Open Access Open Access  Restricted Access Subscription Access

On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Adel Mahmoud Gomaa

Abstract


Assume that $E$ is a Banach space, $B_{r}=\{x\in E:\Vert x\Vert \le r\}$ and $C([-d,0],B_{r})$ is the Banach space of continuous functions from $[-d,0]$ into $B_{r}$. Consider $f:\mathbf{R}^+\times E\to E$; $f^{d}:[0,T]\times C([-d,0],B_{r})\to E$; for each $t\in [0,T]$ the mapping $\theta_{t}\in C([-d,0],B_{r})$ is defined by $\theta_{t}x(s)= x(t+s)$, $s\in [-d,0]$ and let $A(t)$ be a linear operator from $E$ into itself. In this paper we give existence theorems for bounded weak and strong solutions of the nonlinear differential equation 26767 \dot{x}(t)=A(t)x+f(t,x),\qquad t\in \mathbf{R}^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 \dot{x}(t)=L(t)x(t)+f^{d}(t,\theta_{t}x),\qquad \text{if}\quad t\in [0,T] \qquad\qquad(\mathrm{Q}) 26767 has at least one weak solution where $L(t)$ is a linear operator from $E$ into $E$. Moreover, under suitable assumptions, the problem $(\mathrm{Q})$ has a solution. Furthermore under a generalization of the compactness assumptions, we show that $(\mathrm{Q})$ has a solution too.

Full Text:

PDF


DOI: http://dx.doi.org/10.7146/math.scand.a-15242

Refbacks

  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.
OK


ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library