Results of Semigroup of Linear Operator Generating a Quasilinear Equations of Evolution

Abstract

In this paper, results of $\omega$-order preserving partial contraction mapping generating a quasilinear equation of evolution were presented. In general, the study of quasilinear initial value problems is quite complicated. For the sake of simplicity we restricted this study to the mild solution of the initial value problem of a quasilinear equation of evolution. We show that if the problem has a unique mild solution $v\in C([0,T]: X)$ for every given $u\in C([0,T]:X)$, then it defines a mapping $u\to v=F(u)$ of $C([0,T]:X)$ into itself. We also show that under the suitable condition, there exists always a $T',\ 0<T'\leq T$ such that the restriction of the mapping $F$ to $C([0,T']:X)$ is a contraction which maps some ball of $C([0,t']:X)$ into itself by proving the existence of a local mild solution of the initial value problem.