On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$\end{document}, where Φ is a coercive continuous operator from Lp to Lq, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$\end{document}; first- and second-order partial differential equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$\end{document}; and general equations F(x; ..., u″ii, ...., ...., u′i, ...; u) = g(x) of elliptic type.