A higher order nonlinear neutral differential equation

Volume 12, Issue 10, pp 675--698 http://dx.doi.org/10.22436/jnsa.012.10.06

Publication Date: June 15, 2019 Submission Date: February 24, 2019 Revision Date: April 06, 2019 Accteptance Date: May 14, 2019

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1573 Downloads
• 2479 Views

Authors

Guojing Jiang - Basic Teaching Department, Vocational Technical College, Dalian, Liaoning 116035, China. Wei Sun - Jiaokou No.1 Middle School, Lvliang, Shanxi 032400, China. Zhefu An - School of Mathematics, Liaoning University, Shenyang, Liaoning 110036, China. Liangshi Zhao - Center for Studies of Marine Economy and Sustainable Development, Liaoning Normal University, Dalian, Liaoning 116029, China.

Abstract

This paper is concerned with the higher order nonlinear neutral differential equation \[ [a(t)(x(t)+b(t)x(\tau(t)))']^{(n-1)}+f(t, x(g_1(t)),\ldots, x(g_k(t)))=c(t),\quad t\ge t_0. \] By dint of the Leray-Schauder nonlinear alternative, Rothe fixed point theorem and some new techniques, we prove the existence of uncountably many bounded positive solutions for the equation. Several nontrivial examples are given to illustrate the applications and advantages of the results presented in this paper.

Share and Cite

Keywords

• Higher order nonlinear neutral differential equation
• uncountably many bounded positive solutions
• Leray-Schauder nonlinear alternative theorem
• Rothe fixed point theorem

MSC

•  34K40
•  35G20

References

• [1] R. P. Agarwal, S. R. Grace, D. O'Regan, Nonoscillatory solutions for discrete equations, Comput. Math. Appl., 45 (2003), 1297--1302
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985)
  • View Article
  • MathSciNet
  • Google Scholar


• [3] Z. Liu, H. Gao, S. M. Kang, S. H. Shim, Existence and mann iterative approximations of nonoscillatory solutions of nth order neutral delay differential equations, J. Math. Anal. Appl., 329 (2007), 515--529
  • View Article
  • Google Scholar


• [4] Z. G. Zhang, A. J. Yang, C. N. Di, Existence of positive solutions of second order nonlinear neutral differential equations with positive and negative terms, J. Appl. Math. Comput., 25 (2007), 245--253
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] Y. Zhou, Existence for nonoscillatory solutions of second order nonlinear differential equations, J. Math. Anal. Appl., 331 (2007), 91--96
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] X. L. Zhou, R. Yu, Oscillatory behavior of higher order nonlinear neutral forced differential equations with oscillating coefficients, Comput. Math. Anal., 56 (2008), 1562--1568
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] Y. Zhou, B. G. Zhang, Y. Q. Huang, Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations, Czechoslovak Math. J., 55 (2005), 237--253
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH