Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 339-365
  • ISSN: 0011-4642

Abstract

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We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x ˙ ( t ) = ℒ ( t ) x ( t ) + f ( t , x ( t ) ) , t ∈ ℝ ( P ) where { ℒ ( t ) : t ∈ ℝ } is a family of linear operators from a Banach space E into itself and f : ℝ × E → 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 : [ a , b ] × C ( [ - d , 0 ] , E ) → E . Let ℒ ^ : [ a , b ] → L ( E ) be a strongly measurable and Bochner integrable operator on [ a , b ] and for t ∈ [ a , b ] define τ t x ( s ) = x ( t + s ) for each s ∈ [ - d , 0 ] . We prove that, under certain conditions, the differential equation with delay x ˙ ( t ) = ℒ ^ ( t ) x ( t ) + f d ( t , τ t x ) if t ∈ [ a , b ] , ( 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.

How to cite

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Gomaa, Adel Mahmoud. "Existence theorems for nonlinear differential equations having trichotomy in Banach spaces." Czechoslovak Mathematical Journal 67.2 (2017): 339-365. <http://eudml.org/doc/288200>.

@article{Gomaa2017,
abstract = {We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation \[ \dot\{x\}(t)=\mathcal \{L\}( t)x(t)+f(t,x(t)),\quad t\in \mathbb \{R\}\qquad \{\rm (P)\} \] where $\lbrace \mathcal \{L\}(t)\colon t\in \mathbb \{R\}\rbrace $ is a family of linear operators from a Banach space $E$ into itself and $f\colon \mathbb \{R\}\times E\rightarrow E$. 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 $\widehat\{\mathcal \{L\}\}\colon [a,b]\rightarrow L(E)$ 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 \[ \dot\{x\}(t)=\widehat\{\mathcal \{L\}\}(t)x(t)+f^\{d\}(t,\tau \_\{t\}x)\quad \text\{if \}t\in [a,b],\qquad \{\rm (Q)\} \] 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.},
author = {Gomaa, Adel Mahmoud},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonlinear differential equation; trichotomy; existence theorem},
language = {eng},
number = {2},
pages = {339-365},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence theorems for nonlinear differential equations having trichotomy in Banach spaces},
url = {http://eudml.org/doc/288200},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Gomaa, Adel Mahmoud
TI - Existence theorems for nonlinear differential equations having trichotomy in Banach spaces
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 339
EP - 365
AB - We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation \[ \dot{x}(t)=\mathcal {L}( t)x(t)+f(t,x(t)),\quad t\in \mathbb {R}\qquad {\rm (P)} \] where $\lbrace \mathcal {L}(t)\colon t\in \mathbb {R}\rbrace $ is a family of linear operators from a Banach space $E$ into itself and $f\colon \mathbb {R}\times E\rightarrow E$. 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 $\widehat{\mathcal {L}}\colon [a,b]\rightarrow L(E)$ 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 \[ \dot{x}(t)=\widehat{\mathcal {L}}(t)x(t)+f^{d}(t,\tau _{t}x)\quad \text{if }t\in [a,b],\qquad {\rm (Q)} \] 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.
LA - eng
KW - nonlinear differential equation; trichotomy; existence theorem
UR - http://eudml.org/doc/288200
ER -

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