For any u 0 ( t ) ∈ C + [ 0, 1 ] ∖ , the sequence u n ( t ) = ( Q u n − 1 ) ( t ) = ( H F u n − 1 ) ( t ) = T F u n − 1 + ( T G ) T F u n − 1 + ( T G ) 2 T F u n − 1 + ⋯ + ( T G ) n T F u n − 1 + ⋯. ( n = 1, 2, … ) converges uniformly to the unique solution u ⋆, and the rate of convergence is determined by ∥ u n ( t ) − u ⋆ ( t ) ∥ = O ( 1 − d α n ), where 0 < d < 1 is a positive number.
We have a ˆ ∥ u k ∥ T p − C 1 ≤ B q ( q − p ) b ( C 1 + ∥ u k ∥ T ), which implies that the sequence ( u k ) k is bounded in X T. By the compact embedding X T ⊂ C ( [ − T, T ] ), there exist u ∈ X T and the subsequence of ( u k ) k, still denoted by ( u k ) k, such that u k ⇀ u weakly in X T and u k → u strongly in C ( [ − T, T ] ).
On the other hand, by virtue of conditions (2.39), (2.40), and (2.43), from the equalities u k m ″ ( t ) = f ( t, u k m ( t ), u k m ′ ( t ) ) for 0 < t ≤ b k m ( m = 1, 2, … ) (2.45). it follows that the sequence ( u k m ″ ) m = 1 + ∞ is also uniformly converging on every finite closed interval contained in ] 0, + ∞ [. Suppose u ( t ) = lim m → + ∞ u k m ( t ) for t ∈ R +.
Proof Let u 1 ∈ X and the sequence u n be defined by u n + 1 = T u n.
With respect to the operators U (i ) we assign to each f ∈ S i the sequence u (f ) ≔ (u n ) n ≥ 0, called the U -sequence at f, and inductively defined by u 0 ≔ f, u 1 ≔ U (i ) (f ), u 2 ≔ U (1 − i ) U (i ) (f ), u 3 ≔ U (i ) U (1 − i ) U (i ) (f ), and so on.
And it is composed by the case group U T and control group U F. U = U T ∪ U F, and | U T | = l1, | U F | = l2.
We consider the sequence t = ( t k ) defined by t k = { 1 u n v k, if k is odd, 0, if k is even.
Throughout this paper, ℕ denotes the set of all positive integers, I the identity mapping on C and F ( T ) the set of all fixed points of T. The Picard or successive iterative process [2] is defined by the sequence { u n } : { u 1 = u ∈ C, u n + 1 = T u n, n ∈ N. (1.1).
4. NPI implementation is modeled by the decision variable u(t), where 0 ≤ u(t) ≤ b < β.
The expression u ′(f) for the lower chip-rate sequence u ′(t) is similar to u(f) in (2) but with f c substituted by f c′.
