Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth

We establish some new sufficient conditions under which periodic and non-periodic solutions exist for systems of nonlinear, first-order differential inclusions. Our new results are significant for two main reasons: they apply to differential inclusions that may have a right-hand side that grows super-linearly in the second variable and thus our ideas extend known linear-growth results; and our results also apply to systems of first-order differential inclusions, thus our work extends previous research on scalar-valued inclusions and may lead to a deeper understanding of higher-order differential inclusions. Our approach is based on novel differential inequalities and the well-known nonlinear alternative of Leray-Schauder. Some new results for ordinary differential equations with Caratheodory single-valued right-hand sides are also obtained. (C) 2008 Elsevier Inc. All rights reserved.