Recurrent and almost recurrent multivalued maps and their selections. III

Let $(U,\rho)$ be a complete metric space and let $\mathcal R^p(\mathbb R,U),$ $p\geqslant~1$, and $\mathcal R(\mathbb R,U)$ be the spaces of (strongly) measurable functions $f\colon\mathbb R\to U$ for which the Bochner transforms $\mathbb R\ni t\mapsto f^B_l(t;\cdot)=f(t+\cdot)$ are recurrent functions with ranges in the metric spaces $L^p([-l,l],U)$ and $L^1([-l,l],(U,\rho'))$ where $l>0$, and $(U,\rho')$ is the complete metric space with the metric $\rho'(x,y)=\min\{1,\rho(x,y)\}$, $x,y\in U$. Analogously, we define the spaces $\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$ and $\mathcal R(\mathbb R,\mathrm{cl}_bU)$ of functions (multivalued mappings) $F\colon\mathbb R\to\mathrm{cl}_bU$ with ranges in the complete metric space $(\mathrm{cl}_bU,\mathrm{dist})$ of nonempty closed bounded subsets of the metric space $(U,\rho)$ with the Hausdorff metric $\mathrm{dist}$ (while defining the multivalued mappings $F\in\mathcal R(\mathbb R,\mathrm{cl}_bU)$ the metric $\mathrm{dist}'(X,Y)=\min\{1,\mathrm{dist}(X,Y)\}$, $X,Y\in\mathrm{cl}_bU$, is also considered). We prove the existence of selectors $f\in\mathcal R(\mathbb R,U)$ (accordingly $f\in\mathcal R^p(\mathbb R,U)$) of multivalued maps $F\in\mathcal R(\mathbb R,\mathrm{cl}_bU)$ (accordingly $F\in\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$) for which the sets of almost periods are subordinated to the sets of almost periods of multivalued maps $F$. For functions $g\in\mathcal R(\mathbb R,U),$ the conditions for the existence of selectors $f\in\mathcal R(\mathbb R,U)$ and $f\in\mathcal R^p(\mathbb R,U)$ such that $\rho(f(t),g(t))=\rho(g(t),F(t))$ for a.e. $t\in\mathbb R$ are obtained. On the assumption that the function $g$ and the multivalued map $F$ have relatively dense sets of common $\varepsilon$-almost periods for all $\varepsilon>0$, we also prove the existence of selectors $f\in\mathcal R(\mathbb R,U)$ such that $\rho(f(t),g(t))\leqslant\rho(g(t),F(t))+\eta(\rho(g(t),F(t)))$ for a.e. $t\in\mathbb R$, where $\eta\colon[0,+\infty)\to[0,+\infty)$ is an arbitrary nondecreasing function for which $\eta(0)=0$ and $\eta(\xi)>0$ for all $\xi>0$, and, moreover, $f\in\mathcal R^p(\mathbb R,U)$ if $F\in\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$. To prove the results we use the uniform approximation of functions $f\in\mathcal R(\mathbb R,U)$ by elementary functions belonging to the space $\mathcal R(\mathbb R,U)$ which have the sets of almost periods subordinated to the sets of almost periods of the functions $f$.