Note that x ∘ σ is f-bounded for any f-bounded sequence x and any sequence σ : N → N. Example 2.1 If f ( t ) = e t, then f-boundedness of a sequence x is equivalent to the boundedness above of x.
This requires some arguments of Miyagaki and Souto in [20] to prove the boundedness of a certain sequence.
It is generally known that the boundedness of an iteration sequence is the basic requirement in the iteration methods for solving various optimization problems.
As usual, the Ambrosetti-Rabinowitz [21] type condition ( AR ) ∃ ν > 4 : ν F ( x, t ) ≤ t f ( x, t ), | t | large, is assumed to ensure the boundedness of a Palais-Smale sequence.
Indeed, By reflexivity of E and boundedness of the sequence {x n } there exists a weakly convergent subsequence { x n j } ⊂ { x n } such that x n j ⇀ p for some p ∈ C. Now we show that p ∈ Fix ( S ).
It follows from the boundedness of the sequence ({T^{n}0}) that there exist a positive real number r such that (r^{(n)}_{i}leq r) for all n and (i=1,2,ldots,m).
By the boundedness of the sequence ({u_{k}}) in X, we can find an element (u in X) such that (u_{k} rightharpoonup u) in X as (ktoinfty) and bigllangle {f_{j}^,u}bigrrangle =lim _{k to infty}{bigllangle {f_{j}^,u_{k}} bigrrangle }=0 for (j=1,2,ldots ) .
Furthermore, by utilizing the mathematical induction, a sufficient condition is established to ensure the asymptotic boundedness of the sequence of the error covariance.
The boundedness of ( PS ) sequence.
The boundedness of ( PS ) sequence. .
