Sentence examples for under the additional condition from inspiring English sources

Under the additional condition that ∑ n = 1 ∞ φ 1 / 2 ( 2 n ) < ∞, (2.2) holds for some constant functions β s ( x ) and γ s ( x ).

The inequality of type (2.9) was studied by Chang et al. [14] under the additional condition that is a nonempty bounded closed subset of.

Hence condition (2.24) holds for an array of rowwise -mixing random variables under the additional condition that depending on the sequence of random variables in each row are bounded.

Amini et al. (2016 ) prove uniqueness of price-payment equilibria in a model with fire sales under the additional condition on the inverse demand function f that x↦x f(x) is strictly increasing.

In this paper, we will prove this conjecture under the additional condition that μ ( f ) < 1 2. Theorem 1 Let n ∈ N and let f be a transcendental entire function of order ρ ( f ) < 1.

(4) Under the additional condition that there exist θ ∈ ( 0, 1 ) and some k 1 ≥ 0 such that d ( T x, T y ) ≤ θ. d ( x, y ) + k 1. d ( x, T x ) for all  x, y ∈ X, (6)  .

We now deal with an asymptotic behavior of functions in under the additional conditions (3.1) and (3.2).

However, this phenomenon raises in complete graphs under the additional conditions that (1) agents seldom miss the posted articles and (2) they do not comment on articles posted by non-reciprocal agents.

We first prove the theorem under the additional conditions (u^{0}, v^{0}in C^{1} overline{Omega})), u^{0}gg0, qquad v^{0}gg0, (3.1) and for all (xinoverline{Omega}), u^{0}leqbar{u}, qquad v^{0}leqbar{v}, (3.2) where (bar{u}) and (bar{v}) are defined in (1.6).

Moreover, we also find that Theorem 3.1 of [16] holds under the following additional conditions: t − δ i ( t ) ≥ 0 for all  t ≥ 0, i ∈ J n.

In our present paper we propose to consider similar questions on uniformly convex metric spaces under the mildest additional conditions we may impose.