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Comparison theorem
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<h2 id="differential-equations">Differential equations</h2> <p>In the theory of <a href="/facts/Differential_equations/9MGbE3Ca">differential equations</a>, 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.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>One instance of such theorem was used by Aronson and Weinberger to characterize solutions of <a href="/facts/Fisher%2527s_equation/v112v7zY">Fisher's equation</a>, a reaction-diffusion equation.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Other examples of comparison theorems include: </p> <ul><li><a href="/facts/Chaplygin%2527s_theorem/4NaF6GEt">Chaplygin's theorem</a></li> <li><a href="/facts/Gr%25C3%25B6nwall%2527s_inequality/yxbPMPKS">Grönwall's inequality</a>, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations</li> <li>Lyapunov comparison theorem</li> <li><a href="/facts/Sturm_comparison_theorem/ABukvpTe">Sturm comparison theorem</a></li> <li>Hille-Wintner comparison theorem</li></ul> <h2 id="riemannian-geometry">Riemannian geometry</h2> <p>In <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <ul><li><a href="/facts/Rauch_comparison_theorem/IlWdVJDq">Rauch comparison theorem</a> relates the <a href="/facts/Sectional_curvature/GpI9zQDh">sectional curvature</a> of a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> to the rate at which its <a href="/facts/Geodesic/fbWR6l80">geodesics</a> spread apart</li> <li><a href="/facts/Toponogov%2527s_theorem/WF4UXvAW">Toponogov's theorem</a></li> <li><a href="/facts/Myers%2527s_theorem/VGm0ta5q">Myers's theorem</a></li> <li>Hessian comparison theorem</li> <li>Laplacian comparison theorem</li> <li>Morse–Schoenberg comparison theorem</li> <li>Berger comparison theorem, Rauch–Berger comparison theorem<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li><a href="/facts/Berger%25E2%2580%2593Kazdan_comparison_theorem/bbGJhRu0">Berger–Kazdan comparison theorem</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li>Warner comparison theorem for lengths of N-Jacobi fields (<i>N</i> being a submanifold of a complete Riemannian manifold)<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li><a href="/facts/Bishop%25E2%2580%2593Gromov_inequality/hyvvBp2L">Bishop–Gromov inequality</a>, conditional on a lower bound for the <a href="/facts/Ricci_curvature/qujou7QS">Ricci curvatures</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li>Lichnerowicz comparison theorem</li> <li>Eigenvalue comparison theorem <ul><li><a href="/facts/Cheng%2527s_eigenvalue_comparison_theorem/z7YnXK3p">Cheng's eigenvalue comparison theorem</a></li></ul></li> <li><a href="/facts/Comparison_triangle/2t8PW5fh">Comparison triangle</a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Direct_comparison_test/AeHYrl7o">Limit comparison theorem</a>, about convergence of series</li> <li><a href="/facts/Direct_comparison_test/AeHYrl7o">Comparison theorem for integrals</a>, about convergence of integrals</li> <li><a href="/facts/Zeeman%2527s_comparison_theorem/sH0NmJCf">Zeeman's comparison theorem</a>, a technical tool from the theory of <a href="/facts/Spectral_sequences/9S1v2dRh">spectral sequences</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Comparison_theorem&oldid=46412">"Comparison theorem"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Differential_inequality&oldid=46691">"Differential inequality"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Walter, Wolfgang (1970). Differential and Integral Inequalities. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-86405-6. ISBN 978-3-642-86407-0. <a href="978-3-642-86407-0" target="_blank">978-3-642-86407-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lakshmikantham, Vangipuram (1969). Differential and integral inequalities: theory and applications. Mathematics in science and engineering. Srinivasa Leela. New York: Academic Press. ISBN 978-0-08-095563-6. <a href="978-0-08-095563-6" target="_blank">978-0-08-095563-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Aronson, D. G.; Weinberger, H. F. (1975). Goldstein, Jerome A. (ed.). "Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation". Partial Differential Equations and Related Topics. Berlin, Heidelberg: Springer: 5–49. doi:10.1007/BFb0070595. ISBN 978-3-540-37440-4. <a href="978-3-540-37440-4" target="_blank">978-3-540-37440-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975. <a href="/wiki/Jeff_Cheeger" target="_blank">/wiki/Jeff_Cheeger</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>R.L. Bishop & R. Crittenden, Geometry of manifolds <a href="/wiki/Richard_L._Bishop" target="_blank">/wiki/Richard_L._Bishop</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>