} end{aligned}To establish Proposition 1.1, it suffices to prove.
Using the standard method in [20], one can easily establish Proposition 5.2, hence we omit its proof here.
We can now readily establish Proposition 1.
Later, Samet et al. [63] established some propositions to show that many fixed point theorems on (nonsymmetric) G-metric spaces given recently by many authors follow directly from well-known theorems on metric spaces.
Note also that all the structures whose stability below the line ( c=1) has been established in Propositions 4 and 5 are not stable above this line, because the nodes supporting strong links with peripheral nodes of a tree would withdraw support for them as soon as (c>1).
With a state budget deficit of $20 billion and Californians looking for someone to blame, Mr. Darling is not alone in opposing the need for a supermajority for new taxes, which was established by Proposition 13, the antitax measure passed in 1978, the last year he ran.
Thus, by reasoning as in Theorem 3.3, and using the estimates established in Proposition 3.2, the result follows.
In addition, the case of fewer firms also leads to more anger, as established by Proposition 1.
The property of a local valuation problem established in Proposition 7 is fairly general: its validity hinges on the existence and uniqueness of a solution to a common BSDE for the gained value and the ex-dividend price.
Case (i): m ≥ 1 and 0 ≤ n ≤ N. Let v y = V p ′ y + ϕ p ′ y, ∀ y ∈ Ω ̄ ε, where ϕ is the unique solution of the problem (55), which is established in Proposition 3.2.
Let ((h^{epsilon},u^{epsilon})in H^{3}(Omega times (H^{3}(Omega))^{2}) be the smooth solution of system (1.1 - 1.3 1.1 - 1.3hestablishedsinion 1.1 and let u be a classical solution of the rotating lake equations (1.8)-(1.9) with the initial data (u_{0}) which satisfies assumProposition2}) in (L^{infty}([0,T_{ast}];H^{s}(Omega))) and (T_{ast}) is the existence time of (1.1)-(1.9).
