Mild solutions to a class of nonlinear second order evolution equations

Abstract

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form\begin{equation*}\begin{cases} u''(t)+A(u'(t))+B(u(t))\ni f(t), &t\in(0,T),\\u(0)=u_0, \quad u'(0)=g(u')\end{cases}\end{equation*} where $A\colon D(A)\subseteq X\rightarrow 2^{X}$ is an $m$-accretive operator on a Banach space $X,$ $B\colon X\rightarrow X$ is a Lipschitz mapping, $g\colon C([0,T];X)\to X$ is a function and $f\in L^1(0,T,X)$. We obtain sufficient conditions for this problem to have at least a mild solution.