In the sequel, we shall consider a metric d and a quasi-order '⪯' on C ( [ 0, T ], R ) such that ( C ( [ 0, T ], R ), d, ⪯ ) is monotonically complete or mixed-monotonically complete and preserves the monotone convergence.
Theorem 3.8 Suppose that the quasi-ordered metric space ( X, d, ⪯ ) is monotonically complete and preserves the monotone convergence.
Theorem 6.2 Suppose that the quasi-ordered metric space ( C ( [ 0, T ], R ), d, ⪯ ) is monotonically complete and preserves the monotone convergence.
Corollary 6.3 Suppose that the quasi-ordered metric space ( C ( [ 0, T ], R ), d, ⪯ ) is monotonically complete and preserves the monotone convergence.
Theorem 4.12 Suppose that the quasi-ordered metric space ( X, d, ⪯ ) is monotonically complete and preserves the monotone convergence, and that the metrics d and d are compatible in the sense of preserving convergence.
end{aligned} Hence, it follows from the monotone convergence theorem that the sequence ((u_{k})_{kgeq0}) converges to a function (uinLambda) satisfying the integral equation begin{aligned} u=lambda Vbigl(a f u bigr).
Given the fact the failure propagation forms a monotone decreasing sequence bounded by the zero vector, it converges toward its very minimum, as stated by the monotone convergence theorem.
Also the monotone convergence of the new methods is established.
For comparison, the monotone convergence based ILC design method is extended to the situation with more iteration-varying factors.
We also provide proofs for robust monotone convergence of the proposed CO-ILC algorithms for a class of uncertainty models.
