The existence and uniqueness of positive solutions for integral boundary value problems (Q624888)

Consider the equation \[ -u''(t)=f(t,u(t)),\quad t\in(0,1) \] associated with the conditions \[ u(0)=\int_0^1u(t)d\phi(t),\quad u(1)=0. \] The authors use the method of upper and lower solutions and prove the existence of solutions under the following conditions: (H1) \(f(t,u)\) is continuous and decreasing with respect to \(u.\) \(\phi\) is increasing, \(\phi(0)=0\) and \(\int_0^1(1-s)\,d\phi(s)<1\); (H2) \(f(t,\lambda)\) is not equal to zero identically for all \(\lambda>0\) and \(t\in(0,1).\) Moreover, it satisfies \(\int_0^1t(1-t)f(t,\lambda t(1-t))\,dt<+\infty\) for all \(\lambda>0.\) The uniqueness is also discussed.