Equicontinuity


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Equicontinuity

 

an important property of some sets of functions. A set of functions is said to be equicontinuous on a given closed interval [a, b] if, for any number ∊ > 0, there exists a δ > 0 such that ǀf(x2) - x1ǀ < ∊ for any function f(x) of the given set whenever xl and x2 are in [a, b] and ǀx2xlǀ < δ. All functions of an equicontinuous set are uniformly continuous on [a, b].

The property of the equicontinuity of a set of functions finds application in the theory of differential equations and in functional analysis by virtue of the following theorem: for a uniformly convergent sequence of members of a given set of functions to exist, it is necessary and sufficient that the set of functions be equicontinuous and uniformly bounded—that is, that all the functions satisfy on [a, b) the condition ǀf(x)ǀ ≤ M with the same M. The possibility of singling out a uniformly convergent sequence means that the given set forms a relatively compact set in the space C of continuous functions.

References in periodicals archive ?
Next, we verify the equicontinuity of the set [mathematical expression not reproducible].
Our main contribution is to establish, under almost no assumptions, that local equicontinuity (in t) is equivalent to local convergence; i.e., local control of the differences [T.sub.t]f(x) - [T.sub.t]f(y) for all t small is equivalent to local control of the differences [T.sub.t]f(x) - f(x) for all small t.
It follows from the equicontinuity of [([B.sub.n]).sub.n [greater than or equal to] 0] that the set Y is closed.
Equicontinuity of [T.sub.n](D) can be derived from the absolute continuity of Lebesgue integral and (25).
To implement this we need equicontinuity and equiboundedness of {[y.sub.n]} and {[y'.sub.n]}.
Some properties of cellular automata with equicontinuity points.
El Kinani, Equicontinuity of power maps in locally pseudo-convex algebras, Comment.
The remainder of the book proceeds from this theme, discussing equicontinuity, the strong topology, operators, completeness, inductive limits, compactness, and barrelled spaces.
where mod c(D) is the modulus of equicontinuity of the set of functions D given by the formula
To study the equicontinuity of T (K [intersection] [B.sub.[rho]]), let [x.sub.1], [x.sub.2] [member of] (0,1).
We first introduce regularity conditions to ensure the stochastic equicontinuity of the sample HJ-distance and the consistency of [??].
The equicontinuity for case [t.sub.1] < [t.sub.2] [less than or equal to] 0 and [t.sub.1] [less than or equal to] 0 [less than or equal to] [t.sub.2] is obvious.