align*}
in Banach spaces, where $T>0, 0<q<1.$ New results are obtained by using Sadovskii's fixed point theorem and the Banach contraction mapping principle. An example is given to illustrate the theory.

Keywords

• Impulsive Fractional Evolution,Nonautonomous,Measure of noncompactness

References

1. Dr. V.S. GandhiDept. of MathematicsMiddlesex University, London, United Kingdom (UK)Email: [e-mail]
2. Dr. Nita ShahProfessor, Dept of Mathematics Gujarat University Ahmedabad, Gujarat, India
3. Dr. Ravi Chandran Dept. of Mathematics Tamil NaduEmail: [e-mail]