Sentence examples for the following assertions are true from inspiring English sources

Then the following assertions are true.

Let then the following assertions are true: If then (442).

Then the following assertions are true: (1 If is a linear nonzero positive functional, then boundary value problem (4.1), (4.2) is uniquely solvable for each.

Then, for any fixed z ∈ D ( p 4 δ [ t 0, t 0 + p ], D ( g ) ), the following assertions are true: 1.

If the set ⋂ { ε > 0 : C ≠ ∅ } C is not empty then the following assertions are true: If we denote by ε ¯ = inf { ε > 0 : C ≠ ∅ } ε, then ⋂ { ε > 0 : C ≠ ∅ } C = C. ε ¯ = inf t ∈ U sup q ∈ O b j U ( t ) d ( t, q ).

Then the following assertions are true: (1) Spr (T) is a simple eigenvalue of T, having a positive eigenfunction denoted by ψ0 > 0, i.e., ψ0 ∈ int P, and there is no other eigenvalue of T with a positive eigenfunction; (2) For every y ∈ int P, the equation u - r T u = y.

Assume that is a strictly contractive operator, that is, there exists a Lipschitz constant with such that for all Then for a given element one of the following assertions is true: for all ; there exists a nonnegative integer such that.

Assume that is a strictly contractive operator with the Lipschitz constant Then for a given element one of the following assertions is true: (A1) for all ; (A2) there exists a nonnegative integer such that.

Then, for a given element (uin X), exactly one of the following assertions is true: (a) (p(T^{n}u,T^{n+1}u)=infty) for all (nin mathbb{N} cup{0}),   (b) there exists a nonnegative integer ℓ such that (p(T^{n}u,T^{n+1}u)<infty) for all (ngeq ell).  .

Let us note that Theorems 3.2 and 3.7 of [3] show that under some restrictions on the following assertion is true.

Remark 3.3 The proof of Theorem 3.2 could be substantially simplified and also extended to the case f ∈ G ( [ a, b ], X ) if the following assertion was true.