lower semicontinuous function

lower semicontinuous function

[¦lō·ər ‚sem·ē·kən′tin·yə·wəs ‚fənk·shən]
(mathematics)
A real-valued function ƒ(x) is lower semicontinuous at a point x0 if, for any small positive number ε, ƒ(x) is always greater that ƒ(x0) - ε for all x in some neighborhood of x0.
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Let [eta] : K x K [right arrow] R [union] {+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping.
lt;u, [eta](y, x)> is convex and lower semicontinuous function,
N] [right arrow] (-[infinity], +[infinity]] be a proper and lower semicontinuous function.
Let f: X [right arrow] R [union]{+[infinity]} be a proper lower semicontinuous function.
In [7], when A = [partial derivative][phi] where [phi] is a proper, convex and lower semicontinuous function, we proved an ergodic theorem and a weak convergence theorem for solutions to (1), by assuming (2), (3), (4) and that [t.
In this section, we consider the evolution equation (1) when the monotone operator A is the subdifferential [partial derivative][phi] of a proper, convex and lower semicontinuous function [phi]: H [right arrow]] -[infinity], +[infinity]].
If in Theorem 1 A is the subdifferential of a proper, convex and lower semicontinuous function [phi] : H [right arrow] (-[infinity], +[infinity]] and F is nonempty (i.
It is known that if A is the subdifferential of a proper, convex, lower semicontinuous function, [A.
Let [PSI] : [omega] [right arrow] RU{+[infinity]} be a lower semicontinuous function.
where f : E [right arrow] E is a self mapping and [phi] : [0, + [infinity]) [right arrow] [0, + [infinity]) is a lower semicontinuous function from right such that [phi] is positive on (0, + [infinity]) and [phi](0) = 0.
Let (M,[rho]) be a complete metric space and let [phi]: M [right arrow] R be a lower semicontinuous function which is bounded from below.
Let us assume that H is a lower semicontinuous function on [partial derivative][OMEGA].