Sentence examples for much weaker condition from inspiring English sources

For most purposes, including the present one, a much weaker condition suffices, namely that V should be non-constant in the sense that V contains some valuation distinct from vT and vF.

Now we will prove a Filippov-Pliss type theorem under much weaker condition, which gives the estimation only of the difference between (x cdot )) and (y(cdot )) but not between their derivatives.

It is shown in the present paper that the requirement does apply providing that the random point process representing the system natural frequencies is at least locally stationary (which is a much weaker condition than local stationarity of the frequency response function), and a number of the implications of this result are explored.

Here, using same special properties of βk, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ε="Ω(Δxm), where m r,p ⩾2 is the optimal sensitivity order, regardless of critical points.

This has been established using a different way than the one used in the literature and with much weaker conditions.

In addition, it is known that the convergence analysis of three-step can be proved under much weaker conditions.

The 'Banach Contraction Mapping Theorem' was generalized by many authors to mappings that satisfy much weaker conditions (see, for instance, [3 10]).

It was proved first time by Filippov in the case of Lipschitzian right-hand side in [19] and afterward extended by Pliss in [20] under much weaker conditions.

In addition, the existence of a common solution for certain class of functional equations arising in dynamic programming, under much weaker conditions are discussed.

The main novelty of our paper is that we prove the existence of the limit of the solutions of the above equations under much weaker conditions than (1.9).

Remark 3.2 It is clear that the Mazur-Ulam theorem has been proved under much weaker conditions than the main result of Alaca [25] in the framework of 2-fuzzy 2-normed linear spaces.