Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

summary:We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation \dot {x}(t)=\mathcal {L}( t)x(t)+f(t,x(t)),\quad t\in \mathbb {R}\leqno {\rm (P)} where $\{\mathcal {L}(t)\colon t\in \mathbb {R}\}$ is a family of linear operators from a Banach space $E$ into itself and $f\colon \mathbb {R}\times E\to E$. By $L(E)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a0$, we let $C([-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^{d}\colon [a,b]\times C([-d,0],E)\rightarrow E$. Let $\widehat {\mathcal {L}}\colon [a,b]\to L(E)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define $\tau _{t}x(s)=x(t+s)$ for each $s \in [-d,0]$. We prove that, under certain conditions, the differential equation with delay \dot {x}(t)=\widehat {\mathcal {L}}(t)x(t)+f^{d}(t,\tau _{t}x)\quad \text {if }t\in [a,b],\leqno {\rm (Q)} has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too