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Convergence of solutions of nonlinear differential equations

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Summary

Solutions of\(\dot x = f\left( {t,x_t } \right)\) are said to converge if every pair of solutions x(t), y(t) satisfy x(t) − y(t) →0 as t → ∞. An invariance principle of LaSalle is used to determine conditions under which the solutions of\(\dot x = F\left( {t,x} \right) + G\left( {t,x} \right) + e\left( t \right)\) converge. In certain cases the approach used does not require boundedness of solutions as has been required in most previous results on convergence of solutions. The results of this investigation are applied to a number of nonlinear second order differential equations. Sufficient conditions are also found for the convergence of solutions of certain functional differential equations.

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Entrata in Redazione il 10 febbraio 1976.

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Swick, K.E. Convergence of solutions of nonlinear differential equations. Annali di Matematica 114, 1–26 (1977) doi:10.1007/BF02413777

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Keywords

  • Differential Equation
  • Nonlinear Differential Equation
  • Functional Differential Equation
  • Order Differential Equation
  • Invariance Principle