Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 339–365

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

Article

## Abstract

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
where {ℒ(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × EE. 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 fd: [a, b] × C([−d, 0],E) → E. Let $$\hat {\mathcal{L}}:[a,b] \to L(E)$$ be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τtx(s) = x(t + s) for each s ∈ [−d, 0]. We prove that, under certain conditions, the differential equation with delay
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(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

35F31 34D09

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