Reversed <em>S</em> -shaped connected component for second-order periodic boundary value problem with sign-changing weight

Liangying Miao; Zhiqian He; Liangying Miao; Zhiqian He

doi:10.3934/math.2020376

AIMS Mathematics

2020, Volume 5, Issue 6: 5884-5892. doi: 10.3934/math.2020376

Previous Article Next Article

Research article

Reversed S -shaped connected component for second-order periodic boundary value problem with sign-changing weight

Liangying Miao ^{1
,
,},
Zhiqian He ²

1.
School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, P. R. China
2.
Department of Basic Teaching and Research, Qinghai University, Xining 810016, P. R. China

Received: 25 April 2020 Accepted: 06 July 2020 Published: 15 July 2020
MSC : 34B10, 34B18

In this paper, we consider the existence of an reversed S -shaped connected component in the set of positive solutions for second order periodic boundary value problem with a sign-changing weight function. By bifurcation technique, we identify the interval of bifurcation parameter in which the periodic boundary value problem has one or two or three positive solutions according to the asymptotic behavior of f at 0 and ∞.
- positive solution,
- periodic boundary value problem,
- bifurcation
Citation: Liangying Miao, Zhiqian He. Reversed S -shaped connected component for second-order periodic boundary value problem with sign-changing weight[J]. AIMS Mathematics, 2020, 5(6): 5884-5892. doi: 10.3934/math.2020376

Related Papers:

Abstract

In this paper, we consider the existence of an reversed S -shaped connected component in the set of positive solutions for second order periodic boundary value problem with a sign-changing weight function. By bifurcation technique, we identify the interval of bifurcation parameter in which the periodic boundary value problem has one or two or three positive solutions according to the asymptotic behavior of f at 0 and ∞.

References

[1]	F. M. Atici, G. S. Guseinov, On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. Comput. Appl. Math., 132 (2001), 341-356. doi: 10.1016/S0377-0427(00)00438-6
[2]	A. Boscaggin, F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differ. Equations, 252 (2012), 2900-2921. doi: 10.1016/j.jde.2011.09.011
[3]	A. Constantin, A general-weighted Sturm-Liouville problem, Ann. Sci. Ec. Norm. Super., 24 (1997), 767-782.
[4]	G. Dai, R. Ma, H. Wang, Eigenvalues, bifurcation and one-sign solutions for the periodic p-Laplacian, Commun. Pure Appl. Anal., 12 (2013), 2839-2872. doi: 10.3934/cpaa.2013.12.2839
[5]	J. R. Graef, L. Kong, H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. Differ. Equations, 245 (2008), 1185-1197.
[6]	R. Hakl, M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differ. Equations, 263 (2017), 451-469. doi: 10.1016/j.jde.2017.02.044
[7]	X. Hao, L. Liu,Y. Wu, Existence and multiplicity results for nonlinear periodic boundary value problems, Nonlinear Anal., 72 (2010), 3635-3642. doi: 10.1016/j.na.2009.12.044
[8]	Z. He, R. Ma, M. Xu, Positive solutions for a class of semipositone periodic boundary value problems via bifurcation theory, Electron. J. Qual. Theory Differ. Equ., 29 (2019), 1-15.
[9]	Z. He, R. Ma, M. Xu, Three positive solutions for second-order periodic boundary value problems with sign-changing weight, Bound. Value Probl., 93 (2018), 1-17.
[10]	A. Lomtatidze, J. Šremr, On periodic solutions to second-order Duffing type equations, Nonlinear Anal. Real, 40 (2018), 215-242. doi: 10.1016/j.nonrwa.2017.09.001
[11]	R. Ma, Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364-4376. doi: 10.1016/j.na.2009.02.113
[12]	R. Ma, J. Xu, X. Han, Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight, Nonlinear Anal., 74 (2011), 3379-3385. doi: 10.1016/j.na.2011.02.013
[13]	R. Ma, C. Gao, R. Chen, Existence of positive solutions of nonlinear second-order periodic boundary value problems, Bound. Value Probl., 2010 (2010), 1-18. doi: 10.1155/2010/728101
[14]	R. Ma, J. Xu, X. Han, Global structure of positive solutions for superlinear second-order periodic boundary value problems, Appl. Math. Comput., 218 (2012), 5982-5988.
[15]	N. S. Papageorgiou, C. Vetro, F. Vetro, Solutions for parametric double phase Robin problems, Asymptotic Anal., 1 (2020), 1-12.
[16]	N. S. Papageorgiou, C. Vetro, F. Vetro, Parameter dependence for the positive solutions of nonlinear, nonhomogeneous Robin problems, RACSAM, 114 (2020), 1-29. doi: 10.1007/s13398-019-00732-2
[17]	I. Sim, S. Tanaka, Three positive solutions for one-dimensional p-Laplacian problem with signchanging weight, Appl. Math. Lett., 49 (2015), 42-50. doi: 10.1016/j.aml.2015.04.007
[18]	J. Xu, R. Ma, Bifurcation from interval and positive solutions for second order periodic boundary value problems, Appl. Math. Comput., 216 (2010), 2463-2471.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)