Ввиду непрерывности решения
![](data:image/gif;base64,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)
как функции от
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оно должно бесконечное число раз проходить через нуль. А это означает, что существует последовательность значений
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, по которой это решение стремится к нулю. Это невозможно (по теореме Майергофера-Еругина).
Допустим, что
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,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)
. Так как
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--- решение уравнения , то
![](data:image/gif;base64,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)
в промежутке
![](data:image/gif;base64,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)
. Допустим, что
![](data:image/gif;base64,R0lGODlhQwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgA9AA8AhQAAAAAAAAAAHR4AHh0AHR0AAAAAMx0AMgAcSB0dSAAzWh0zWgAzWx1GbDMAADIAHTMAHTMAMzIAMjIyADNGbjNZfzNbgEgcAEgdHUceM1ozAFszAE9HHEhOM1l/WUhuf11/f113d0Rqe1V7amxGHX9ZM39ZSHd3XX9/XX9uSHtqRHd3ZmZ/ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbtQIBwSCwaj8ikUkkKOIdOATGyrA6p0yzSJfgMHUWDteoSF1MDYhp5GWaKknE1XqQID0Q8EgHAFDVHBEsFAC5PQg1FgkIkRQpIAVJFD0ZOAUlmQ2KUQxuWQ3p3SGBGmUSJSJkWACkEHaaIRA6rQqRFEROPRCSwAJxHbUJvQmKwv7WyqFcAvKVFjUitXilDgM1D0Mlft0OLRLDBSN60ZqYXyshftACw6L5FeMNGDo9v10JeovJGeg5OqP4iFUEnUEgBWAU+EdlApOAQBtjkDIEgIomJI+uIQEhCR6KRNfMgwkkCUpZJjyhdLMGyDEkQADs=)
не меняет знак. Тогда
![](data:image/gif;base64,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)
Проинтегрируем обе части по отрезку
![](data:image/gif;base64,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)
, где
![](data:image/gif;base64,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)
получим
![](data:image/gif;base64,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)
Произведем замену
![](data:image/gif;base64,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)
. Получим
![](data:image/gif;base64,R0lGODlhewA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgB2ADMAhgAAAAAAAB0AAAAAHR0AHRwcHB4AHh0AMgAAMx0dSAAdSAAcSAAzWh0zWgAzWx4zRx1GbDMAHTMAADIAHTIAMjMAMzQeSDMeRzQ0HTIyADMzWzNdXTNGbjNGbDNZfzNbgEgdAEkdHUgcAEgdHUceM0geNFozHVozAFszAEYzRkZGWVtbW0ZGblV7VV13d11/f0huf1l/bkRqe1V7amxGHW5GM39ZM39ZSG5GRnd3XX9uSG5/WX9/XWpqRHtqRG5uRHd3ZmpqamZ/ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2OjQGPkpOUg5GVmIeXmZydAJuemKChpI6jpY+nhBaorZ+uk6qCJLCosrWaiESDAgEfADS4lbfChJsRGMQAHMWSys2XtCkAFwAmg9WCOs3O3IqXCd4AAr+ez8WXIAGbu4KbwZxE66TnwpvThCG1AvTiiJfbDLFyxS9UPVwHO51Yt65cAAaDIMTylyuRjQADQimwBGAXPGuUEsIiJmFeJwKECgKYMOhASIqGiEn0RANBykELBkGcCNMYt4y8fp0YVGBYT5+JZnqSYFOQShkAHgBo8fKou0RDS5VkGKnkRgFNu1l9lSjn2EUiXd06YfZsorSD/5iZU9TWbUVKKJ0x3MgRkYiwdmMW4srQEIrCjhwAGHH3EODASCkpdRRQsN/HkK9SYknoMOEAihkRwzz4s+nTqFNz1SzIM+HQgz5KovAv0d/MliWJmOxoZ27HuAdTcklLkkQNvwvdDt7XGdBHNAIABrGRWN3gcG1hvY47eynr0Bn+io4RRi3v/cpKIvIcgIRCREg7MpAcJjHfjkQMKr5IbqPKzR2F3iE5MWYIb3lBh9+AcxkiAUTmrYfRIZxZYhJ8D0XUUmQCHqIObJK8B1xiH10ziIisdVhKBQDgN8hjFSoCoHsc9sTgICzWZIiOsXFHSFv4SaDUjZwkxJ8hLAqSIIshj/FGSFYAFEUIimRZJVI4hgC2myExEocIZw8cCYBLAdqIiJODlCPIVjNt1R4AKAwiz3NgTRnAV4CBSGQmlyCTYWlvVhJBJ7TVaB8iER7SQCb0OdgIlSmaGU4Hg9SAS5KTYGooRZesA5tSh8Gm5ll7ioKIPocMZFepRi3TSAmZsVoVc6KRSiskZwUCADs=)
Тогда
![](data:image/gif;base64,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)
Таким образом получаем
![](data:image/gif;base64,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)
Теперь пусть
![](data:image/gif;base64,R0lGODlhKQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABgAlAA4AhAAAAAAAAB0AHQAAHR0AMh0AMwAcSB0zWh1GbDIAHTNGbjNZfzNbgEgcAEceM1ozHVszAFozAFszHURqe0huf1V7al1/f2xGHW5GM3tqRH9uSH9/XXd3ZmaIiIBbM4iIZgVcICCOZGmeKJml5sMCCiq85SyfUKAz9DicusAp0is1hCZEilLyBJ9QXSmRMhRJx9LFempcSb/k6BJQAggAx7dwQqtHLkBgwCxKTs4wgOf5+v+AewB9gVdxhS85SiEAOw==)
. Учтем, что с заменой
![](data:image/gif;base64,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)
и получаем
![](data:image/gif;base64,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)
по условию теоремы. Это неравенство противоречиво, так как слева стоит конечная величина.
Рассмотрим общий случай, когда
![](data:image/gif;base64,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)
может менять знак. Тогда
![](data:image/gif;base64,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)
Так как
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,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)
, то с некоторого момента величина
![](data:image/gif;base64,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)
станет положительной и знак модуля можно будет опустить. Тогда получим
![](data:image/gif;base64,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)
Проинтегрируем обе части от
![](data:image/gif;base64,R0lGODlhEAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABwALAA0AhAAAAAAAAAAAMwAdSB0dSAAzWwAzWh1GbDIAMjMzWzNbgEgdHUgdAEceM1ozAFszAEhuf11/f113d2xGHX9uSH9/XXd3XWpqand3ZmaIiIBbM4iIZgECAwECAwECAwECAwUmICCO5BicwVAWwFKWxEsqsojUgFYywpuMjdEk0MOVgkYjUvZIAUIAOw==)
до
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAGAAkAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbDIAHTNGbjNbgEgcAEceM1szAFozAERqe0huf1V7al1/f2xGHXtqRH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUYICBOYimUTKAmYlOWz1u8ygsQwFIGQwyEADs=)
, где
![](data:image/gif;base64,R0lGODlhEAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABwALAA0AhAAAAAAAAAAAMwAdSB0dSAAzWwAzWh1GbDIAMjMzWzNbgEgdHUgdAEceM1ozAFszAEhuf11/f113d2xGHX9uSH9/XXd3XWpqand3ZmaIiIBbM4iIZgECAwECAwECAwECAwUmICCO5BicwVAWwFKWxEsqsojUgFYywpuMjdEk0MOVgkYjUvZIAUIAOw==)
--- значение, после которого
![](data:image/gif;base64,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)
становится положительным.
Сделаем замену
![](data:image/gif;base64,R0lGODlhOAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAyAA8AhQAAAAAAAB4AHh0AAB0AHRwcHAAAHQAAMx0AMh0dSAAcSAAdSAAzWgAzWx0zWh4zRx1GbDIAHTMAADMAHTMAMzIAMjMzWzNGbjNZfzNbgEgcAEceM1ozAFszAFozHU9HHEhOM1l/WVV7VV1/f113d0Rqe0huf1V7amxGHX9ZM3d3XX9/XX9uSHtqRG5uRHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwasQIBwSCwaj8ikMkkxNpfQKPGAFJSk2GRFichGWwMIIBDAADjFFuEqRHmhVEACgKJGiB1yIDPs4vWADUUBQ24AMXGGAAQAcUNoRRJ8WYJiRBqWG0MHH45Cd5FvAKAAfoxCDkNingAKRhKWQ4B6RTGPQ6eKcZ4aRxJvHJYFQ79TQ6QsR8lvpJoWQgN6sUKxAQZGgl4SAQsAA56QRhNsR1uiWkkCSMXnSNlYT+1FQQA7)
, получим
![](data:image/gif;base64,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)
Устремим
![](data:image/gif;base64,R0lGODlhKQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABgAlAA4AhAAAAAAAAB0AHQAAHR0AMh0AMwAcSB0zWh1GbDIAHTNGbjNZfzNbgEgcAEceM1ozHVszAFozAFszHURqe0huf1V7al1/f2xGHW5GM3tqRH9uSH9/XXd3ZmaIiIBbM4iIZgVcICCOZGmeKJml5sMCCiq85SyfUKAz9DicusAp0is1hCZEilLyBJ9QXSmRMhRJx9LFempcSb/k6BJQAggAx7dwQqtHLkBgwCxKTs4wgOf5+v+AewB9gVdxhS85SiEAOw==)
и учтем
![](data:image/gif;base64,R0lGODlhhwA9AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABwCDADIAhQAAAAAAAAAAHR0AAB0AHRwcHB4AHgAAMx0AMgAcSAAdSB0dSB0zWgAzWgAzWx4zRx1GbDIdADMAADIAHTMAHTQdNDIAMjMAMzQ0NDMzWzNdXTNGbjNZfzNbgEgdHUgcAEgdAEceM1ozAFozHVszAEhIW0ZGblV7VUhuf11/f113d0Rqe1V7amxGHW5GM39ZM2xZNW5GRn9ZSG5dXXtqRH9uSH9/XW5uRHd3XWZ3d2pqand3ZmaIiIBbM4iIZgECAwb/QIBwSCwaj8ikcslsOp/QqPTZCggAtEEg0Jl6v+CwtxJJiM/odBg1DAxr6rh8jjQPTfS8HswWHvaAgV9uDYKGh0g9W21ciI6PYQOQk4dbXZSYe5KZnHNZl52hYCJbi0IBhaKhVn1LIkOkXQ4AEEOtqo8SpkwERJsAE0MIuJM9UX9CNQNdxkKvxJO1Tj1XQ7+1PQXQkM9PEsgAAxzAWAAhABnbjqnepVsaugrh4Opg0k3d9U0bhm74dvqW9AKjqFQTRfKUfNhWpZqWRkVIGAzjQcoAekU+YBQk0d2WWUbI2Kl45B6YBVJ0JdEIrZU/AHCMBFPS0SNIWFEuLmGJK8aQ/2ZJWiBC2IRnEY9Ikyr1aCRpko1EPtyrYvIJu1NHhD4xCo1QkmHnYgK4ha8IAwAjwnAVVZARqDbVwgRAppVWU6Q3M0LVByIhERdD+IkR62VhQER1q0oBiKTKFr9DLiSRjEgsWSln0446KIBsj73JDKxAJFHxlJdgGCdR/WTY4Tl1FZZsEvu1GtNFBhKZ2WZIbde2z0gAIKOJ6s9EZt27JeFt8C8gAuRFYrgIRt7AhQx/bqh5kRb0gObTjpu7HKj0REjTVmS7+T3Vhbie0CoYDXPo3mR/P+WsEqdCkEBEPPMUMR1/i31BQRMWIPgFKWEYsIR7DkoBCgigLUHZERtWGB6Ff0Oc42EmyqA2oiglhHgiJ3GtSIyILnIiEWSvBQEAOw==)
Последнее неравенство противоречиво, что говорит о том, что не существует решения, которое не является неограниченно продолжимым вправо.
Метод функций Ляпунова дал довольно сильный и гибкий аппарат исследования устойчивости решений дифференциальных уравнений. Модификации этого используют сейчас и для выявления других свойств решений дифференциальных уравнений. Например, японский математик Окамура использовал идеи, сходные с идеями второго метода Ляпунова, для изучения продолжимости решений, а затем Йошизава применил этот метод для получения сведений об ограниченности решений.
Как известно, Теоремы Ляпунова дают возможность судить об устойчивости по знаку производной
![](data:image/gif;base64,R0lGODlhEAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAA8AhAAAAAAAAAAAHR0AHQAAMwAcSAAzWh0zWh1GbDMAADQeSDNbgEgdAEgcAEceM1ozAFszAEgzM1l/WUhuf11/f2xGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwUsICCOZCkK5qgFS+oyQdxWMUmQTWmo5eOOiF/iBxhMAEHTTZGqACK/wA/yCwEAOw==)
, где
![](data:image/gif;base64,R0lGODlhEAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwALAAwAhAAAAAAAAB0AHQAAMwAcSAAzWh0zWh1GbDMAADQeSDNbgEgdAEgcAEceM1ozAFszAEgzM1l/WV1/f0huf2xGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwUnICCOpLgEqAJQKDmQTFmMWemU+IGPyD4KE4AO90rsKACIDxBYPpYhADs=)
--- положительно определенная функция. Таким образом изучается неравенство
![](data:image/gif;base64,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)
. После работ русского ученого С.А. Чаплыгина началось широкое применение дифференциальных неравенств в теории дифференциальных уравнений. Развитие теории привело к сочетанию метода функций Ляпунова с методом дифференциальных неравенств: начали рассматривать функции Ляпунова в дифференциальных неравенствах вида
![](data:image/gif;base64,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)
что позволяет получить, в частности, интересные выводы относительно продолжимости и ограниченности решений. Остановимся кратко на этом вопросе .
Если рассмотреть систему
![](data:image/gif;base64,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)
то ее решение
![](data:image/gif;base64,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)
может быть ограниченным, иметь конечное время определения или существовать для всех
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAGAAkAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbDIAHTNGbjNbgEgcAEceM1szAFozAERqe0huf1V7al1/f2xGHXtqRH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUYICBOYimUTKAmYlOWz1u8ygsQwFIGQwyEADs=)
.
В неравенстве нас будут интересовать только его положительные решения. Сами неравенства могут быть двух типов:
а) неравенства, не имеющие ни одного положительного решения с конечным временем определения;
б) неравенства, не имеющие ни одного положительного неограниченного решения. Заметим, что в дальнейшем, если под
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWwAzWh1GbDMAADIAHTNZfzNGbjNbgEgcAEgdAFozAEhZf11/f0huf0Rqe113d2xGHXtqRH9/XX9uSHd3XWZ3d3d3ZmaIiIBbM4iIZgU6ICCOZAlcQqAGSwkFxugFhRwg5XyIDmECjoEo8RMFhrVfoAEsAoQiaMlR8pEgP8WIKmIAUgEpYAUIAQA7)
понимается некоторое множество, то через
![](data:image/gif;base64,R0lGODlhGQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAUAA8AhQAAAAAAAB0AAAAAHR0AMgAAMwAdSAAcSAAzWgAzWx1GbDMAADIAHTIAMjNZfzNGbjNbgEgcAEgdAFozAFtISEhZf0huf11/f113d0Rqe2xGHWxZRn9uSH9/XXtqRHd3XWZ3d3d3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZSQIBwSCwaj8iAAclcMpGNIYXoEQSuAcdwhJwEEtvAYagwjgJl4hnBjBSOkYFQc3UCFswAczE+BiBIEUxyg3BPQm9EE4dDDEOCQg9IVgGEQlhIQQA7)
обозначается дополнение этого множества в пространстве.
Приведем без доказательства несколько утверждений .
Теорема
Предположим, что
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWwAzWh1GbDMAADIAHTNZfzNGbjNbgEgcAEgdAFozAEhZf11/f0huf0Rqe113d2xGHXtqRH9/XX9uSHd3XWZ3d3d3ZmaIiIBbM4iIZgU6ICCOZAlcQqAGSwkFxugFhRwg5XyIDmECjoEo8RMFhrVfoAEsAoQiaMlR8pEgP8WIKmIAUgEpYAUIAQA7)
--- ограниченное множество пространство
![](data:image/gif;base64,R0lGODlhFQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAASAA0AhQAAAAAAAAAAHR0AAB0AMgAcSAAdSAAzWh1GbDMAADMzWzNZfzNbgEgdHUceM1ozAEgzM1szM0ZGRkhIW1tISFl/WUhuf11/f113d2xGHX9ZM25GM25GRn9uSH9/XWZ3d2aIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZDQIBwSCwaj0MCIIEgZgLQaIABkAAqxpCA2Cg2i4VjM2J8HMlLQzGxAGiQRimcaBY65sME3rglbuYPYUMZcwNSUV9GQQA7)
, содержащее начало координат, и что функция
![](data:image/gif;base64,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)
определена во всем множестве
![](data:image/gif;base64,R0lGODlhGQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAUAA8AhQAAAAAAAB0AAAAAHR0AMwAAMwAdSAAcSAAzWgAzWwAyWh1GbB1IbjMAADIAHTIAMjNZfzNGbjNbgEgcAEgdAFoyAFszAFtISEhZf0huf11/f1V3d0Rqe113d2xGHW5IHX9ZM2xZRn9uSH9/XXtqRHd3VWZ3d3d3ZmaIiIBbM4BdM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZVQIBwSCwaj8iAAclcMpGP4YVIEgSuAcgwhawEEkNV4DBcGEEBRjEVUDAnhSNlIPxcnYAGM8BskI8BEkgUTHSFRxNPQnFEFopDDkOJQhFIVgGGQlhIQQA7)
и при всех
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
. Допустим далее, что
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,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)
равномерно на каждом интервале изменения времени
![](data:image/gif;base64,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)
. Наконец, предположим, что
![](data:image/gif;base64,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)
, во всем
![](data:image/gif;base64,R0lGODlhGQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAUAA8AhQAAAAAAAB0AAAAAHR0AMwAAMwAdSAAcSAAzWgAzWwAyWh1GbB1IbjMAADIAHTIAMjNZfzNGbjNbgEgcAEgdAFoyAFszAFtISEhZf0huf11/f1V3d0Rqe113d2xGHW5IHX9ZM2xZRn9uSH9/XXtqRHd3VWZ3d3d3ZmaIiIBbM4BdM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZVQIBwSCwaj8iAAclcMpGP4YVIEgSuAcgwhawEEkNV4DBcGEEBRjEVUDAnhSNlIPxcnYAGM8BskI8BEkgUTHSFRxNPQnFEFopDDkOJQhFIVgGGQlhIQQA7)
и для
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
. Если неравенство не имеет ни одного положительного решения с конечным временем определения, то каждое решение
![](data:image/gif;base64,R0lGODlhHQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAYAA8AhQAAAAAAAB4AHh0AHQAAHR0AAAAAMx0AMgAdSAAcSB0dSAAzWgAzWx1GbDIAHTMAADMAHTMAMzIAMjQ0HTNGbjNbgEgcAEgdAEceM1ozAFszAF1/f113d0Rqe0huf1V7amxGHW5GM3d3XX9/XX9uSHtqRG5/WWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZwQIBwSCwaj0hApLhMFlMGI0ngJEqMFOGhKmwUB0MQMUAYBhAAB1ETaA+3YYBYCDV6h48KsVCMEtVED0Z6QiB+hUYPd0ITfUQWi0KCQ14KACFDHm8AGEYgAQxCFwFDGmNlRKFVEElXXABUia9DTbREQQA7)
системы неограниченно продолжаемо.
Для применения результатов такого рода часто полагают
![](data:image/gif;base64,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)
, то есть неравенство записывается в виде
![](data:image/gif;base64,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)
Лемма
Если
![](data:image/gif;base64,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)
, то неравенство , при непрерывности
![](data:image/gif;base64,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)
для всех
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
и положительности и непрерывности
![](data:image/gif;base64,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)
для
![](data:image/gif;base64,R0lGODlhKAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAjAAwAhAAAAAAAAB0AHR0AAAAAMwAcSAAdSAAzWh0zWh1GbDMAADQeSDNZfzNbgEgdAEgcAEceM1szAFozAEgzM1l/WV1/f0huf2xGHW5GM39/XX9uSHd3XWZ3d2aIiIBbM4iIZgVkICCOZGmeqBOsDXCtaCy+gUmQjxx7N6CUh5HnFNGJWqLCSGIERAJBlIA0HSWaopqJR/qJvFjRoDci+0YCC+CKfbEB3NGgDFiEtaWqqHoBTJoKAUgng0pZRhFmJx6GYEVhOjQjIQA7)
, не имеет ни одного положительного решения с конечным временем определения.
Лемма
Если
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,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)
, то неравенство не имеет ни одного положительного неограниченного при
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
решения.
Теорема
Пусть
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWwAzWh1GbDMAADIAHTNZfzNGbjNbgEgcAEgdAFozAEhZf11/f0huf0Rqe113d2xGHXtqRH9/XX9uSHd3XWZ3d3d3ZmaIiIBbM4iIZgU6ICCOZAlcQqAGSwkFxugFhRwg5XyIDmECjoEo8RMFhrVfoAEsAoQiaMlR8pEgP8WIKmIAUgEpYAUIAQA7)
и
![](data:image/gif;base64,R0lGODlhEAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwALAAwAhAAAAAAAAB0AHQAAMwAcSAAzWh0zWh1GbDMAADQeSDNbgEgdAEgcAEceM1ozAFszAEgzM1l/WV1/f0huf2xGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwUnICCOpLgEqAJQKDmQTFmMWemU+IGPyD4KE4AO90rsKACIDxBYPpYhADs=)
имеют тот же смысл, что и в теореме ,
![](data:image/gif;base64,R0lGODlhMQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAsAAwAhQAAAAAAAB4AHgAAHR0AAB0AHQAAMx0AMwAdSAAcSAAzWh0zWx1Ibh1GbDIdADMAADQdNDQeSDQ0NDNZfzNGbjNdgDNbgEgdAEgcAEkdHUgeNFszHVozAFoyAFszAEgzM0ZGblV7VV1/f0Rqe0huf113d25IHWxGHX9ZM25GM25GRntqRH9uSH9/XWZmVXd3XXd3ZmaIiIBbM4BdM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZwQIBwSCwaj8jkJcCsAEzMpHR6NBAv1GxWMURpv0kO2ChAngIDISFKbIyJpsGoyBpCAIghSfh4FzEBE0kBRwV7DINMiouLTkIqXUUgQlYRfkKASR1WRScAH5cAJwN7RHVChFxCpYShB4lCHmwAHqGhQQA7)
при
![](data:image/gif;base64,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)
равномерно по
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
и
![](data:image/gif;base64,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)
. Если неравенство
![](data:image/gif;base64,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)
не имеет ни одного положительного неограниченного при всех
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
решения, то система устойчива в смысле Лагранжа.