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Various maximum and
monotonicity properties of some initial boundary value problems for classes of linear
second order hyperbolic partial differential operators in two independent variables are
established. For example, let M be such an operator in Cartesian coordinates (x,y)
and let T be a domain bounded by a characteristic curve of M with everywhere
negative slope, and segments OA and OB of the positive x-axis and the positive
y-axis, respectively; under certain restrictions on the coefficients of the operator M, if
Mu ≦ 0 in T, u = 0 on OA ∪ OB and ∂u∕∂y ≦ 0 on OA then u(x,y) ≦ 0 in
T.
Such maximum and monotonicity properties also have applications to ordinary
differential equations; the above mentioned maximum property yields a comparison
theorem on the distance between zeros of solutions to some ordinary differential
equations.
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