View source for Contingency equation

A relation

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255002.png" /> is the contingency of a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255003.png" />, i.e. the set of all partial limits of the ratio

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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255005.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255006.png" /> is a given non-empty set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255007.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c0255009.png" /> (cf. [[#References|[1]]], [[#References|[2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c02550010.png" /> is a bounded closed convex set that is upper semi-continuous relative to the inclusion function of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025500/c02550011.png" />, the contingency equation is equivalent to the paratingency equation (defined analogously to the contingency equation, but using the ratio

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cf. [[#References|[3]]]) and the [[Differential inclusion|differential inclusion]]

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(cf. [[#References|[4]]]). The properties of a contingency equation are similar to those of a differential inclusion.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Marchaud,   "Sur les champs de demi-cônes et les équations différentielle du premier ordre"  ''Bull. Soc. Math. France'' , '''62'''  (1934)  pp. 1–38</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.A. Barbashin,   Yu.I. Alimov,   "On the theory of relay differential equations"  ''Izv. Vysshikh Uchebn. Zaved. Mat.'' :  1  (1962)  pp. 3–13  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.K. Zaremba,   "Sur les équations au paratingent"  ''Bull. Sci. Math. (2)'' , '''60''' :  5  (1936)  pp. 139–160</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T. Waźewski,   "Sur une condition équivalente à l'équation au contingent"  ''Bull. Acad. Sci. Polon. Sci. Ser. Math.'' , '''9''' :  12  (1961)  pp. 865–867</TD></TR></table>