Asymptotics of the Solution of a Variational Problem on a Large Interval

Abstract The variational problem of minimizing the energy functional that results in a second- order nonlinear differential equation of pendulum type on a finite interval with natural boundary conditions is analyzed. It is shown that the number of solutions of the boundary-value problem depends on the length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} of the interval and unboundedly increases as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\to\infty$$\end{document}. The solutions on which the energy minimum is realized converge as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\to\infty$$\end{document} to the solution of a variational problem in the class of periodic functions.