Abstract
A class of second-order singular differential equations with a linear difference operator is investigated in this paper. The novelty of the present article is that for the first time we show that weak and strong singularities enable the achievement of a new existence criterion of positive periodic solutions through an applications of a fixed point theorem of Krasnoselskii’s, i.e., our results of the existence of positive periodic solutions reveal a delicate relation between the value of external force e(t) and the velocity of nonlinear term \(f(t,x(t-\tau (t)))\) approaching towards infinity when \(x(t-\tau (t))\) tending to zero.
Similar content being viewed by others
References
Nagumo, M.: On the Periodic Solution of an Ordinary Differential Equation of second Order. Zenkoku Shijou Suugaku Danwakai 19944; 54–61. English translation in Mitio Nagumo collected papers, Springer, Berlin (1993)
Lazer, A.C., Solimini, S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)
Torres, P.J.: Mathematical Models with Singularities—A Zoo of Singular Creatures. Atlantis Press, Paris (2015)
Cheng, Z., Ren, J.: Periodic and subharmonic solutions for Duffing equation with a singularity. Discrete Contin. Dyn. Syst. 32, 1557–1574 (2012)
Cheng, Z., Ren, J.: Studies on a damped differential equation with repulsive singularity. Math. Methods Appl. Sci. 36, 983–992 (2013)
Cheng, Z., Ren, J.: Existence of positive periodic solution for variable-coefficient third-order differential equation with singularity. Math. Methods Appl. Sci. 37, 2281–2289 (2014)
Chu, J., Torres, P.J., Zhang, M.: Periodic solution of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239, 196–212 (2007)
Fonda, A., Manásevich, R., Zanolin, F.: Subharmonic solutions for some second-order differential equations with singularities. SIAM J. Math. Anal. 24, 1294–1311 (1993)
Fonda, A., Toader, R.: Radially symmetric systems with a singularity and asymptotically linear growth. Nonlinear Anal. 74, 2485–2496 (2011)
Hakl, R., Torres, P.J.: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248, 111–126 (2010)
Ma, R., Chen, R., He, Z.: Positive periodic solutions of second-order differential equations with weak singularities. Appl. Math. Comput. 232, 97–103 (2014)
Pino, M., Manásevich, R., Montero, A.: \(T\)-periodic solutions for some second order differential equations with singularities. Proc. R. Soc. Edinb. Sect. A 120, 231–243 (1992)
Pino, M., Manásevich, R.: Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 130, 269–277 (1993)
Torres, P.J.: Existence of one signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed theorem. J. Differ. Equ. 190, 643–662 (2003)
Torres, P.J.: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 232, 277–284 (2007)
Wang, H.: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 249, 2986–3002 (2010)
Wang, Z., Ma, T.: Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity 25, 279–307 (2012)
Xia, J., Wang, Z.: Existence and multiplicity of periodic solutions for the Duffing equation with singularity. Proc. R. Soc. Edinb. Sect. A 137, 625–645 (2007)
Zhang, M.: A relationship between the periodic and the Dirichlet BVPs of singular differential equations. Proc. Roy. Soc. Edinb. Sect. A 128, 1099–1114 (1998)
Evans, G.W., Ramey, G.: Adaptive expectations, under parameterization and the Lucas critique. J. Monet. Econ. 53, 249–264 (2006)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New Work (1993)
Ardjouni, A., Djoudi, A.: Existence, uniqueness and positivity of solutions for a neutral nonlinear periodic differential equation. Comput. Appl. Math. 34, 17–27 (2015)
Cheung, W.S., Ren, J., Han, W.: Positive periodic solution of second-order neutral functional differential equations. Nonlinear Anal. 71, 3948–3955 (2009)
Du, B., Liu, Y., Abbas, I.A.: Existence and asymptotic behavior results of periodic solution for discrete-time neutral-type neural networks. J. Frankl. Inst. 353, 448–461 (2016)
Luo, Y., Luo, Z.: Existence of positive periodic solutions for neutral multi-delay logarithmic population model. Appl. Math. Comput. 216, 1310–1315 (2010)
Ren, J., Cheng, Z., Siegmund, S.: Neutral operator and neutral differential equation. Abstr. Appl. Anal. 2011, 1–22 (2011)
Wu, J., Wang, Z.: Two periodic solutions of second-order neutral functional differential equations. J. Math. Anal. Appl. 329, 677–689 (2007)
Agarwal, P.R., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research is supported by NSFC Project (No. 11501170), China Postdoctoral Science Foundation funded project (No. 2016M590886), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).
Rights and permissions
About this article
Cite this article
Cheng, Z., Li, F. Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator. J. Fixed Point Theory Appl. 21, 48 (2019). https://doi.org/10.1007/s11784-019-0687-x
Published:
DOI: https://doi.org/10.1007/s11784-019-0687-x
Keywords
- Positive periodic solution
- linear difference operator
- weak and strong singularities
- Krasnoselskii’s fixed point theorem