an:05950493
Zbl 1221.34059
El-Shahed, Moustafa
Positive solutions for nonlinear singular third order boundary value problem
EN
Commun. Nonlinear Sci. Numer. Simul. 14, No. 2, 424-429 (2009).
00286524
2009
j
34B16 34B18 47N20
third order boundary value problem; Krasnoselskii's fixed-point theorem; Green's function; positive solution
Summary: We investigate the problem of existence of positive solutions for the nonlinear third order boundary value problem
\[
u'''(t)+\lambda a(t)f(u(t))=0,\quad t\in(0,1),
\]
\[
u(0)=u'(0)=0,\quad \alpha u'(1)+\beta u''(1)=0,
\]
where \(\lambda \) is a positive parameter. By using Krasnoselskii's fixed-point theorem in cones, we establish various results on the existence of positive solutions of the boundary value problem. Under various assumptions on \(a(t)\) and \(f(u(t))\), we give the intervals of the parameter \(\lambda \) which yield the existence of the positive solutions. An example is also given to illustrate the main results.