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Transactions of the American Mathematical Society

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Structure of the solution set of some first order differential equations of comparison type


Authors: T. G. Hallam and J. W. Heidel
Journal: Trans. Amer. Math. Soc. 160 (1971), 501-512
MSC: Primary 34.42
MathSciNet review: 0281995
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Abstract: The comparison principle is a powerful tool that has a wide variety of applications in ordinary differential equations. The results of this article describe the geometric structure of the solution space of some first order scalar differential equations that may arise in the comparison method. A quite general class of differential equations is found to have a similar solution set configuration as the differential equation of separable variable type. One of the main results establishes, under certain conditions, that there is a unique unbounded solution of the first order differential equation which exists on an interval of the form [ $ {t_0},\infty $). Furthermore, this unbounded solution separates the solutions that are bounded on [ $ {t_0},\infty $) from those that are not continuable to all $ t > {t_0}$.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0281995-2
Keywords: Comparison principle, separation of variables, asymptotic behavior, solution set configuration
Article copyright: © Copyright 1971 American Mathematical Society