JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (8): 84-94.doi: 10.6040/j.issn.1671-9352.0.2017.582

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Existence of periodic solutions of a class of third order delay differential equations in Banach spaces

CHEN Yu-jia, YANG He*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2017-11-14 Online:2018-08-20 Published:2018-07-11

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Abstract: Applying the monotone iterative method of upper and lower solutions, we discuss the existence of 2π-periodic solutions for the third-order differential equation with delays in ordered Banach space Eu(t)+M0u(t-τ0)=f(t,u(t), u(t-τ1), u(t-τ2)), t∈R,where f:R×E3→E is a continuous function which is 2π-periodic in t, and τ0,τ1,τ2 are positive constants. By establishing a new maximum principle, a monotone iterative procedure for the equation is constructed. Some existence and uniqueness results of 2π-periodic solutions for this equation are obtained.

Key words: ordered Banach space, differential equation with delays, monotone iterative technique, maximum principle, periodic solution

CLC Number: 

  • O175.15
[1] JUAN J Nieto, YU Jiang, YAN Jurang. Monotone iterative method for functional-differential equations[J]. Nonlinear Analysis, 1998, 32(6):741-747.
[2] JIANG Daqing, WEI Junjie. Monotone method for first-and second-order periodic boundary value problems and periodic solutions of functional differential equations[J]. Nonlinear Analysis. 2002, 50(7): 885-898.
[3] JIANG Daqing, JUAN J Nieto, ZUO Wenjie. On monotone method for first and second-order periodic boundary value problems and periodic solutions of functional differential equations[J]. Journal of Mathematical Analysis and Applications. 2004, 289(2): 691-699.
[4] WU Yuexiang. Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter[J]. Nonlinear Analysis, 2009, 70(1): 433-443.
[5] WANG Youyu, LIAN Hairong, GE Weigao. Periodic solutions for a second order nonlinear functional differential equation[J]. Applied Mathematics Letters, 2007, 20(1): 110-115.
[6] LI Qiang, LI Yongxiang. Existence and multiplicity of positive periodic solutions for second-order functional differential equations with infinite delay[J]. Electronic Journal of Differential Equations, 2014(93): 1-14.
[7] CABADA A. The method of lower and upper solutions for second, third, fourth and higher order boundary value problem[J]. Journal of Mathematical Analysis and Applications, 1994, 185(2): 302-320.
[8] JUAN J Nieto. Nonlinear second-order periodic boundary value problems[J]. Journal of Mathematical Analysis and Applications, 1988, 130(1): 22-29.
[9] PEDRO J Torres. Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem[J]. J Differential Equations, 2003, 190(2):643-662.
[10] LI Yongxiang. Positive periodic solutions for fully third-order ordunary differenttial equations[J]. Computers and Mathematics with Applications, 2010, 59(11): 3461-3471.
[11] CHU Jifeng, ZHOU Zhongcheng. Positive solutions for singular non-linear third-order periodic boundary value problems[J]. Nonlinear Analysis, 2006, 64(7):1528-1542.
[12] GUO Dajun, LIU Xinzhi. Extremal solutions for a boundary value problem of nth-order impulsive integro-differential equations in a banach space[J]. Mathematical Analysis, 2006, 13(5): 599-619.
[13] HANS Peter Heinz. On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions[J]. Nonlinear Analysis, 1983, 7(12):1351-1371.
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