MULTIPLICITY RESULTS OF FOURTH-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATION WITH A PARAMETER

Yun Xin; Xuefeng Han; Zhibo Cheng

doi:10.11948/2017029

2017 Volume 7 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(2): 455-477

Next Article Previous Article

Yun Xin, Xuefeng Han, Zhibo Cheng. MULTIPLICITY RESULTS OF FOURTH-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATION WITH A PARAMETER[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 455-477. doi: 10.11948/2017029

Citation:

Yun Xin, Xuefeng Han, Zhibo Cheng. MULTIPLICITY RESULTS OF FOURTH-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATION WITH A PARAMETER[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 455-477. doi: 10.11948/2017029

MULTIPLICITY RESULTS OF FOURTH-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATION WITH A PARAMETER

1 College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, China;
2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China;
3 Department of Mathematics, Si Chuan University, Cheng Du, 610064, China

Fund Project:

Abstract

Abstract

In this paper, we investigate a class of fourth-order singular nonlinear differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green's function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.
- Fourth-order /
- superlinear and sublinear /
- singular /
- Krasnoselskii fixed point theorem /
- periodic solution
MSC: 34C25;34B16;34B18

FullText(HTML)

References

[1]	D. Bonheure and C. De Coster, Forced singular oscilla tors and the method of loeer and upper solutions, Topol. Methods Nonlinear Anal., 2003, 22, 927-938. Google Scholar
[2]	Z. Cheng, Existence of positive periodic solutions for third-order differential equation with strong singularity, Adv. Difference Equ., 2014, 2014(162), 1-12. Google Scholar
[3]	Z. Cheng and J. Ren, Periodic solutions for a fourth-order Rayleigh type pLaplacian delay equation, Nonlinear Anal. TMA, 2009, 70, 516-523. Google Scholar
[4]	Z. Cheng and J. Ren, Periodic and subharmonic solutions for Duffing equation with singularity, Discrete Contin. Dyn. Syst. A, 2012, 32, 1557-1574. Google Scholar
[5]	Z. Cheng and J. Ren, Studies on a damped differential equation with repulsive singularity, Math. Meth. Appl. Sci., 2013, 36, 983-992. Google Scholar
[6]	Z. Cheng and J. Ren, Positive solutions for third-order variable-coefficient nonlinear equation with weak and strong singularities, J. Difference Equ. Appl., 2015, 21, 1003-1020. Google Scholar
[7]	Z. Cheng and J. Ren, Existence of periodic solution for fourth-order Liénard type p-Laplacian generalized neutral differential equation with variable parameter, J. Appl. Anal. Comput., 2015, 5, 704-720. Google Scholar
[8]	J. Chu, P. Torres and M. Zhang, Periodic solution of second order nonautonomous singular dynamical systems, J. Differential Equations, 2007, 239, 196-212. Google Scholar
[9]	M. Conti, S. Terracini and G. Verzini, Infinitely many solutions to fourth-order superlinear periodic problem, Trans. Amer. Math. Soc., 2003, 356, 3283-3300. Google Scholar
[10]	Y. Cui and Y. Zou, Existence and uniqueness theorems for fourth-order singular boundary value problems, Comput. Math. Appl., 2009, 58, 1449-1456. Google Scholar
[11]	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. Google Scholar
[12]	R. Hakl and P. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 2010, 248, 111-126. Google Scholar
[13]	M. Krasnoseskill, Positive Solutions of Operator Equation, Noordhoff, Groningen, 1964. Google Scholar
[14]	A. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 1987, 99, 109-114. Google Scholar
[15]	X. Li and B. Wu, A novel method for nonlinear singular fourth order four-point boundary value problems, Comput. Math. Appl., 2011, 62, 27-31. Google Scholar
[16]	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. Google Scholar
[17]	J. Ren, Z. Cheng and Y. Chen, Existence results of periodic solutions for thirdorder nonlinear singular differential equation, Math. Nachr., 2013, 286, 1022-1042. Google Scholar
[18]	J. Ren, S. Siegmund and Y. Chen, Positive periodic solutions for third-order nonlinear differential equations, Electron. J. Differential Equations, 2011, 66, 1-19. Google Scholar
[19]	J. Sun and Y. Liu, Multiple positive solutions of singular third-order periodic boundary value problem, Acta Math. Sci., 2005, 25, 81-88. Google Scholar
[20]	P. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 2007, 232, 277-284. Google Scholar
[21]	H. Wang, Positive periodic solutions of singular systems with a parameter, J. Differential Equations, 2010, 249, 2986-3002. Google Scholar
[22]	H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl., 2003, 281, 287-306. Google Scholar
[23]	J. Xia and Z. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. R. Soc. Edinb. Sect. A, 2007, 137, 625-645. Google Scholar
[24]	Y. Xin and Z. Cheng, Some results for fourth-order nonlinear differential equation with singularity, Bound. Value Probl., 2015, 2015(200), 1-20. Google Scholar
[25]	M. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 1996, 203, 254-269. Google Scholar
[26]	K. Zhang, Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms, Topol. Methods Nonlinear Anal., 2012, 40, 53-70. Google Scholar