Existence Results for Nonautonomous Impulsive Fractional Evolution Equations

Year 2018, Volume: 1 Issue: 3, 133 - 147, 14.11.2018

Dimplekumar Chalishajar , Duraisamy Senthil Raja Kulandhaivel Karthikeyan Ponnusamy Sundararajan

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

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