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Oscillation of nonlinear vector differential equations

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Summary

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.

References

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    Ju. I. Domšlak,On the oscillation of solutions of vector differential equations, Dokl. Akad. Nauk SSSR,193 (1970), pp. 21–23; Soviet. Math. Dokl.,11 (1970), pp. 839–841.

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    E. S. Noussair -C. A. Swanson,Oscillation theorems for vector differential equations, Utilitas Math.,1 (1972), pp. 97–109.

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    E. S. Noussair - C. A. Swanson,Oscillation theory for semilinear Schrödinger equations and inequalities, Proc. Roy. Soc. Edinburgh, Sect. A (to appear).

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    G. O. Okikiolu,Aspects of the Theory of Bounded Integral Operators in L D-Spaces, Academic Press, London-New York, 1971.

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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

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Scalar Product
  • Exterior Domain
  • Differential Inequality