Positive periodic solutions to super-linear second-order ODEs

článek v časopise ve Web of Science, Jimp

en

We study the existence and uniqueness of a positive solution to the problemu ''=p(t)u+q(t,u)u+f(t);u(0)=u(omega),u '(0)=u '(omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u<^>{\prime \prime }} = p(t)u + q(t,u)u + f(t);\,\,\,\,\,u(0) = u(\omega ),\,\,\,{u<^>\prime }(0) = {u<^>\prime }(\omega )$$\end{document}with a super-linear nonlinearity and a nontrivial forcing term f. To prove our main results, we combine maximum and anti-maximum principles together with the lower/upper functions method. We also show a possible physical motivation for the study of such a kind of periodic problems and we compare the results obtained with the facts well known for the corresponding autonomous case.

We study the existence and uniqueness of a positive solution to the problemu ''=p(t)u+q(t,u)u+f(t);u(0)=u(omega),u '(0)=u '(omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u<^>{\prime \prime }} = p(t)u + q(t,u)u + f(t);\,\,\,\,\,u(0) = u(\omega ),\,\,\,{u<^>\prime }(0) = {u<^>\prime }(\omega )$$\end{document}with a super-linear nonlinearity and a nontrivial forcing term f. To prove our main results, we combine maximum and anti-maximum principles together with the lower/upper functions method. We also show a possible physical motivation for the study of such a kind of periodic problems and we compare the results obtained with the facts well known for the corresponding autonomous case.

second-order differential equation; super-linearity; positive solution; existence; uniqueness

01.03.2025

SPRINGER HEIDELBERG

HEIDELBERG

0011-4642

75

1

257–275

19