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Next, by using Theorem 3.1 which ensures the existence of a solution in the sense of distributions to, we will deduce, by applying Proposition 2.6, the existence of a second one greater than or equal to the first one on the whole interval ; in order to do this, we will assume that satisfy,, as well as the following conditions.

The relationship between any two variables will be deduced by applying the previous statistical concepts especially for the carbonate aquifer at this study which is controlled by several factors such as porosity (fractures) including the density of fractures, TDS, water resistivity, shale content, and amount of water.

In this study, a general equation for dust removal in the centrifugal dust removal section (CDRS) of ECP fan is deduced by applying the classic force equilibrium principle, which states that the centrifugal force on a dust particle should be equal to the aerodynamic resistance of the corresponding spiral airflow at a given spatial point.

In conventional MRE experiments the wave propagation speed and the wave-damping coefficient can be deduced by repeatedly applying variable driving frequencies.

It is straightforward to deduce the stated relation by applying the definition of the difference operator, Lemma 3.1 and the telescopic identity where we recall the notation z m = Z m −1, s, and that z0=0 by assumption.

The following theorem can be directly deduced from Theorem 3.1 by applying Lemma 2.3.

end{aligned} (7.9 One can deduce Theorem 7.17 from Lemma 7.18 by applying Proposition 3.12, but there is a hitch: one has to insure that when (mathcal {C}) is a model prefibration, the latching and matching functors (L_n), (M_n) are left resp.

We start this section by applying Theorem 1 to deduce the soft version of Banach's contraction principle cited above.

By applying Theorem 3.8 we deduce that the system is uniformly exponentially stable and the proof is complete.

In the following, by applying the result, we deduce the resolvent set of A on the imaginary axis.

We conclude the paper by applying this fact to deduce the existence of solution for the three algorithms mentioned at the beginning of this section.