Замечание. Для автономной системы вместо
![](data:image/gif;base64,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)
используется функция
![](data:image/gif;base64,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)
.
Рассмотрим систему вида
![](data:image/gif;base64,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)
где
![](data:image/gif;base64,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)
определена и непрерывна на
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAhAAwAhQAAAAAAAAAAHR0AAB0AHQAAMwAdSB0dSQAcSB0dSAAzWgAzWx1GbDMAADMAMzMeRzNbgEgcAEgdAFszAFozAEYzRltuSEhuf113d11/f0Rqe2xGHX9ZM25GM39/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZmQIBwSCwaj8hJYMlAOinLgAJQKW4ETmeguDVOs8ZuEVJsGB3FA3Fw7BQ/hIvzU6QUwEMEEq0mRvRDElF3QxF4ZUdiQwROaAllX0MbhESAc0UTAhpFkUJmR2hODZQAbERRh0ZKqERBADs=)
, где
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--- некоторый промежуток прямой, а
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--- область
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-мерного пространства
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.
Определение. Будем говорить, что вектор-функция
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удовлетворяет на множестве
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локальному условию Липшица по
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, если для каждой точки
![](data:image/gif;base64,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)
найдется такая окрестность
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и постоянная Липшица
![](data:image/gif;base64,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)
, что для любой из двух точек
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
из этой окрестности выполняется неравенство
![](data:image/gif;base64,R0lGODlhsAAlAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgCqACAAhQAAAAAAAB4AHgAAHR0AHR0AAAAAMx0AMh0AMwAdSAAcSB0dSAAzWwAzWh1GbDMAADIAHTMAHTMAMzIAMjQ0HTNGbjNbgEgcAEgdAEceM1ozAFszAF1/f113d0huf0Rqe1V7amxGHX9ZM25GM39ZSHd3XX9/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsOp/QaDQAoEqv2OzVatUOqV2vEQwWm89eLrqMJpKr7fg6QK8f1Wd2ts4XvsNygYJwhGZ6cX+DioF4hoWIVYeGfHSLTZSVY35rcJiASp6blZ+WpVubeY9tiaatWI1ikpyjSQ+AEq5Htg5muG6oUnZfqrOyQhBCCUMGWgJiyCzKQsxZzsPETwNFxo60R7xEE0UouRpN5FAH16RDLEkhmuyxkdjUQ+ZFBFLwWfZD4ET0RWmAip0GeQAuABQliJWbAP4geBiyoY6FOwMmsgggDYA6KRjoRCxSkc7FMRkBbFR2Yp2RkCeTCPzFJeXKJwFscuxkjCAR/38A8ClpwI/Ig5hQhC4zohQJ0SIPChrVxiSEgnhEnu4TwmCYN6NFQgAFMFFJAV3bMCE5WmQs2SVnofKChbCIT5pE4qaldESvKFkffxa5uuSCroVPAgsRa4SwEsNQpW7ruPbOI8hXMP8l5lgIC3+ak1BAu9UI0NBIRkO9CCuvW5X+3HUiovpK7c2AUANANkRdBiYLjCh+otAI794Afi8JTuRja6hdi1y465EhEeZReAUfgZvIiZlFAOYsW8BeiADRVaWHAp7itpQAyg85n75L0edKNmKKDob+tSb+AeMQV0hEIIY4pR1hoBcIElIXFNzM4w1BCCRxggAfGPHAerVEAXUPOklYoyGHusSE3yvYdFPGRq8R4YsUL0LY4hAxQlHjiac8iOJXufTIBI7BpChhhD4WKVk3DdGjo5GuADmFkGkoyeSU2wAjYZI8UsmkkxBCqcWAWhrJJU5e7iFlmFOO6QSRX56JpphWRrlkkFm+mYual5S5YxAAOw==)
.
Введем обозначения.
Рассмотрим отношение
![](data:image/gif;base64,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)
.
Рассмотрим верхний (нижний) предел последнего отношения
![](data:image/gif;base64,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)
Этот предел будем называть производной функции в силу системы .
Теорема
Пусть функция
![](data:image/gif;base64,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)
определена, непрерывна и локально липшицева относительно
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAKAAgAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUcICCOJDAFwhiUgAEcLDQgrLjUwNPi9SoyARogBAA7)
на произведении
![](data:image/gif;base64,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)
.
Тогда для продолжимости всех решений системы на промежутке
![](data:image/gif;base64,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)
необходимо и достаточно, чтобы на множестве
![](data:image/gif;base64,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)
существовали две функции Ляпунова
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
, обладающие свойствами:
1)
![](data:image/gif;base64,R0lGODlhkwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgCPABMAhQAAAAAAAAAAHR4AHh0AHR0AAAAAMx0AMx0AMgAcSAAdSB0dSAAzWwAzWh0zWwAyWh1Ibh1GbDMAADIAHTMAHTMAMzIAMjQeSDQ0HTNZfzNGbjNdgDNbgEgdAEgcAEgeNEceM1ozAFszAFozHUgzM1tISFV7VUhuf11/f1V3d0Rqe113d1VmZlV3am5IHWxGHX9ZM25GM25ISH9ZSHd3VX9/XX9uSHd3XXtqRHd3ZmaIiIBbM4BdM4iIZgECAwECAwb/QIBwSCwaj8ikcslsOp/QpiBKrVan1qx2yAtstmBw9xsum8/otBrZCbi/Lvexsl6+5Mr2GxAPJOl1SXd+RQZEHUeGWQNPDEQwihJ2ikKISJRVjEcRTJFFDUMwc1s2TYRFHEMJS6BCokiAWaVIp0ipQqtCIUu7RQRULq1GIgEQR79DyEghwkm9RMpPwUkBCsfQRJxKJ0PEe0YBAioAMNVDCEe1RTuYkkraS9xC3l7p4uTmQuhKBZjsRO4ABLxU5BkSF0IcAQSnrtDCJAOTYAJg8AjCFwqHRDQiQV27ZNyMGXkxsUmBIxuHNATwb8hJJDaQiTxCksrLIimF9DPSUucQxUMXJBaxtsQDSnhG4iAFEC2aEaBLJhJVYpTjUgD5riUT8gIACYdCegKougQDyiYZhWQYMnWkV46nxJJVYpZjEYRM1gqZupLIBCL7lBhbACAGYCo8iObU1ReAhSF/zzEZXPgwFcXdjoAoAi8ckZ18AmRUlza0m9Oi7+IBYAN05m1DOmPRqSjO6CKl+6BOTaQPk81EKICJFQhJieDDiwMfYmPAOLtMFleRUFrJK+bOz0YvnoY49yLXi3h/Ml5N4+9mdJNB/yQIADs=)
;
2)
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,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)
равномерно относительно
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAGAAkAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbDIAHTNGbjNbgEgcAEceM1szAFozAERqe0huf1V7al1/f2xGHXtqRH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUYICBOYimUTKAmYlOWz1u8ygsQwFIGQwyEADs=)
на каждом конечном сегменте,
![](data:image/gif;base64,R0lGODlhLAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAoAA4AhQAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWgAzWx0zWh1GbDMAADIAMjNZfzNbgEgcAEgdHUceM1ozAFszAFozHV1dRkhZf11/f0huf2xGHX9ZM39ZSHd3XX9/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZnQIBwSCwaj8ikEvlQLgKBBjFDWB6fBWREmAlYv5kkZtj8WrPWg/DpKELf0GPYWhZuzUa00jD04I1zS2NJE3BvCEd6SG1DEH9FgUgKQxYADI9Fim5vVQBemABPAQJDA52GoKmqSwtKQQA7)
.
Замечание. Вместо условия 1) в теореме может быть взято условие
![](data:image/gif;base64,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)
.
Следствие. Если
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
непрерывны во всем пространстве, то для продолжимости каждого решения системы на
![](data:image/gif;base64,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)
необходимо и достаточно, чтобы в пространстве
![](data:image/gif;base64,R0lGODlhHwAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAgAcAA8AhQAAAAAAAB0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GXx1GbDMAADMzWzNbgEgdAEgcAEgdHUceM1ozAFszAEgzM1szM1tISEZGRllGRll/WV1/f113d0Rqe2xGHX9uSH9/XWZ3d3d3ZmaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZoQIBwSCwaj8jk8UFQOo2SISRArS6KBQADGRVCiqSAo1gBNI5dIaL4JUYAJMCj2mQfDcO4Mg1ZE/tGDAdCF0MUAYMAbUMeYk9HU1VUj0iLAAJ+lGyZAAFXmn9FFAMdoEScWqZDD0cBqkEAOw==)
существовали две непрерывно дифференцируемые функции Ляпунова
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
, обладающие свойствами:
1)
![](data:image/gif;base64,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)
;
2)
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,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)
равномерно относительно
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAGAAkAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbDIAHTNGbjNbgEgcAEceM1szAFozAERqe0huf1V7al1/f2xGHXtqRH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUYICBOYimUTKAmYlOWz1u8ygsQwFIGQwyEADs=)
на каждом конечном сегменте,
![](data:image/gif;base64,R0lGODlhLAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAoAA4AhQAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWgAzWx0zWh1GbDMAADIAMjNZfzNbgEgcAEgdHUceM1ozAFszAFozHV1dRkhZf11/f0huf2xGHX9ZM39ZSHd3XX9/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZnQIBwSCwaj8ikEvlQLgKBBjFDWB6fBWREmAlYv5kkZtj8WrPWg/DpKELf0GPYWhZuzUa00jD04I1zS2NJE3BvCEd6SG1DEH9FgUgKQxYADI9Fim5vVQBemABPAQJDA52GoKmqSwtKQQA7)
.
Поскольку одна из целей данной дипломной работы --- показать на примере применение функций Ляпунова к исследованию продолжимости решений дифференциальных систем, мы ставим перед собой задачу применить функции Ляпунова для решения вопроса продолжимости на
![](data:image/gif;base64,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)
всех решений некоторых нелинейных уравнений третьего порядка.
Рассмотрим уравнение
![](data:image/gif;base64,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)
эквивалентное системе
![](data:image/gif;base64,R0lGODlhsQBgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCvAFwAhgAAAAAAAAAAHR0AAB0AHR4AHgAAMx0AMh0AMwAdSAAcSB0dSAAzWwAzWh0zWx0zWh1JSR1GbDMAACgGFTMLCzIAHTMAHTMAMzIAMjQeSDQ0HTNdXTNGbjNZfzNbgDduf0gcAEgdAEkdHUceM0geNFozAFozHVszAEk0AFszM0YzRk9HHEZGM0hOM1tbW0ZGbllZf1l/WUhuf11/f0B/f1luf113d0Rqe1V7amxGHW5GM39ZM25bNW5GRn9ZSH9ZWX9/XX9uSH9uWXtqRHd3XWZ3d2Z/Znd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2Oj5CRkosCk5aXmJmam5ISnJ+goaKjgx6kp6ipqoMmq66vsLGyipWzto0pt5xJAbq+v8DBqrnCkgG1AAEJgycBzs8BDIYBCoPVxdiwDQDSlgaDKtmTxOKPAxGYBOXrqyCHzdDO3YWe7JLk9ok6GtuX9fkANyUBEGSQjEnuMgTEtXBatxAB0CH8RmgAxYYYU7k45CPjIHweMS0AEOKQhHkhU2IKMgCZSkMjXsqcSdNezJo4c+qUdXOnz59AN/UMSrSoUURDjypd+nMCU6JO17l8ailquV5Us2pVZLVYs2sAwAqK5+wQRGdTHR1joQAFNWtb/y0luZipxKRuEtDNDZi0mDpNdgnBi4eyxDx3gv4CeBHXkgRTPQoNhoZyUOBIF4cQegwgMru+wv5ZugyJNIh+AEQ3tgTh0uUfkfpJ6AVjUOt8oIOBoAspBzRjAb5JuLi7Im9hH1ZP6piNQrmSwUZC31yOQvJiLQ8Gy77QufLv4H85CP9yGfmQIVCfD6l+fUaJ7uPL5wRyvv37jlrluxCJ/6bZhfh3iICfECiIgYhoV0hatxzXSAGbHGQeAA4SAuEuFV6YiHeF8BIMBpccEAqIjIiIFCQkEsLBICbaM8Q5yQTQAQCkTdIeJLzVqMiNk8CXWCE8CtLXWcdIRlY0jnwFl/8gOVA0UpMAVDANMsq4BRZYLUJyFl1SFlLkWOZlSUhugoDgDDpdCjIYIWI+Ygooe3VIUQ6DqOMgnXQSclE4gqgGSXt3ApDnIH4ykpegAPBwnI+pJUKmJZM9M49ig4DA6FwrVDiAIZQSCl+k8phkiKaGHKrmkSidVghvaRIqyZtGklXZIpx5BoBiDwyCToUAwNqnqJIUNGoivjYKSS0nCUrXoNSdCIuffw1KUYX80CPIiq5OIpYg1LZXKACg+RacnoRYWiokM6JyW7aDXNSqrgCMpEOlACjEJkKHvCsIOvKyiNkgF1agoIhJtTlIUpSNUlwhAzwjGXy+PYRVmcfN2kifp2pCjCRJE3ODCAmInDXhCYR8OcigIUyIjY6RbMSJxbJYcMmjokxnSIqGaHiIdOfGBgACiugMLCdCs+sLd4jAvAhLDErCC6+2INgIzfhVXTXVVv/SVdbiYM311+R5DfbYq4lN9tlZcYg2MNatHYzabvtyXdy6wE33LG3ffYvdescyd9+x8A24KwoO/op+hicOFOKKu8Jo45DX1HTkmAQCADs=)
Теорема
Пусть функции
![](data:image/gif;base64,R0lGODlhHAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAXAA8AhQAAAAAAAB4AHh0AHQAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDIAHTMAADMAHTMAMzIAMjNGbjNbgEgcAEgdAEceM1ozAFszAF1/f113d0Rqe0huf1V7amxGHW5bNXd3XX9/XX9uSHtqRHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZnQIBwSCwaj8ZHElksFZAC5hAiNUgBi6JooBkmiJNAIMsoXsSBoVXYyHYAnmfRUmwIJ1+AJSsnloltAARsCgAdfUMHdYZiiIgAE0Z2AH1RV2pCFGIIQxdFAYNDHYVMDpdXlpISp0pSQQA7)
,
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
удовлетворяют следующим условиям:
а)
![](data:image/gif;base64,R0lGODlhHAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAXAA8AhQAAAAAAAB4AHh0AHQAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDIAHTMAADMAHTMAMzIAMjNGbjNbgEgcAEgdAEceM1ozAFszAF1/f113d0Rqe0huf1V7amxGHW5bNXd3XX9/XX9uSHtqRHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZnQIBwSCwaj8ZHElksFZAC5hAiNUgBi6JooBkmiJNAIMsoXsSBoVXYyHYAnmfRUmwIJ1+AJSsnloltAARsCgAdfUMHdYZiiIgAE0Z2AH1RV2pCFGIIQxdFAYNDHYVMDpdXlpISp0pSQQA7)
непрерывна при
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEceM1szAFozAF1/f0Rqe0huf1V7amxGHX9/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkMRMIKJ/FgpBR3BydSsjwQBQkA4ijdzieNANa5EA6BLGnQwqUgR2+AYw04JJcG91EA1tIYEljQmMRS1AAelwABgASjEaPVEIBBHxETGdnCkWdQ0EAOw==)
,
б) функция
![](data:image/gif;base64,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)
ограничена для достаточно больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBsABRhogeLdFKvZIBIkACQAyargCQwAjcPyVvMuX6fYlp0IAOw==)
, то есть
![](data:image/gif;base64,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)
для больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBsABRhogeLdFKvZIBIkACQAyargCQwAjcPyVvMuX6fYlp0IAOw==)
;
в) функция
![](data:image/gif;base64,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)
непрерывна и имеет непрерывную производную по
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAKAAgAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUcICCOJDAFwhiUgAEcLDQgrLjUwNPi9SoyARogBAA7)
и, кроме того, удовлетворяет условиям:
1)
![](data:image/gif;base64,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)
для достаточно больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBsABRhogeLdFKvZIBIkACQAyargCQwAjcPyVvMuX6fYlp0IAOw==)
и
![](data:image/gif;base64,R0lGODlhEwAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwANABQAhAAAAAAAAB0AHR0AAAAAHQAAMwAcSB0dSB1JSTMAADQeSDNGbjNbgEgcAEgdAFszAEYzRltbW0ZGbl1/f0huf25GRn9uSH9/XWZ/ZmaIiIBbM4iIZgECAwECAwECAwECAwU8ICCO5BiUaHCipMq26wu48hy/tJwHxtjbpMIIIsoBBKlbIklalowNgAJGagipowjzAHCwVJYBAXf7llMhADs=)
,
2)
![](data:image/gif;base64,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)
для достаточно больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBsABRhogeLdFKvZIBIkACQAyargCQwAjcPyVvMuX6fYlp0IAOw==)
и
![](data:image/gif;base64,R0lGODlhEwAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwANABQAhAAAAAAAAB0AHR0AAAAAHQAAMwAcSB0dSB1JSTMAADQeSDNGbjNbgEgcAEgdAFszAEYzRltbW0ZGbl1/f0huf25GRn9uSH9/XWZ/ZmaIiIBbM4iIZgECAwECAwECAwECAwU8ICCO5BiUaHCipMq26wu48hy/tJwHxtjbpMIIIsoBBKlbIklalowNgAJGagipowjzAHCwVJYBAXf7llMhADs=)
;
тогда все решения системы неограниченно продолжаемы.
Доказательство
Рассмотрим функцию
![](data:image/gif;base64,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)
Ее производную в силу системы для достаточно больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAAAHR0AAAAdSB0dSAAzWgAzWx1GbDQ0HTNbgEgcAEgdAEhuf2xGHW5GM39uSGZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBsABRhogeLdFKvZIBIkACQAyargCQwAjcPyVvMuX6fYlp0IAOw==)
,
![](data:image/gif;base64,R0lGODlhEwAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwANABQAhAAAAAAAAB0AHR0AAAAAHQAAMwAcSB0dSB1JSTMAADQeSDNGbjNbgEgcAEgdAFszAEYzRltbW0ZGbl1/f0huf25GRn9uSH9/XWZ/ZmaIiIBbM4iIZgECAwECAwECAwECAwU8ICCO5BiUaHCipMq26wu48hy/tJwHxtjbpMIIIsoBBKlbIklalowNgAJGagipowjzAHCwVJYBAXf7llMhADs=)
и
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAgwAAAAAAAAAAHQAzWzNZfzNbgEgdAFozAEhuf1luf11/f2xGHX9ZWX9uWYiIZgECAwQqEMhJQag0XCw17xvnfaMYYqOhBQKIFS55wlOJbsRtrUN9VjbgLzP0fSwRADs=)
легко оценить:
![](data:image/gif;base64,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)
Получили дифференциальное неравенство вида
![](data:image/gif;base64,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)
,
где
![](data:image/gif;base64,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)
, а
![](data:image/gif;base64,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)
. По лемме это неравенство не имеет ни одного положительного решения с конечным временем определения. В качестве множества
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWwAzWh1GbDMAADIAHTNZfzNGbjNbgEgcAEgdAFozAEhZf11/f0huf0Rqe113d2xGHXtqRH9/XX9uSHd3XWZ3d3d3ZmaIiIBbM4iIZgU6ICCOZAlcQqAGSwkFxugFhRwg5XyIDmECjoEo8RMFhrVfoAEsAoQiaMlR8pEgP8WIKmIAUgEpYAUIAQA7)
, о котором говорится в теореме, можно взять любое ограниченное множество, содержащее начало координат и такое, что вне его выполняются условия, наложенные на функции
![](data:image/gif;base64,R0lGODlhMQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAsAA8AhQAAAAAAAB4AHh0AAAAAHR0AHQAAMx0AMh0dSAAdSAAcSAAzWgAzWx0zWh1JSR1GbDIAHTMAADMAHTMAMzIAMjQeSDQ0HTNZfzNGbjNbgEgcAEgdAEceM1ozAFszAEYzRk9HHEhOM1tbW0ZGbll/WV1/f113d0Rqe0huf2xGHX9ZM25GM25GRn9ZSHd3XX9/XX9uSHtqRHd3ZmZ3d2Z/ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwatQIBwSCwaj8ikcjhZOgHN59AgXcIEVUr1eSjGBg9AIHABdLbPVJEKQABSVIgxQBgGEkZPYGhTELsANmxqQgUAbEVqhEmIH0QRGQAaYQAcUyCIRQNOhkcRhUMNQ2GZRZFKn55hnYtspUMWTqkYRampU0NyRmFuK0kaABVKA2MBlKNDKQEMQht7AAOvGq8AzENnRxInTyhFIkdaROFHWKjWbQAbnkbnaEYDdUZRQkEAOw==)
и
![](data:image/gif;base64,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)
.
Применяя теорему , приходим к требуемому выводу.
Замечание. Если вместо требований, наложенных на функцию
![](data:image/gif;base64,R0lGODlhMQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAsAA8AhgAAAAAAAB4AHh0AAAEAAAEAHR0AHR4AAAAAHgAAMx0AMx0AMgAAMh0dSQAdRwAcSAAdSB0dRwAAVQAAqgAA/wAzWgAyWB0zWx1ISB1Ibh1GbB1IajIAHTMAADMAHTIAADMAMzIAMjQeSDQ0HTFZfTNZfzVIbjNdgDNbgEgcAEcdAEgdAFUAAEkdAEceM0UeM1ozAFgyAkYzRk9HHEhOM1tbW0hIblV7VV1/f1V3d0Rqe1V3f0huf113d1VublVmZkhzgG5IHWpGHWxGHWpIHX9ZM31ZMW5GM25GRn9ZSHd3VX9/XX93VXtqRH9uSHd3XWZmVXd3ZmZ3ZmaIiIBdM4BbM4BzSIiIZqoAAP8AAAECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwfDgACCg4SFhoeIiYqDIIuOAI2PgwmSi1YClZGViwqFTQMZAAEEJAAwm49BhZQADQBBlByGAQWDAQ6GMAGDVBCEC4JFrEODBgCsharEisgyhB0oACmhAC6TM8iFA47Ghx2C3QAXg6HZhSWL394aAN3Lx4LmgyOO6iaF6uqDDIOyhhsAIgA4kigFABGKDhAIEAAgIXaCiASoIEjFLgAH5K3gZ4jioFOIdDwCUqjGIU2CQiTCpOiDx4AAWnjriCrRAQQnCQUCADs=)
, потребовать
![](data:image/gif;base64,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)
при достаточно больших
![](data:image/gif;base64,R0lGODlhEQAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwALABQAhAAAAAAAAAEAHR0AAAAdRx0dRwAAqgAyWAAzWh1GbB1IajQ0HTNZf0gcAEcdAERqe1VmZkhzgG5IHWxGHWpIHX9ZM25GM39uSHd3ZoBdM4iIZv8AAAECAwECAwECAwECAwU7ICCOJBCUZHCiosq2K+u+sxyjcyCMAVEDBwACRhokeLdGKmZZDImACuAyirgCT0dAYbqVfl/v8tUlB0IAOw==)
,
![](data:image/gif;base64,R0lGODlhEwAbAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwANABQAhQAAAAAAAB0AHR4AAAAAHgAAMwAAMgAcSAAdSB0dRwAAqh1ISDMAADQeSDNZfzVIbjNbgDNdgEgcAEgdAEkdAFozAEYzRltbW0hIbl1/f0huf0hzgG5GRn9uSHd3XX9/XXtqRGZ3ZmaIiIBdM4BbM4BzSIiIZqoAAAECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZBQIBwSBwGisjAEUlUMpvLJ8ApnUafVGk2cBgihFlAYWgBXwXJKyNNXBfDEkADSpwY3tcLOwGgMJUlAwRYV3+FSUEAOw==)
, то, взяв
![](data:image/gif;base64,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)
, получим