Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈Em, where G is an exterior domain in En, and b is a continuous nonnegative valued function in G × Em. A solution v: G→Em is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in En satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G.
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Entrata in Redazione il 26 giugno 1975.
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Noussair, E.S., Swanson, C.A. Oscillation of nonlinear vector differential equations. Annali di Matematica 109, 305–315 (1976). https://doi.org/10.1007/BF02416966
- Differential Equation
- Partial Differential Equation
- Scalar Product
- Exterior Domain
- Differential Inequality