A continuous and bounded map (Phi: overline{Omega}rightarrow F) is called k-set contractive if, for any bounded set (Ssubset D), we have alpha_{F}bigl(Phi(S bigr)leq kalpha_{E}bigl(Phi(S) bigr).
Let E, F be two Banach spaces and (Dsubset E), a continuous and bounded map (Phi: overline{Omega}rightarrow F) is called k-set contractive if for any bounded set (Ssubset D), we have alpha_{F}bigl(Phi(S bigr)leq kalpha_{E}bigl(Phi(S) bigr).
In fact for any bounded set, there exists a constant such that for any.
For any bounded set M, T (M) is uniformly bounded and equicontinuous.
For any bounded set (Ssubset{mathcal{C}}), we assert that (overline{P}(S)) is the bounded set in ({mathcal{C}}).
For any bounded set (Ssubset{mathcal{C}}), we assert that ({T}(S)) is the bounded set in ({mathcal{C}}).
Now, for any bounded set (Dsubset K), we need to prove that ({ Q_{m}(D }) is relatively compact in E.
Furthermore, for any bounded set B, the set A ( t ) = ⋃ B Λ ( B, t ) ¯ X is a pullback attractor.
For any bounded set (Dsubset P), (A(D)) is bounded, so that the functions in (A(D)) are uniformly bounded.
x ∈ R 3. Proof For any bounded set K ⊂ E, there exists a positive constant M 0 such that ∥ u ∥ ≤ M 0 for all u ∈ K.
Then, there exists a bounded uniform absorbing set in for the family of processes, that is, for any bounded set, there exists, (312).
