Existence and multiplicity of positive solutions for nonlinear boundary value problems with a parameter

In this paper, the nonlinear boundary value problem {−(p(t)u′)′+q(t)u=λf(t,u),0≤t≤ω,u(0)=u(ω),p(0)u′(0)=p(ω)u′(ω), is studied. By using the fixed point index theory, some existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values of λλ. The results obtained herein generalize and improve the main results of [J.R. Graef, L. Kong, H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. Differential Equations 245 (2008) 1185–1197; X. Hao, L. Liu, Y. Wu, Existence and multiplicity results for nonlinear periodic boundary value problems, Nonlinear Anal. 72 (2010) 3635–3642] and some other known results.