Some relations between nonexpansive and order preserving mappings

Authors:
Michael G. Crandall and Luc Tartar

Journal:
Proc. Amer. Math. Soc. **78** (1980), 385-390

MSC:
Primary 47H07

MathSciNet review:
553381

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Abstract: It is shown that nonlinear operators which preserve the integral are order preserving if and only if they are nonexpansive in and that those which commute with translation by a constant are order preserving if and only if they are nonexpansive in . Examples are presented involving partial differential equations, difference approximations and rearrangements.

**[1]**Haïm Brézis and Walter A. Strauss,*Semi-linear second-order elliptic equations in 𝐿¹*, J. Math. Soc. Japan**25**(1973), 565–590. MR**0336050****[2]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, 10.1090/S0025-5718-1980-0551288-3**[3]**A. Douglis,*Lectures on discontinuous solutions of first order nonlinear partial differential equations in several space variables*, North British Symposium on Partial Differential Equations, 1972.**[4]**Ignace I. Kolodner,*On the Carleman’s model for the Boltzmann equation and its generalizations*, Ann. Mat. Pura Appl. (4)**63**(1963), 11–32. MR**0168930****[5]**S. N. Kružkov,*Generalized solutions of nonlinear equations of the first order with several variables. I*, Mat. Sb. (N.S.)**70 (112)**(1966), 394–415 (Russian). MR**0199543****[6]**Thomas G. Kurtz,*Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics*, Trans. Amer. Math. Soc.**186**(1973), 259–272 (1974). MR**0336482**, 10.1090/S0002-9947-1973-0336482-1**[7]**Michel Pierre,*Un théorème général de génération de semi-groupes non linéaires*, Israel J. Math.**23**(1976), no. 3-4, 189–199. MR**0428142****[8]**G. Pólya and G. Szegö,*Isoperimetric Inequalities in Mathematical Physics*, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR**0043486****[9]**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****[10]**Michael B. Tamburro,*The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation*, Israel J. Math.**26**(1977), no. 3-4, 232–264. MR**0447777**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0553381-X

Keywords:
Nonlinear operator,
order preserving,
nonexpansive,
rearragement

Article copyright:
© Copyright 1980
American Mathematical Society