Abstract
In this paper, we look for periodic solutions of planar Hamiltonian systems \t\t\t{x′=f(y)+p1(t,y),y′=−g(x)+p2(t,x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\left \\{ \\textstyle\\begin{array}{l} x'=f(y)+p_{1}(t,y),\\\\ y'=-g(x)+p_{2}(t,x). \\end{array}\\displaystyle \\right . $$\\end{document} By using the Poincaré-Birkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.
Highlights
1 Introduction In this paper, we are concerned with the existence and multiplicity of periodic solutions of planar Hamiltonian systems x = f (y) + p (t, y), ( . )
In the case when f (y) ≡ y, p (t, y) ≡ and p (t, x) = p(t), system ( . ) becomes x = y, y = –g(x) + p(t), which is equivalent to the differential equation x + g(x) = p(t)
In [ ], Fonda and Sfecci studied the periodic solutions of the planar Hamiltonian systems of the type x = g (t, y), y = –g (t, x)
Summary
) has at least one π -periodic solution provided that there exists an integer n > such that. We use the time map to study the periodic solutions of system Assume that the time map τ (c) satisfies the condition:. B are two positive constants, e, p : R → R are continuous and conditions (hi) (i = , ) and (τ ) are satisfied. From condition (h ) we know that f can be written in the form f (y) = ay+ – by– + h(y), where h : R → R is continuous and satisfies h(y). Assume that conditions (hi) (i = , , ) hold, and let (r , θ ) = φ( π , r , θ ) – θ ;. We obtain d (t) ≤ δd(t) + η x(t) + y(t) + cη
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
