Eigenvalue of fourth-order mm-point boundary value problem with derivatives

This paper deals with the existence of positive solutions for the fourth-order nonlinear ordinary differential equation x⁗(t)=λg(t)f(t,x(t),x″(t)),00ρ=αδ+αγ+βδ>0, and ξi∈(0,1),ηi∈[0,+∞)(i=1,2,…,m−2). By means of a fixed-point theorem due to Krasnaselskii, some new existence results of positive solutions for the above multi-point boundary value problem are obtained, which improve the main results of Graef et al. [J.R. Graef, C. Qian, B. Yang, A three-point boundary value problem for nonlinear fourth-order differential equations, J. Math. Anal. Appl. 287 (2003) 217–233]. An example is given to demonstrate the main results of this paper.