Positive Solutions To Some Nonlinear Eigenvalue Problems Of Third-order Ordinary Differential Equations

Content: In this paper, we investigate the problem of existence of positivesolutions for the nonlinear singular third-order two-point boundaryvalue problem (BVP)(E - C_i)(i = 1,…, 7)withλ> 0, where a(t) is a nonnegative continuous function defined on(0,1) and F : [0,1]×[0,∞)→[0,∞) is continuous. Here, by a positive solution of the (BVP)(E - C_i)(i = 1,…, 7) we mean a function u_i^*(t) which is positive on (0,1) and satisfies differential equation (E) and the boundary conditions (C_i)(i = 1,…, 7). The results in the presentpaper are based on the Krasnosel'skii' fixed point theory.In Section 3, we discuss the existence of a single positive solution of the (BVP)(E - C_i)(i = 1,…,7), and in Section 4, various conditionson the existence of multiple positive solutions to boundary value problem are discussed. For every result, an open interval of eigenvalue is determined in an explicitly way. Especially, we allow the function a(t) of nonlinear term to have suitable singularities, which extend the corresponding results in Sun^[1] and Yao^[5].