Ambrosetti-Prodi type results of the nonlinear first-order periodic problem
-
摘要: 研究了一阶周期问题\left\{\!\!\!\begin{array}{ll}u'(t)=a(t)g(u(t))u(t)-b(t)f(u(t))+s, t\in {\mathbb{R}},\\[2ex] u(t)=u(t+T)\end{array}\right.\eqno 解的个数与参数\,s\,(s\in{\mathbb{R}})\,的关系,其中\,a\in C({\mathbb{R}},[0,\infty)), b\in C({\mathbb{R}},(0,\infty))\,均为\,T\,周期函数, \int_0^T a(t){\rmd}t0; f, g\in C({\mathbb{R}},[0,\infty)). 当\,u0\,时, f(u)0, 当\,u\geqslant0\,时, 0l\leqslant g(u)L\infty.运用上下解方法及拓扑度理论, 获得结论:存在常数\,s_{1}\in{\mathbb{R}}, 当\, ss_{1}\,时, 该问题没有周期解; s=s_{1}\,时, 该问题至少有一个周期解; ss_{1}\,时, 该问题至少有两个周期解.
-
关键词:
- Ambrosetti-Prodi问题 /
- 上下解方法 /
- 拓扑度
Abstract: This paper shows the relationship between the parameter~s~and the number of solutions of the first-order periodic problem \left\{\!\!\!\begin{array}{ll} u'(t)=a(t)g(u(t))u(t)-b(t)f(u(t))+s,~~\ \ \ t\in {\mathbb{R}},\\[2ex] u(t)=u(t+T)\end{array}\right.\eqno where a\in C({\mathbb{R}},[0,\infty)),~b\inC({\mathbb{R}},(0,\infty)) are T-periodic, \int_0^T a(t){\rm d}t0; f, g\in C({\mathbb{R}},[0,\infty)), and f(u)0 foru0, 0l\leqslant g(u)L\infty for u\geqslant0. By using the method of upper and lower solutions and topological degree techniques, we prove that there exists s_{1}\in{\mathbb{R}}, such that the problem has zero, at least one or at least two periodicsolutions when ss_{1}, s=s_{1}, ss_{1}, respectively. -
[1] [1]AMBROSETTI A, PRODI G. On the inversion of some differentiable mappings with singularities between Banach spaces [J]. Ann Mat Pura Appl, 1972, 93: 231-24[2]FABRY C, MAWHIN J, NKASHAMA M N. A multiplicity result for periodic solutoins of forced nonlinear second order ordinary differental equations [J]. Bull London Math Soc, 1986, 18: 173-180.[3]RACHUNKOVA I. Multiplicity results for four-point boundary value problems [J]. Nonlinear Anal, 1992, 18: 497-505.[4]BEREANU C, MAWHIN J. Existence and multiplicity results for periodic solutions of nonlinear difference equations [J]. J Difference Equ Appl, 2006, 12: 677-695.[5]BEREANU C, MAWHIN J. Existence and multiplicity results for some nonlinear problems with singular phi-Laplacian [J]. J Differential Equations, 2007, 243(2): 536-557.[6]WANG H Y. Positive periodic solutions of functional differentialequations [J]. J Differential Equations, 2004, 202: 354-366.[7]GRAEF J R, KONG L J. Existence of multiple periodic solutions for first order functional differential equations [J]. Math Comput Modelling, 2011, 54: 2962-2968.[8]GRAEF J R, KONG L J. Periodic solutions of first order functionaldifferential equations [J]. Appl Math Lett, 2011, 24: 1981-1985.[9]BAI D Y, XU Y T. Periodic solutions of first order functionaldifferential equations with periodic deviations [J]. Comput Math Appl, 2007, 53: 1361-1366.[10]KANG S G, SHI B, WANG G Q. Existence of maximal and minimal periodicsolutions for first-order functional differential equations [J].Appl Math Lett, 2010, 23: 22-25.
点击查看大图
计量
- 文章访问数: 577
- HTML全文浏览量: 17
- PDF下载量: 818
- 被引次数: 0