This paper studies the relationship between vector-valued $\text{BMO}$ functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbb{T}$, respectively. For $1\,<\,q\,<\,\infty $ and a Banach space $B$, we prove that there exists a positive constant $c$ such that

$$\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{D}}{{\left( 1-\left| z \right| \right)}^{q-1}}{{\left\| \nabla f\left( z \right) \right\|}^{q}}{{P}_{{{Z}_{0}}}}\left( z \right)dA\left( z \right)\le {{c}^{q}}\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{\mathbb{T}}}{{\left\| f\left( z \right)-f\left( {{z}_{0}} \right) \right\|}^{q}}{{P}_{{{z}_{0}}}}\left( z \right)dm\left( z \right)$$

holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where

$${{P}_{{{z}_{0}}}}\left( z \right)=\frac{1-|{{z}_{0}}{{|}^{2}}}{|1-{{{\bar{z}}}_{0}}z{{|}^{2}}}.$$

The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Let $E$ be a real uniformly smooth Banach space and let $A$ be a nonlinear $\phi$-strongly quasi-accretive operator with range $R(A)$ and open domain $D(A)$ in $E$. For a given $f\in E$, let $A$ satisfy the evolution system $\rd u(t)/\rd t+Au(t)=f$, $u(0)=u_0$. We establish the strong convergence of the Ishikawa and Mann iterative methods with appropriate error terms recently introduced by Xu to the equilibrium points of this system. Related results deal with the strong convergence of the iterative methods to the fixed points of $\phi$-strong pseudocontractions defined on open subsets of $E$.

AMS 2000 Mathematics subject classification: Primary 47H06; 47H15; 47H17