where B ∈ Λ, is pre-compact in Λ.
Then 1. B is pre-compact if and only if β X ( B ) = 0 ; 2.
If it is continuous and maps bounded sets into pre-compact sets, an operator is called completely continuous.
An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.
Next, we show that (tilde{e}(theta,t)) and (tilde{i}(a,t)) remain in a pre-compact subset of (L_^{1}) independent of (X_{0}).
An operator (T Xrightarrow X) is completely continuous if it is continuous and maps bounded sets into pre-compact sets (or relatively compact sets).
A set (Msubset C J,R)) is said to be a pre-compact set provided that the following two conditions are satisfied: (i) All the functions in the set M are uniformly bounded.
Next, we need to show that (tilde{y}_{i} t,a)) remains in a pre-compact subset of (L^{1}_) that is independent of ((x_{i0},y_{i0}(cdot),v_{0} in Gamma).
σ ∈ H w ( g ) ) absorbing set obtained in Theorem 4.2, then we need only to show that for any { u τ n } ⊂ B 0, { σ n } ⊂ H w ( g ) and t n → ∞, { U σ n ( t n, τ n ) u τ n } n = 1 ∞ is pre-compact in ( W 1, p ∩ L q × L q . Thanks to Lemma 4.2, it is sufficient to verify that for any { u τ n } ⊂ B 0, { σ n } ⊂ H w ( g ) and t n → ∞, { U σ n ( t n, τ n ) u τ n } n = 1 ∞ is pre-compact in W 1, p.
Let E be a Banach space and ({u_{n}} subset E) a sequence such that mathcal{F} u_{n})quad mbox{is bounded}quad mbox{and}quad mathcal{F}' u_{n}) to0quad mbox{as } ntoinfty, (3.2) then ({u_{n}}) is pre-compact, i.e., ({u_{n}}) has a convergent subsequence.
The gouge was axially pre-compacted under the test conditions (5 MPa) for 30 min.
