Sentence examples for in 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.
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.
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.
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).
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.
NOTE: The names of the principals in the foregoing article have been changed, for obvious reasons.
The assumption in the foregoing discussion was that money wages were at the equilibrium level.
In the foregoing analysis, the system's stability has been determined precisely using the interface impedance.
