equicontinuous family of functions

equicontinuous family of functions

[¦ē·kwē·kən′tiŋ·yə·wəs ′fam·lē əv ′fəŋk·shənz]
(mathematics)
A family of functions with the property that given any ε > 0 there is a δ > 0 such that whenever | x-y | < δ,="">x) - ƒ(y)| < ε="" for="" every="" function="">x) in the family. Also known as uniformly equicontinuous family of functions.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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equicontinuous on [[0, T].sup.N], if, for each [epsilon] > 0, there exists a closed subset [D.sub.[epsilon]] [subset] [[0, T].sup.N] such that m([D'.sub.[epsilon]]) [less than or equal to] [epsilon] and the restriction of all functions from X to the set [D.sub.[epsilon]] form an equicontinuous family of functions. Furthermore, X is said to be countable a.e.
A Family of Measures of Noncompactness in the Locally Sobolev Spaces and Its Applications to Some Nonlinear Volterra Integrodifferential Equations

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