On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

Adel Mahmoud Gomaa. "On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces." Commentationes Mathematicae 47.2 (2007): null. <http://eudml.org/doc/291525>.

@article{AdelMahmoudGomaa2007,
abstract = {In the present work we give an existence theorem for bounded weak solution of the differential equation \[ \dot\{x\}(t) = A(t)x(t) + f (t, x(t)),\quad t \ge 0 \] where $\lbrace A(t) : t \in I\mathbb \{R\}^+ \rbrace $ is a family of linear operators from a Banach space $E$ into itself, $B_r = \lbrace x \in E : \Vert x\Vert \le r\rbrace $ and $f \colon \mathbb \{R\}^+ \times B_r \rightarrow E$ is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay \[ \dot\{x\}(t) = \hat\{A\}(t) x(t) + f^d (t, θ\_t x)\quad \text\{if\}\ t \in [0, T], \] where $T, d 0$, $C_\{B_r\} ([-d, 0])$ is the Banach space of continuous functions from $[-d, 0]$ into $B_r$, $f_d\colon [0, T] \times C_\{B_r\} ([-d, 0]) \rightarrow E$ weakly-weakly continuous function, $\hat\{A\}(t)\colon [0,T] \rightarrow L(E)$ is strongly measurable and Bochner integrable operator on $[0,T]$ and $θ_t x(s) = x(t + s)$ for all $s \in [-d, 0]$.},
author = {Adel Mahmoud Gomaa},
journal = {Commentationes Mathematicae},
keywords = {Nonlinear differential equations; weak solutions; measures of noncompactness; delay},
language = {eng},
number = {2},
pages = {null},
title = {On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces},
url = {http://eudml.org/doc/291525},
volume = {47},
year = {2007},
}

TY - JOUR
AU - Adel Mahmoud Gomaa
TI - On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces
JO - Commentationes Mathematicae
PY - 2007
VL - 47
IS - 2
SP - null
AB - In the present work we give an existence theorem for bounded weak solution of the differential equation \[ \dot{x}(t) = A(t)x(t) + f (t, x(t)),\quad t \ge 0 \] where $\lbrace A(t) : t \in I\mathbb {R}^+ \rbrace $ is a family of linear operators from a Banach space $E$ into itself, $B_r = \lbrace x \in E : \Vert x\Vert \le r\rbrace $ and $f \colon \mathbb {R}^+ \times B_r \rightarrow E$ is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay \[ \dot{x}(t) = \hat{A}(t) x(t) + f^d (t, θ_t x)\quad \text{if}\ t \in [0, T], \] where $T, d 0$, $C_{B_r} ([-d, 0])$ is the Banach space of continuous functions from $[-d, 0]$ into $B_r$, $f_d\colon [0, T] \times C_{B_r} ([-d, 0]) \rightarrow E$ weakly-weakly continuous function, $\hat{A}(t)\colon [0,T] \rightarrow L(E)$ is strongly measurable and Bochner integrable operator on $[0,T]$ and $θ_t x(s) = x(t + s)$ for all $s \in [-d, 0]$.
LA - eng
KW - Nonlinear differential equations; weak solutions; measures of noncompactness; delay
UR - http://eudml.org/doc/291525
ER -