Abstract
In this paper we study differential equations of the formx′(t) + x(t)=f(x(t)), x(0)=x 0 ε 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: C→C is less than or equal to one, then limt→∞∥f(x(t))−x(t)∥=0 and, if {x(t):t ⩾ 0} is precompact, then limt→∞x(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 ε 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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Ishikawa, S., Nussbaum, R.D. Some remarks on differential equations of quadratic type. J Dyn Diff Equat 3, 457–490 (1991). https://doi.org/10.1007/BF01049742
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DOI: https://doi.org/10.1007/BF01049742