POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE

Jie Liu; Zhibo Cheng; Yi Wang

doi:10.11948/20190305

2020 Volume 10 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(4): 1636-1650

Next Article Previous Article

Jie Liu, Zhibo Cheng, Yi Wang. POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1636-1650. doi: 10.11948/20190305

Citation:

Jie Liu, Zhibo Cheng, Yi Wang. POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1636-1650. doi: 10.11948/20190305

POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE

1.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2.
Department of Mathematics, Sichuan University, Chengdu, 610064, China
3.
School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Corresponding author: Email address: czb_1982@126.com (Z. Cheng)
Fund Project: Research is supported by National Natural Science Foundation of China (11501170, 11571201, 11601048, 11601270), Key Research Funds for the Universities of Henan Province (19A110018, 20B110006), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302, NSFRF180320), Young backbone teachers of colleges and universities in Henan Province (2017GGJS057) and Henan Polytechnic University Doctor Fund (No. B2016-58)

Abstract

Abstract

This paper is devoted to investigate the following second-order nonlinear differential equation with singularity of attractive type $ \begin{equation*} x''-a(t)x = f(t,x)+e(t), \end{equation*} $ where the nonlinear term $ f $ has a singularity at the origin. By using the Green's function of the linear differential equation with constant coefficient and Schauder's fixed point theorem, we establish some existence results of positive periodic solutions.
- Positive periodic solutions /
- singularity of attractive type /
- attractiverepulsive singularities
MSC: 34B16, 34B18, 34C25

FullText(HTML)

References

[1]	Z. Cheng and J. Ren, Periodic solution for second order damped differential equations with attractive-repulsive singularities, Rocky Mountain J. Math., 2018, 48, 753–768. doi: 10.1216/RMJ-2018-48-3-753 CrossRef Google Scholar
[2]	Z. Cheng and F. Li, Positive periodic solutions for a kind of second-order neutral differential equations with variable coefficient and delay, Mediterr. J. Math., 2018, 15, 19. doi: 10.1007/s00009-017-1064-x CrossRef Google Scholar
[3]	W. Cheng, J. Ren and W. Han, Positive periodic solution of second-order neutral functional differential equations, Nonlinear Anal., 2009, 71, 3948–3955. doi: 10.1016/j.na.2009.02.064 CrossRef Google Scholar
[4]	J. Chu, P. Torres and M. Zhang, Periodic solution of second order non-autonomous singular dynamical systems, J. Differential Equations, 2007, 239, 196–212. doi: 10.1016/j.jde.2007.05.007 CrossRef Google Scholar
[5]	J. Chu and P. Torres, Applications of Schauder¡s fixed point theorem to singular differential equations, Bull. Lond. Math. Soc., 2007, 39, 653–660. doi: 10.1112/blms/bdm040 CrossRef Google Scholar
[6]	A. Fonda, R. Manásevich and F. Zanolin, Subharmonics solutions for some second order differential equations with singularities, SIAM J. Math. Anal., 1993, 24, 1294–1311. doi: 10.1137/0524074 CrossRef Google Scholar
[7]	R. Hakl and P. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 2010, 248, 111–126. doi: 10.1016/j.jde.2009.07.008 CrossRef Google Scholar
[8]	R. Hakl and M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations, 2017, 263, 451–469. Google Scholar
[9]	D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 2005, 211, 282–302. doi: 10.1016/j.jde.2004.10.031 CrossRef Google Scholar
[10]	A. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 1987, 99, 109–114. doi: 10.1090/S0002-9939-1987-0866438-7 CrossRef Google Scholar
[11]	R. Ma, R. Chen and Z. He, Positive periodic solutions of second-order differential equations with weak singularities, Appl. Math. Comput., 2014, 232, 97–103. Google Scholar
[12]	J. Ren, D. Zhu and H. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 2019, 24, 1843–1865. Google Scholar
[13]	I. Rachunková, M. Tvrdý and Vrkoč, Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 2001, 176, 445–469. doi: 10.1006/jdeq.2000.3995 CrossRef Google Scholar
[14]	P. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 2007, 232, 277–284. doi: 10.1016/j.jde.2006.08.006 CrossRef Google Scholar
[15]	P, Torres, Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 2007, 137, 195–201. doi: 10.1017/S0308210505000739 CrossRef Google Scholar
[16]	A. Urena, Periodic solutions of singular equations, Topol. Methods Nonlinear Anal., 2016, 47, 55–72. Google Scholar
[17]	H. Wang, Positive periodic solutions of singular systems with a parameter, J. Differential Equations, 2010, 249, 2986–3002. doi: 10.1016/j.jde.2010.08.027 CrossRef Google Scholar
[18]	Z. Wang and T. Ma, Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities, Nonlinearity, 2012, 25, 279–307. doi: 10.1088/0951-7715/25/2/279 CrossRef Google Scholar
[19]	Z. Wang, Periodic solutions of Liénard equation with a singularity and a deviating argument, Nonlinear Anal. Real World Appl., 2014, 16, 227–234. doi: 10.1016/j.nonrwa.2013.09.021 CrossRef Google Scholar
[20]	Y. Xin and Z. Cheng, Positive periodic solution to indefinite singular LišŠnard equation, Positivity, 2019, 23, 779–787. doi: 10.1007/s11117-018-0637-7 CrossRef Google Scholar
[21]	S. Yao and J. Liu, Study on variable coeffcients singular differential equation via constant coeffcients differential equation, Bound. Value Probl., 2019, 2019(3), 24 pp. Google Scholar
[22]	E. Zeidler, Applied functional analysis, New York: Springer-Verlag, 1995. Google Scholar
[23]	M. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 1996, 203, 254–269. doi: 10.1006/jmaa.1996.0378 CrossRef Google Scholar