Sentence examples for the foregoing theorem from inspiring English sources

In addition, the condition imposed in the foregoing theorem only needs the function to be lower semi-continuous, a rather weak condition in optimization.

To prove lim z → 0 + A M, j ( z ) = 0, we use the duality principle as in the proof of the foregoing theorem.

The foregoing theorem shows that the concept of weak sharp minima can be converted into a class of optimization problems with the same optimal value.

In the foregoing theorem, if we set,, and, then we get the following result which improves and generalizes the result of Jungck [16, Corollary  3.2] in metric space.

Now repeating the proof of the foregoing theorem with respect to (3.19) with the kernel K ˜ z j ( t, x ), we can also get a similar estimate as in (3.11) in the form n C ≥ ∑ l = 1 n ∥ K ∗ f l, z j ∥ L p ′ ≥ c j ∑ l = 1 n ( ∫ 0 z | B l, z j ( t ) | p ′ d t ) 1 p ′ ( ∫ z ∞ | A ˜ l, z j ( x ) | 2 d x ) 1 2. (3.20).

Now, repeating the proofs of the foregoing theorems with respect to (3.10) with the kernel K ˜ z j ( x, t ), we can also get a similar estimate as in (3.2) in the form n C ≥ ∑ l = 1 n ∥ K f l, z j ∥ L q ≥ c j ∑ l = 1 n ( ∫ z ∞ | A l, z j ( x ) | q d x ) 1 q ( ∫ 0 z | B ˜ l, z j ( t ) | 2 d t ) 1 2. (3.11).

For quasi-uniform meshes with N elements and 2D BEM, this corresponds to O (N − 3 / 2 ) and hence s = 3 / 2. With the foregoing proposition and according to Theorem 4.5, the adaptive algorithm attains any possible convergence order 0 < s ≤ 3 / 2 and the generically quasi-optimal rate is thus achieved.

Notice that, in the foregoing example, all the conditions of Theorem 2.1 are fulfilled and (x=0) is a unique common fixed point of the mappings f and g (see Fig. 5).

Disregarding the term ((mathopen[0, inftymathclose) : X)) and taking into consideration the previously defined spaces AAP and (operatorname {AAPS}^{p}), we have the following inclusion diagram of asymptotically almost periodic function spaces (see Theorem 4.9): By the foregoing, any inclusion of this diagram can be strict.

As already pointed out in the foregoing section, these classes will be shown to coincide by Theorem 1 below, whose proof is just based on Lemma 1.

Remark 4 In view of Proposition 1 and its analogy for q.p. sequences mentioned in the foregoing section, a discrete (i.e., restricted to ℤ) analogy of Theorem 2 holds for sequences.