| Title:
|
Existence theorems for nonlinear differential equations having trichotomy in Banach spaces (English) |
| Author:
|
Gomaa, Adel Mahmoud |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
339-365 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $a<b$ and $d>0$, 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. (English) |
| Keyword:
|
nonlinear differential equation |
| Keyword:
|
trichotomy |
| Keyword:
|
existence theorem |
| MSC:
|
34D09 |
| MSC:
|
35F31 |
| idZBL:
|
Zbl 06738523 |
| idMR:
|
MR3661045 |
| DOI:
|
10.21136/CMJ.2017.0592-15 |
| . |
| Date available:
|
2017-06-01T14:26:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146760 |
| . |
| Reference:
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