Levitan Almost Periodic and Almost Automorphic Solutions of V-monotone Differential Equations

In the present article we consider a special class of equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=f(t, x)\quad \quad \quad (1)$$\end{document}when the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f : \mathbb R \times E \to E$$\end{document} (E is a strictly convex Banach space) is V-monotone with respect to (w.r.t.) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in E$$\end{document} , i.e. there exists a continuous non-negative function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V: E\times E \to \mathbb R_{+}$$\end{document} , which equals to zero only on the diagonal, so that the numerical function α(t):= V(x1(t), x2(t)) is non-increasing w.r.t. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \mathbb R_{+}$$\end{document} , where x1(t) and x2(t) are two arbitrary solutions of (1) defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R_{+}$$\end{document} . The main result of this article states that every V-monotone Levitan almost periodic (almost automorphic, Bohr almost periodic) Eq. (1) with bounded solutions admits at least one Levitan almost periodic (almost automorphic, Bohr almost periodic) solution. In particulary, we obtain some new criterions of existence of almost recurrent (Levitan almost periodic, almost automophic, recurrent in the sense of Birkgoff) solutions of forced vectorial Liénard equations.