Journal of East China Normal University(Natural Sc ›› 2015, Vol. 2015 ›› Issue (6): 53-58.doi: 10.3969/j.issn.1000-5641.2015.06.008

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Ambrosetti-Prodi type results of the nonlinear first-order periodic problem

 MA  Lu-Yi   

  • Received:2014-11-03 Online:2015-11-25 Published:2015-12-23

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}t>0; f, g\in C({\mathbb{R}},[0,\infty)), and f(u)>0 foru>0, 0<l\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  s<s_{1}, s=s_{1}, s>s_{1}, respectively.

Key words: Ambrosetti-Prodi problem, upper and lower solutions, topological degree

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