where the mapping [PHI] : [R.sup.m] x [R.sup.m] [right arrow] [R.sup.m] is bounded on a bounded set
and [phi] is a L-Lipschitz continuous mapping on [R.sup.m].
Let P be a bounded set
in [L.sup.1] ([[0, T].sup.N]).
Let S be a bounded set
in a real Banach space E; the Kuratowski noncompactness measure of S is given by
(ii) For each bounded set
X, Y, Z [subset] [D.sub.r], and each for each closed interval J [subset] I, t [member of] I, there exists positive constant l [greater than or equal to] 0 such that
To this aim, we must prove that A is continuous and it transforms every bounded set
into a relatively compact set.
Let the class of all bounded set
of P(L) be denoted by [ALEPH].
Assuming that D [subset] X the mapping A : D [right arrow] X is said to be a condensing operator if A is continuous and bounded (sends bounded sets
into bounded sets
), and, for any nonrelatively compact and bounded set
S [subset] D,
A Borel probability measure P on Conf(S), the space of locally finite configurations, is called determinantal if there exists an operator K [member of] [I.sub.1,loc](S, [mu]) such that for any bounded measurable function g, for which g - 1 is supported in a bounded set
B, we have
A multimap [member of] : Z [right arrow] E is called [PSI]-condensing if for every bounded set
[OMEGA] [subset] Z, the relation [PSI](G([OMEGA])) [greater than or equal to] [PSI]([OMEGA]) implies the relative compactness of [OMEGA].
(Mansour and Dillon, 2011) proposed hybrid (Path and state based) reliability models is used to check the reliability aspects and simulation results are also presented and bounded set
approach is used to evaluate the reliability of composite web services and only two state model is done effectively and it also extended to three state model to determine reliability.
In her view, patent law more correctly should be seen as merely creating opportunities to bargain over the definition of rights, as patents are established individually for each patent in the face of rapidly changing knowledge and meaning and thus could never provide a definitive and clearly bounded set
then [A.sub.0] is a bounded set
. To prove the compactness of A(X), we just need to prove that A(X) is close.