IJPAM: Volume 78, No. 5 (2012)

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF
SEMILINEAR EQUATIONS WITH ANALYTIC
SEMIGROUPS IN BANACH SPACES

Richard Olu Awonusika
Department of Mathematics and Industrial Mathematics
Adekunle Ajasin University
P.M.B. 001, Akungba Akoko, Ondo State, NIGERIA


Abstract. This paper studies the existence and uniqueness of mild solutions of the semilinear equation \begin{equation*}
\begin{split}
\frac{du(t)}{dt} &=Au(t)+f(t,u(t)), \hspace{0.5cm} t_{0}<t<t_{1}\\
u(t_{0}) &=x_{0},
\end{split}\end{equation*} where the linear operator $A$ is the infinitesimal generator of an analytic semigroup $T(t)$ satisfying the exponential stability and the function $f(t,u)$ is locally Hölder continuous in $t$ and locally Lipschitz continuous in $u.$ The basic tool in this paper is the use of fractional power of operators.

Received: May 8, 2012

AMS Subject Classification: 35B08, 35B09, 35B10, 35B15, 35B65

Key Words and Phrases: semilinear equations in Banach spaces, infinitesimal generator of analytic semigroup, fractional powers

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 5