comparison theorem

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations.{{Cite book |last=Walter |first=Wolfgang |url=http://link.springer.com/10.1007/978-3-642-86405-6 |title=Differential and Integral Inequalities |date=1970 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-86407-0 |location=Berlin, Heidelberg |doi=10.1007/978-3-642-86405-6}}{{Cite book |last=Lakshmikantham |first=Vangipuram |title=Differential and integral inequalities: theory and applications |date=1969 |publisher=Academic Press |others=Srinivasa Leela |isbn=978-0-08-095563-6 |series=Mathematics in science and engineering |location=New York}}

One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation.{{Cite journal |last=Aronson |first=D. G. |last2=Weinberger |first2=H. F. |date=1975 |editor-last=Goldstein |editor-first=Jerome A. |title=Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation |url=https://link.springer.com/chapter/10.1007/BFb0070595 |journal=Partial Differential Equations and Related Topics |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=5–49 |doi=10.1007/BFb0070595 |isbn=978-3-540-37440-4}} Other examples of comparison theorems include:

• Chaplygin's theorem
• Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations
• Lyapunov comparison theorem
• Sturm comparison theorem
• Hille-Wintner comparison theorem

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.

• Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart
• Toponogov's theorem
• Myers's theorem
• Hessian comparison theorem
• Laplacian comparison theorem
• Morse–Schoenberg comparison theorem
• Berger comparison theorem, Rauch–Berger comparison theoremM. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
• Berger–Kazdan comparison theorem{{mathworld|urlname=Berger-KazdanComparisonTheorem|title=Berger-Kazdan Comparison Theorem}}
• Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
• Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvaturesR.L. Bishop & R. Crittenden, Geometry of manifolds
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
• Cheng's eigenvalue comparison theorem
• Comparison triangle

• Limit comparison theorem, about convergence of series
• Comparison theorem for integrals, about convergence of integrals
• Zeeman's comparison theorem, a technical tool from the theory of spectral sequences

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• {{SpringerEOM|oldid=46412|title=Comparison theorem}}
• {{SpringerEOM|oldid=46691|title=Differential inequality}}

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Category:Mathematical theorems