一类带非线性边界条件的二阶差分方程正解的多解性
Multiplicity of Positive Solutions for a Class of Second-Order Difference Equation with Nonlinear Boundary Conditions
DOI: 10.12677/AAM.2022.117440, PDF, HTML, 下载: 206  浏览: 268  国家自然科学基金支持
作者: 景证棋*, 路艳琼#：西北师范大学，数学与统计学院，甘肃 兰州
关键词: 多解性；正解；上下解方法；拓扑度理论；Multiplicity； Positive Solution； Upper and Lower Solutions； Topological Degree Theory
摘要: 带非线性边界条件的差分方程边值问题在采用聚氨醋水泥钢丝绳加固桥梁以及求解环形域上椭圆 型方程组正径向解等方面有着重要的应用。本文运用不动点指数定理和上下解方法，当非线性项为正函数且在无穷远处超线性增长时，对充分小的参数，建立了上述问题正解的存在性及多解性的结果，这为微分方程边值问题的数值解提供了理论方法。 最后通过一个例子说明定理结论的有效性。
Abstract: Boundary value problems of difference equations with nonlinear boundary conditions have many important applications such as in strengthening Bridges with polyurethane cement wire ropes and solving positive radial solutions of elliptic equations in annular domain. In this paper, by using the fixed point index theorem and the method of upper and lower solutions, we obtain the existence and multiplicity of positive solutions to the above problems for sufficiently small parameters when the nonlinear term is a positive function and superlinear growth at infinity. The results provide a theoretical method for numerical solution of boundary value problems of differential equations. Finally, we give an example to illustrate the validity of the main results.
文章引用：景证棋, 路艳琼. 一类带非线性边界条件的二阶差分方程正解的多解性[J]. 应用数学进展, 2022, 11(7): 4129-4141. https://doi.org/10.12677/AAM.2022.117440

参考文献

[1] Dunninger, D.R. and Wang, H.Y. (1998) Multiplicity of Positive Solutions for a Nonlinear Differential Equation with Nonlinear Boundary Conditions. Annales Polonici Mathematici, 69, 155-165.
https://doi.org/10.4064/ap-69-2-155-165
[2] Umezu, K. (2000) Global Positive Solution Branches of Positone Problems with Nonlinear Boundary Conditions. Differential Integral Equations, 13, 669-686.
[3] Hu, L. and Wang, L.L. (2007) Multiple Positive Solutions of Boundary Value Problem for Systems of Nonlinear Second-Order Differential Equations. Journal of Mathematical Analysis and Applications, 335, 1052-1060.
https://doi.org/10.1016/j.jmaa.2006.11.031
[4] Zheng, D.M. and Lu, S.P. (2011) Positive Solutions of Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations. Chinese Quarterly Journal of Mathematics, 26,: 179-184.
[5] Abdou, K.D. and David, G.C. (2012) On Positive Solutions of One-Dimensional Semipositone Equations with Nonlinear Boundary Conditions. Applied Mathematics Letters, 25, 2411-2416.
https://doi.org/10.1016/j.aml.2012.07.015
[6] Mallick, M., Sankar, L., Shivaji, R. and Sundar, S. (2018) Infinite Semipositone Problems with a Falling Zero and Nonlinear Boundary Conditions. Electronic Journal of Differential Equations, 193, 1-13.
[7] Goddard, J., Morris, Q., Shivaji, R. and Son, B. (2018) Bifurcation Curves for Singular and Nonsingular Problems with Nonlinear Boundary Conditions. Electronic Journal of Differential Equations, 2018, 1-12.
[8] Su, X.X. (2019) Existence of Positive Solutions for a Class of Singular Second-Order Ordinary Differential Equations with Nonlinear Boundary Condition. Journal of Sichuan University (Natural Science Edition), 56, 1019-1025.
[9] Ma, R.Y. and Wang, S.Y. (2020) Positive Solutions for Some Semi-Positone Problems with Nonlinear Boundary Conditions via Bifurcation Theory. Mediterranean Journal of Mathemat- ics, 17, Article No. 12.
https://doi.org/10.1007/s00009-019-1443-6
[10] 李艳, 唐淑瑰. 二阶非线性差分方程边值问题正解的存在性[J]. 生物数学学报, 2013, 28(2): 279-301.
[11] Lu, Y.Q. and Ma, R.Y. (2014) Global Structure of Positive Solutions for Second-Order Differ- ence Equation with Nonlinear Boundary Value Condition. Advances in Difference Equations, 2014, Article No. 188.
https://doi.org/10.1186/1687-1847-2014-188
[12] 马如云, 高承华, 马慧莉, 路艳琼. 差分方程理论及其应用[M]. 北京: 科学出版社, 2018.
[13] Kelley, W.G. and Peterson, A.C. (2001) Difference Equations. Academic Press, San Diego, CA.
[14] Guo, D.J. and Lakshmikantham, V. (1988) Nonlinear Problems in Abstract Cones. Academic Press, New York, San Diego.

为你推荐