By Leeb B.

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10) that φ(t) ≤ C0 (1 + t)−η with η = min1≤i≤m {( i − θi )/(1 − i )}. 11. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup φ(s)1+γ ≤ K 0 (1 + t)γ {φ(t) − φ(t + 1)} t ≤s≤t +1 for some constants K 0 > 0, γ > 0, β < 1. Then φ(t) has the decay property: φ(t) ≤ C0 (1 + t) − (1−β) γ ; and if γ = 0, then φ(t) ≤ C0 exp{−λt 1−β } where C0 > 0, λ > 0 are constants. 3. 12. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup t ≤s≤t +T φ(s)1+γ ≤ g(t)[φ(t) − φ(t + T )] with constants T > 0, γ > 0 and g(t) is a non-decreasing function.

A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is dissipative if and only if (λI − A)x ≤ λ x f or all x ∈ D(A), λ > 0. 6. The motivation for the use of the word “dissipative” comes from the case where X is a Hilbert space. ) denotes the scalar inner product on X. 11. A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is m-dissipative if A is dissipative and R(λI − A) = X f or all λ > 0 that is, for any given g ∈ X, there is f ∈ D(A) such that (λI − A) f = g. 1. Every m-dissipative operator is a dissipative operator.

Preliminary where h(t) ≥ 0 with +∞ 0 h(t)dt ≤ C2 < +∞ and f is a nondecreasing function from R+ into R+ . Then lim y(t) = 0. 6. 5. Then lim y(t) = 0. t →+∞ From the above context of this subsection, we only know that the non-negative function (y(t), say) goes to zero as time tends to inﬁnity. We have no information on the decay rate of y(t). In fact, the decay rate of y(t) depends on some factors which include some terms in the inequality. , Mu˜noz Rivera [275]). 7. Suppose that y(t) ∈ C 1 (R+ ), y(t) ≥ 0, ∀t > 0 and satisﬁes y (t) ≤ −C0 y(t) + C1 e−γ t where C0 , C1 and γ are positive constants.

### 3-manifolds with(out) metrics of nonpositive curvature by Leeb B.

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