JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (7): 16-21.doi: 10.6040/j.issn.1671-9352.0.2019.788

Previous Articles    

Existence and uniqueness of solutions for a class fourth-order periodic boundary value problems

WANG Tian-xiang, LI Yong-xiang*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2020-07-08

Abstract: By using Leray-Schauder fixed point theorem, this paper discusses the existence and uniqueness of solutions for the fourth-order periodic boundary value problem{u(4)(t)=f(t,u(t),u'(t)), t∈[0,1],u(i)(0)=u(i)(1), i=0,1,2,3,where f:[0,1]×R2→R continuous. existence and uniqueness of solutions is obtained under the inequality conditions that allow the nonlinear term f(t,x,y) to grow superlinearly on x and y.

Key words: fourth-order periodic boundary value problem, existence and uniqueness, superlinear growth, Leray-Schauder fixed point theorem

CLC Number: 

  • O175.8
[1] CABADA A, LOIS S. Maximum principles for fourth and sixth order periodic boundary value problems[J]. Nonlinear Analysis, 1997, 29(2):1161-1171.
[2] CABADA A. The method of lower and upper solutions for second, third, fourth and higher order boundary value problems[J]. Mathematical Analysis and Applications, 1994, 185(5):302-320.
[3] LI Yongxiang. Positive periodic solutions of fourth order periodic boundary value problems[J]. Nonlinear Analysis, 2003, 54(6):1069-1078.
[4] YAO Qingliu. Existence of multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problems[J]. Nonlinear Analysis, 2005, 63(1):237-246.
[5] JIANG Daqing, LIU Zhaoli, ZHANG Lili. Optimal existence theory for single and multiple positive solutions to fourth order periodic boundary value problems[J]. Nonlinear Analysis, 2006, 7(6):841-852.
[6] 刘仁义. 一类四阶周期边值问题的正解存在性与多解性[J]. 兰州大学学报, 2005, 41(1):112-114. LIU Renyi. Existence and multiplicity of positive solutions for a class of fourth order periodic boundary value problems[J]. Journal of Lanzhou University, 2005, 41(1):112-114.
[7] BEREANU C. Periodic solutions of some fourth order nonlinear differential equations[J]. Nonlinear Analysis, 2009, 71(4):53-57.
[8] JIANG Daqing, GAO Wenjie, WAN Aying. A monotone method for constructing extremal solutions to fourth order periodic boundary value problems[J]. Computational and Applied Mathematics, 2002, 132(3):411-421.
[9] WENG Shiyou, GAO Haiyin, JIANG Daqing, et al. Upper and lower solutions method for fourth order periodic boundary value problems[J]. Journal of Applied Analysis, 2008, 14(1):53-61.
[10] 马如云. 一类四阶周期边值问题的可解性[J]. 数学物理学报, 1995, 15(3):315-318. MA Ruyun. Solvability of a class of fourth order periodic boundary value problems[J]. Journal of Mathematical Physics, 1995, 15(3):315-318.
[11] LI Yongxiang. Existence of positive solutions for a fourth order periodic boundary value problems[J]. Abstract and Applied Analysis, 2011, 16(4):1-12.
[12] ZHAO Changhong, CHEN Wei, ZHOU Jinglei. Periodic solutions for a class of fourth order nonlinear differential equations[J]. Nonlinear Analysis, 2010, 72(3):1221-1226.
[13] FAN Qiyi, WANG Wentao, ZHOU Jinglei. Periodic solutions of some fourth order nonlinear differential equations[J]. Computational and Applied Mathematics, 2009, 233(2):121-126.
[1] LI Zhao-qian. Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 93-100.
[2] . Uniqueness of positive solutions of singular p-biharmonic equations with Hardy terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 75-80.
[3] WANG Jun-fang, ZHAO Pei-hao. Comparison principles for viscosity solution of fully nonlinear parabolic equations with superlinear gradient nonlinearities [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(8): 77-83.
[4] WANG Su-yun, LI Yong-jun. Solvability of nonlinear second-order boundary value problems with nonlinearities which cross the resonance points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 53-56.
[5] XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.
[6] ZHANG Sha, JIA Mei, LI Yan, LI Xiao-chen. Existence and uniqueness of solutions for three point boundary value problems of impulsive fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 66-72.
[7] WANG Xian-fei, JIANG Long, MA Jiao-jiao. Multidimensional backward doubly stochastic differential equations with generators of Osgood type [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(08): 24-33.
[8] CHEN Qiang, JIA Mei, ZHANG Hai-bin. Existence and uniqueness of solutions for nonlinear fractional four-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 42-48.
[9] SHI Xue-jun1, JIANG Long2*. Lp-solutions for one-dimensional reflected backward stochastic differential equations with continuous generators [J]. J4, 2012, 47(11): 119-126.
[10] ZHANG Xiang, HUANG Shu-xiang. Monotone iterative technique, existence and uniqueness results for the nonlinear fractional reaction-diffusion equation [J]. J4, 2011, 46(2): 9-14.
[11] FANG Bao-yan1, WANG Zhi-gang2, TIAN Shuang-liang1*, SU Li-jun2. The existence and uniqueness of the solution of parabolic equations with wavelet collocation [J]. J4, 2010, 45(6): 65-69.
[12] WANG Zhi-Gang, QIN Xin-Qiang, DANG Fa-Ning, SU Li-Jun. The existence and uniqueness of the solution to meshless methodwith ridge basis functions [J]. J4, 2010, 45(2): 44-49.
[13] WANG Jian-guo,GAO Qing-ling . Existence and uniqueness of solution for singular (k,n-k) conjugate boundary value problems [J]. J4, 2008, 43(3): 43-47 .
[14] WANG Ying and HUANG Zhen . Adapted solution of a backward stochastic evolution equation with nonLipschitz coefficients [J]. J4, 2007, 42(6): 22-26 .
[15] HAN Bao-yan,ZHU Bo and SHI Yu-feng . The limitation theorem of g-supersolution for BSDEs under non-Lipschitzian coefficient [J]. J4, 2007, 42(1): 8-13 .
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!