Positive solutions of some nonlinear BVPs involving singularities and integrals BCS (Q932387)

The author establishes new criteria for the existence of multiple positive solutions of the nonlocal boundary value problem (BVP) \[ -u''(t)=g(t)h(u(t)),\quad t\in (0,1), \] \[ u'(0)+\alpha[u]=0,\, \beta u'(1)+u(\eta)=0,\quad \eta\in [0,1], \] where the nonlinearity \(h\) is allowed to be singular and \(\alpha[u]\) is a positive functional given by \[ \alpha[u]=A_0+\int_0^1u(s)dA(s). \] The method of proofs involves rewriting the BVP as a perturbed Hammerstein integral equation \[ u(t)=\gamma(t)\alpha[u]+\int_0^1k(t,s)g(s)h(u(s))ds \] and using the fixed point index theory in cones.