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

# Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

#### 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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