Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Czechoslovak Mathematical Journal

Adel Mahmoud Gomaa¹ 

关键词: nonlinear differential equation;    trichotomy;    existence theorem;   

DOI  :  10.21136/CMJ.2017.0592-15

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  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} \tag{P}where $\{\mathcal{L}(t)t\in\mathbb{R}\}$ is a family of linear operators from a Banach space $E$ into itself and $f \mathbb{R}\times E\to E$. 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 $f^d[a,b]\times C([-d,0],E)\rightarrow E$. Let $\widehat{\mathcal{L}}[a,b]\to L(E)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in[a,b]$ define $\tau_tx(s)=x(t+s)$ 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_tx)\quad\text{if }t\in[a,b], \tag{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.

【 授权许可】

Unknown