Liapunov functions and boundedness and global existence of solutions |
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Authors: | Haddock R John |
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Institution: | Department of Mathematics , Memphis State University , Tennessee, Memphis, 38111, U.S.A |
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Abstract: | In this paper the properties of a Liapunov function and its derivatives are combined in obtaining sufficient conditions for the boundedness and global existence of solutions of the system of differential equations x=f(t,x) (1) where f: 0,∞)XRn→ Rn is continuous. The following terms are introduced: (i) a scalar function which is mildly unbounded relative to a subset of Rn,(ii) a scalar function which is radially unbounded relative to a set, (iii) a Liapunov function with mildly negative definite derivative in a set, (iv) a Liapunov function with strongly negative definite derivative in a set, and (v) a set which is unbounded-departing with respect to system (1). As a special case the autonomous system of differential equations which corresponds to (1) is considered and the semi-invariance property of the positive limiting set of a solution is used in the results |
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