Global Behavior of the Components for the Second Order m-Point Boundary Value Problems

We consider the nonlinear eigenvalue problems u ′′ rf u 0, 0 < t < 1, u 0 0, u 1 ∑m−2 i 1 αiu ηi , where m ≥ 3, ηi ∈ 0, 1 , and αi > 0 for i 1, . . . , m − 2, with ∑m−2 i 1 αi < 1; r ∈ R; f ∈ C1 R,R . There exist two constants s2 < 0 < s1 such that f s1 f s2 f 0 0 and f0 : limu→0 f u /u ∈ 0,∞ , f∞ : lim|u|→∞ f u /u ∈ 0,∞ . Using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of the above problems.