Otherwise there is a sequence such that and is bounded as, where.
If b a + T 1 q ∥ x ′ ∥ p ≥ h, then ∥ x ′ ∥ p ≤ b − a h h T 1 q, that is, ∥ x ′ ∥ p is bounded as well.
(Boundedness) The solution ((P t),S t))^{T}) of (2.6) satisfying the initial condition (3.1) is bounded as long as it exists.
If ({z(n)}) is bounded, then, by Lemma 2, ({x_{1}(n)}) is bounded as well and, therefore, there exists a finite (lim_{ntoinfty}x_{1}(n)).
They also supposed that the sequence ({Y_{n},ngeq 1}) is bounded, as well as the density (f(x)) of (X_{1}).
Assume also that |bar{p}| < 1. (14) If the sequence ({z(n)}) defined by (12) is bounded, then the sequence ({x_{1}(n)}) is bounded as well.
If Φ: Ω → G is k-smooth transformation, k ≥ 1, then A is a bounded transformation from Wk,p(x) onto Wk,p(Ψ y))(G) and the inverse transformation of A is bounded as well.
This means its Frobenius norm is bounded as well and by the equivalence of norms, so is any norm, in particular (||left (boldsymbol {C_{G}^{T}}right)^{i}||_{tau }).
Thus, φ i ( T u ) ≥ 0, therefore as desired we conclude T ( K ) ⊂ K. Step 2. In this part we turn to the proof that T in the sequel is bounded as well as equicontinuous with the help of the Arzelà-Ascoli theorem.
end{aligned} Furthermore, we have theta P t)+S t)+I t)< frac{C_{0}}{beta}+ biggl(theta P 0)+S(0)+I(0)- frac{C_{0}}{beta} biggr)e^{-beta t}, which implies that ((P t),S t))) is bounded as long as it exists.
Its energy is bounded as follows.
