New results on the positive solutions of nonlinear second-order differential systems

Yingxin Guo, Qufu Normal University, Qufu, Shandong, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 3. (2009), pp. 1-16.

Abstract: In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form
$$
\left\{\aligned &u''=-f(t,v), \ \ 0< t< 1,\\&v''=-g(t,u), \ \ 0< t< 1\\&u(0)=v(0)=0,\varsigma u(\zeta)=u(1),\varsigma v(\zeta)=v(1),\endaligned\right.
$$
where $f:(0,1)\times [0,+\infty)\to [0,+\infty),g:[0,1]\times [0,+\infty)\to [0,+\infty),0<\zeta<1, \varsigma>0,$ and $\varsigma\zeta< 1,f$ may be singular at $t = 0$ and/or $t = 1.$ Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.

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