Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators DD with 0∈kerD0∈kerD is shown for which the fact that a non-zero t↦u(t)t↦u(t) belongs to a cone and −Du−Du to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0Du=0 are given.