Existence for a second-order differential equation in a Banach space governed by an m-accretive operator

Abstract

In the framework of uniformly smooth Banach spaces, we derive the existence and uniqueness of bounded solutions for the general differential equation (inclusion) $p\left(t\right)u^{″}\left(t\right)+q\left(t\right)u^{\prime }\left(t\right)\in Au\left(t\right)+f\left(t\right)$, almost everywhere on $R_{+}=[0,\infty )$, with the initial condition $u\left(0\right)=x\in \overline{D\left(A\right)}$. Here, A is a nonlinear m-accretive operator with $0\in R\left(A\right)$, $f:R_{+}\to X$ is a given suitable function, and $p,q$ are continuous functions. By developing new methods, we extend several previously known results in the literature, including the works of Poffald-Reich 1986 and Moroşanu 2014, and prove the existence of solutions to the aforementioned differential equation for the first time in Banach spaces. We apply our results to investigate the weak and strong $L^p$-valued solutions for certain wave equations on bounded domains. Most of the results are new, even for Hilbert spaces.