Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 339-365, 2017

Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa

Received October 31, 2015.   First published March 31, 2017.

Adel Mahmoud Gomaa, Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Universities Road, P.O. Box: 344 42353 Medina, Kingdom of Saudi Arabia, e-mail: mohameda59@yahoo.com

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} \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 $a<b$ and $d>0$, we let $C([-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d [a,b]\times C([-d,0],E)\rightarrow E$. 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.

Keywords: nonlinear differential equation; trichotomy; existence theorem

Classification (MSC 2010): 35F31, 34D09

DOI: 10.21136/CMJ.2017.0592-15

Full text available as PDF.


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