### Uniqueness of solution for singular boundary value problems of fourth-order differential equations

1. College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
• Received:2016-01-19 Online:2017-02-20 Published:2017-01-18

Abstract: We investigate the uniqueness of solution for{x(4)(t)-h(t)f(x(t))=0, 0h(t) is allowed to be singular at both t=0 and t=1. The main novelty of this paper is that the Lipschitz constant is related to the first eigenvalues corresponding to the relevant operators. We show the uniqueness of solution by applying the u0-norm and contraction mapping principle.

CLC Number:

• O175.8
  MA Ruyun, WU Hongping. Positive solutions of a fourth-order two-point boundary value problem[J]. Applied Mathematics and Computation, 2002, 22(2):407-420. 崔玉军,孙经先,邹玉梅. 奇异四阶微分方程边值问题的正解[J]. 数学研究与评论, 2009, 65(2):376-380. CUI Yujun, SUN Jingxian, ZOU Yumei. Positive solutions of singular boundary value problems of fourth-order differential equations[J]. Journal of Mathematical Research and Exposition, 2009, 65(2):376-380. YAO Qingliu. Positive solutions for eigenvalue problems of fourth-order elastic beam equations[J]. Applied Mathematics Letters, 2004, 17(2):237-243. 张文丽. 四阶变参数微分方程边值问题解的存在性[J]. 山西师范大学学报(自然科学版), 2010, 24(2):20-23. ZHANG Wenli. Existence of Solutions to fourth-order boundary value problem with variable parameters[J]. Journal of Shanxi Normal University(Natural Science Edition), 2010, 24(2):20-23. CRASTA G, MALUSA A. Existence and uniqueness of solutions for a boundary value problem arising from granular matter theory[J]. Journal of Differential Equations, 2015, 259(8):3656-3682. ABBAS M. Existence and uniqueness of solution for a boundary value problem of fractional order involving two Caputos fractional derivatives[J]. Advances in Difference Equations, 2015, 2015(1):1-19. CUI Yujun. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Applied Mathematics Letters, 2016, 51:48-54. 盖功琪,苑成军. 奇异非线性四阶微分方程边值问题正解的存在唯一性[J]. 数学的实践与认识, 2009, 39(7):172-177. GAI Gongqi, YUAN Chengjun. Existence and uniqueness of positive solutions for singular fourth-order boundary value problems[J]. Math Pract Theory, 2009, 39(7):172-177. GUO Dajun, LAKSHMIKANTHAM V. Nonlinear problems in abstract cones[M]. Boston: Academic Press, 1988. 郭大钧. 非线性泛函分析[M]. 2版.济南: 山东科学技术出版社, 2002. GUO Dajun. Nonlinear functional analysis[M]. 2nd. Jinan: Shandong Science and Technology Press, 2002. ZHANG Xinguang, LIU Lishan, WU Yonghong. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term[J]. Mathematical and Computer Modelling, 2012, 55(3-4):1263-1274. ZHANG Guowei, SUN Jingxian. Positive solutions of m-point boundary value problems[J]. J Math Anal Appl, 2004, 291(2):406-418. KRASNOSELSKII M A. Positive solutions of operator equations[M]. Netherlands: P Noordhoff Groningen, 1964. GUO Dajun, SUN Jingxian. Nonlinear integral equations[M]. Jinan: Shandong Science and Technology Press, 1987. FAZIO R. A numerical test for the existeence and uniqeness of solution of free boundery problems: uniqueness of solution of free boundery problems[J]. Applicable Analysis, 1997, 66(1-2):89-100. E卡姆克. 常微分方程手册[M]. 北京: 科学出版社, 1977. KAMKE E. Handbook of ordinary differential equations[M]. Beijing: Science Press, 1977.
  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.  XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.  FANG Hai-qin, LIU Xi-ping*, LIN Le-gang. Existence of a solution for anti-periodic boundary value problems of  fractional differential equations [J]. J4, 2012, 47(6): 5-9.  WANG Xin-hua, ZHANG Xing-qiu. Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions [J]. J4, 2010, 45(8): 76-80.  DAI Li-Mei. Positive solutions of singular boundary value problems of second order impulsive differential equations [J]. J4, 2010, 45(2): 75-78.  LIU Shu-kuan. Existence of multiple positive solutions for singular second order  Neumann boundary value problems [J]. J4, 2010, 45(10): 98-103.  ZHOU Shao-lin. Solvability of singular third-order m-point boundary value problems [J]. J4, 2010, 45(10): 93-97.  CHEN Xiangping, LI Rengui. [J]. J4, 2009, 44(3): 45-49 .  DAI Li-Mei. The PC1 ([0,1],R+) positive solutions of singular boundary value problems of impulsive Emden-Fowler equations with a positive exponent [J]. J4, 2008, 43(12): 10-14.  TANG Qiu-yun,YANG Jin and LIU Yan-sheng . Positive solutions of the nonlinear singular impulsive [J]. J4, 2007, 42(1): 14-18 .
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