Sentence examples for bounded set such that from inspiring English sources

Let Ω ⊂ C [-r, T], X) be a nonempty, bounded set suC [-raT]β ( F . (3.7).

Let E be a Banach space and K ⊂ E a non-empty, open, convex, bounded set such that 0 ∈ K.

Let K be an unbounded closed convex subset of H (without loss of generality, we assume that x = 0 ∈ K from now on), let Ω ⊂ H be an open bounded set such that K ∩ Ω ≠ ∅.

Let be a bounded set such that and (245).

Without of generality, assume that there exists a bounded set such that (3.8).

Now consider a bounded set such that there exists a satisfying or condition.

In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn:n∈N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn+Dn+2⊆2Dn+1 for every n∈N.

By Lemma 4.3 there exists such that for all, and a bounded (or compact) set such that for all.

Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in, for all and some sufficiently large finite with.

Assume that is an open bounded set in such that the following conditions hold.

Moreover, if ρ ( F ) < 1, then the attractor is unique and the basin is R n ; if ρ ( F ) > 1 and F is irreducible, then there does not exist a nonempty bounded set A such that F ( A ) = A. Theorem 5.4 states that ρ ( F ) = 1 is the transition value between F having and not having an attractor.