Throughout this subsection we assume, in addition to (A1)–(A5), that (A6)(_{scriptscriptstyle +}) holds.
Assume in addition to (S 1) and (S 2), the following conditions are satisfied, there exists 0 < r < ξ 1 ρ T < ρ < ∞ such that.
Moreover the relevant calculations will be critical for us in the proof of tangential boundary estimates when G is nonlinear in p. Assume in addition to the hypotheses of Theorem 1.1 that the operator ({mathcal {F}}) is orthogonally invariant.
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions: A2. { z ∈ P : z ≼ u for some u ∈ F ( x ) } is an inductive subset of P, for each x ∈ P. A2′.
Assume in addition to (S 1), (S 2) there exist positive constants a < T η a < b < T ξ m - 2 b < c such that the following conditions hold.
Example 4.2 Let us assume, in addition to the definitions of G and P of Example 4.1, ħ = e x 1 sin x 2, Θ = { x = ( x 1, x 2 ) : | 0 ≤ x 1 2 + x 2 2 < 1 }.
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions: A2. S F ( x ) is an inductive subset of P, for each x ∈ P. A2′.
In order to implement a cooperation plan between the players, we assume, in addition to the individual CSI assumption, that every player is able to know the power of the received signal at each game stage, which is denoted by [18]: P_{y} = sigma^{2} + sum_{substack{i=1}}^{N} p_{i}|g_{i}|^{2}.
Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, and the measure μ in Theorem 2.4, that q is any integer satisfying 1 < q < ∞, v ∈ C∞, ℓ = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ and ∣ v ∣ ∈ L l o c q.
Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, the measure μ, and the Young function φ in Theorem 2.4, that ν ∈ C∞ Θ, Λ k ), k = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded L φ -averaging domains Θ and φ ∈ L1.
