The authors study the existence of solutions to first-order differential inclusions of the form \(y'(t)\in F(t,y(t)) \) a.e. \(t\in [0,T],\) with initial (\(y(0)=a\)) or periodic (\(y(0)=y(T)\)) condition, where \(F: [0,T]\times \mathbb{R}\to \mathbb{R}\) is a closed, bounded and convex-valued multivalued map. They convert the problem to an appropriate fixed-point problem and use a fixed-point theorem for condensing maps combined with upper and lower solutions.