An Invariance Property of Solutions to Second Order Differential Inequalities in Ordered Banach Spaces

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Let B be a Banach space, K be a cone in B and $I = [a,b]$. Conditions are imposed on ${\bf f}(x,{\bf u},{\bf u}')$, ${\bf f}:I \times B \times B \to B$ such that the following invariance property holds: ${\bf u}'' + {\bf f}(x,{\bf u},{\bf u}') \in - K$, ${\bf u}(x_1 ) \in K$, ${\bf u}(x_2 ) \in K$, $a \leqq x_1 < x_2 \leqq b$ implies ${\bf u}(x) \in K$ for all $x \in [x_1 ,x_2 ]$. From this property, comparison theorems for solutions to differential inequalities are obtained. These comparison results give sufficient conditions for $C^2 $ functions to be subfunctions with respect to two point boundary value problems. These results extend earlier results in $R^2 $ by Heimes to more general types of partial orderings in both infinite and finite dimensional spaces. The approach taken in this paper relaxes certain coupling restrictions present in earlier results of this type.