The asymptotic behavior of a class of nonlinear differential equations of second order
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- by Jing Cheng Tong PDF
- Proc. Amer. Math. Soc. 84 (1982), 235-236 Request permission
Abstract:
Let $u + f(t,u) = 0$ be a nonlinear differential equation. If there are two nonnegative continuous functions $\upsilon (t)$, $\varphi (t)$ for $t \geqslant 0$, and a continuous function $g(u)$ for $u \geqslant 0$, such that (i) $\smallint _1^\infty \upsilon (t)\varphi (t)\;dt < \infty$; (ii) for $u > 0$, $g(u)$ is positive and nondecreasing; (iii) $\left | {f(t,u)} \right | < \upsilon (t)\varphi (t)g(\left | u \right |/t)$ for $t \geqslant 1$, $- \infty < u < \infty$, then the equation has solutions asymptotic to $a + bt$, where $a$, $b$ are constants and $b \ne 0$. Our result generalizes a theorem of D. S. Cohen [3].References
- Richard Bellman, Stability theory of differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0061235
- I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94 (English, with Russian summary). MR 79154, DOI 10.1007/BF02022967
- Donald S. Cohen, The asymptotic behavior of a class of nonlinear differential equations, Proc. Amer. Math. Soc. 18 (1967), 607–609. MR 212289, DOI 10.1090/S0002-9939-1967-0212289-3
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 235-236
- MSC: Primary 34C11; Secondary 34E05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0637175-4
- MathSciNet review: 637175