Existence theorems for nonlinear differential equations having trichotomy in Banach spaces. (English) Zbl 1458.34037

Gomaa, Adel Mahmoud

Czech. Math. J. 67, No. 2, 339-365 (2017).

Summary: We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation \[ \dot{x}(t)=\mathcal{L}(t)x(t)+f(t,x(t)),\quad t\in\mathbb{R}\tag{P} \] 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 \(\tau_{t}x(s)=x(t+s)\) for each \(s\in [-d,0]\). We prove that, under certain conditions, the differential equation with delay \[ \dot{x}(t)=\widehat{\mathcal{L}}(t)x(t)+f^{d}(t,\tau_{t}x)\quad\text{if }t\in [a,b],\tag{Q} \] 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.

MSC:

34A34	Nonlinear ordinary differential equations and systems
34D09	Dichotomy, trichotomy of solutions to ordinary differential equations
34G20	Nonlinear differential equations in abstract spaces
35F31	Initial-boundary value problems for nonlinear first-order PDEs

Keywords:

nonlinear differential equation; trichotomy; existence theorem