The paper deals with the first-order differential inclusion subject to the periodic conditions
and also with the first-order differential inclusion subject to the non-periodic conditions
Here, , is a Carathéodory multifunction, is continuous and does not vanish on the whole interval . Further, , are matrices with real elements such that , and either , , or , . The authors provide new sufficient conditions under which solutions of problem (1) or problem (2) exist. The results apply to differential inclusions that may have a right-hand side with a super-linear growth in its second variable and also apply to systems of first-order differential inclusions. The proofs are based on novel differential inequalities and the Leray-Schauder nonlinear alternative. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained.