By the level compactness of on, there exist a subsequence of of and such that.
Assume that, then there exist a subsequence of and a real number, such that (2.6).
Since is bounded in, there exist a subsequence of and with and on, such that (3.10).
By the boundedness of, there exist a subsequence of such that.
By the proof of Theorem 3.1, there exist a subsequence { x 1, n i } of { x 1, n } and x 1 ∈ P such that lim n i → + ∞ x 1, n i ( t ) = x 1 ( t ), t ∈ [ 0, 1 ].
Moreover, for any sequence of real numbers there exist a subsequence and a well-defined function such that for each, one can find such that (3.2). whenever for, and (3.3).
(36) Since A is approximatively compact with respect to B, there exist a subsequence ({Ty_{m_{s}}}) of ({Ty_{m}}) and an element (x_{0}in A) such that lim_{stoinfty}{Ty_{m_{s}}}=x_{0}.
Contrarily, we assume that (y_{i}nrightarrow z), then there exist a subsequence ({y_{i_{j}}}) of ({y_{i}}) and (alpha >0 ) such that d y_{i_{j}},z geqfrac{alpha}{2}quad text{for all }j.
This implies that, there exist a subsequence of { x n k }, denoted also by { x n k }, and a sequence { z k } in F ( T ) such that d ( x n k, z k ) < 1 2 k, ∀ k ∈ N. (3.8).
If y m ↛ y, then there exist a subsequence { y m j } of {y m } and M > 0 such that d ( y m j, y ) ≥ M 2 for all j.
A mapping (T Clongrightarrow H) is said to be demi-compact at a point (zin H) if, for any bounded sequence ({x_{n}}) in C such that ((I-T x_{n}rI-T xrrow z) as (ntoinfty), then there exist a subsequence ({x_{n}rightarrow a point (pin C) such that (x_{n_{j}}rightarrow p) as (jtoinfty) and ((I-T)p=z).
