Some remarks on differential equations of quadratic type |
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Authors: | Ishikawa Shiro Nussbaum Roger D |
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Affiliation: | (1) Department of Mathematics, Keio University, 3-14-1, Hiyoshi, 223 Yokohama, Japan;(2) Department of Mathematics, Rutgers University New Brunswick, 08903, New Jersey |
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Abstract: | In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x
0
C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR
n, and thel
1 norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then limtf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then limtx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) limt f(x(t))–x(t)=0 and that limt
x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR
n into itself such that limt
x(t) fails to exist for mostx
0
C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations. |
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Keywords: | Quadratic differential equation model Boltzmann equation nonexpansive semigroups |
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