Multiple positive solutions to some second-order integral boundary value problems with singularity on space variable

摘要

This article deals with integral boundary value problems of the second-order differential equations{u″(t)+a(t)u′(t)+b(t)u(t)+f(t,u(t))=0,t∈J+,u(0)=∫01g(s)u(s)ds,u(1)=∫01h(s)u(s)ds,$$left { extstyleegin{array}{lcl} u''(t)+a(t)u'(t)+b(t)u(t)+f(t,u(t))=0,quad tin J_{+}, u(0)= int_{0}^{1}g(s)u(s),ext{d}s,qquad u(1)=int _{0}^{1}h(s)u(s),ext{d}s, end{array}displaystyle ight .$$wherea∈C(J)$ain C(J)$,b∈C(J,R−)$bin C(J, R_{-})$,f∈C(J+×R+,R+)$fin C(J_{+}imes R_{+}, R^{+})$andg,h∈L1(J)$g, hin L^{1}(J)$are nonnegative. The result of the existence of two positive solutions is established by virtue of fixed point index theory on cones. Especially, the nonlinearity f permits the singularity on the space variable.