Step 2 (weak-type (p, p)

boundedness for [p.sub.L] < p < 2).

The

boundedness of G in Assumption 4.1 and the stability estimates in (6.2) and (6.3) allow us to control Ge and G[[partial derivative].sub.t]e.

It is noted the

boundedness of [[alpha].sub.Ui], [[??].sub.Ui], [[alpha].sub.[psi]Ti], and [[??].sub.[psi]Ti] ensures the

boundedness of [[alpha].sup.c.sub.Ui], [[??].sup.c.sub.Ui] [[??].sup.c.sub.Ui], and [[??].sup.c.sub.[psi]Ti] by Lemma 9.

Furthermore, because [[theta].sub.[sigma]] and [[eta].sub.[sigma]] are bounded, the

boundedness of the estimated parameters [[??].sub.[sigma]](k) and [[??].sub.[sigma]](k) is guaranteed by [mathematical expression not reproducible].

So, V(t) is uniform ultimate

boundedness. Combining (17) and (36), there is

Medina, "Existence and

boundedness of solutions for nonlinear volterra difference equations in banach spaces," Abstract and Applied Analysis, vol.

In this paper, we consider some characterizations of

boundedness in [N.sub.*](D) and [N.sup.p](D) (p > 1).

For [II.sub.1], [L.sup.2]

boundedness of SL and the Holder inequality tell us

If we assume that [mathematical expression not reproducible] and 0 [less than or equal to] [[alpha].sub.p] [??] 1 then, defining the norms according to (2.5), the

boundedness of both [mathematical expression not reproducible] and (div u, q) + (div v, q) is obvious.

Next, we prove the

boundedness of solutions by considering the following function:

It follows from (41) and the

boundedness of {([u.sub.n], [v.sub.n])} in [D.sub.K]([OMEGA]) that

Ljubisa D.R Koeinac shared his thoughts and research work with audience on 'Function Spaces and

Boundedness Properties'.