Abstract
In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type(x(t)−cx(t−σ))″+f(x(t))x′(t)−φ(t)xμ(t)+α(t)xγ(t)=e(t),\\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}$$ \\bigl(x(t)-cx(t-\\sigma)\\bigr)''+f\\bigl(x(t) \\bigr)x'(t)-\\varphi(t)x^{\\mu}(t)+ \\frac{\\alpha(t)}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+infty)rightarrow R, varphi(t)>0 and alpha(t)>0 are continuous functions with T-periodicity in the t variable, c, γ are constants with |c|<1, gammageq1. Many authors obtained the existence of periodic solutions under the condition 0<muleq1, and we extend the result to mu>1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.
Highlights
The second order differential equations with singularities have a wide range of applications in many subjects, such as physics, engineering, mechanics, and so on
Many authors obtained the existence of periodic solutions under the condition 0 < μ ≤ 1, and we extend the result to μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates
1 Introduction The second order differential equations with singularities have a wide range of applications in many subjects, such as physics, engineering, mechanics, and so on
Summary
The second order differential equations with singularities have a wide range of applications in many subjects, such as physics, engineering, mechanics, and so on (see [1,2,3,4,5]). Α(t) xγ (t) e(t), where f : (0, +∞) → R, φ(t) > 0 and α(t) > 0 are continuous functions with T-periodicity in the t variable, c, γ are constants with |c| < 1, γ ≥ 1. Many authors obtained the existence of periodic solutions under the condition 0 < μ ≤ 1, and we extend the result to μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates.
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.
