Existence of positive solutions of fourth-order problems with integral boundary conditions.

Ma, Ruyun; Chen, Tianlan

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In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $- u^{''} + M u = r g (t) f (u), u (0) = u (ε), u (1) = u (1 - ε)$ and $u^{''} + M u = r g (t) f (u), u (0) = u (ε), u (1) = u (1 - ε),$ where $ε \in (0, 1 / 2)$ , $M \in (0, \infty)$ is a constant and $r > 0$ is a parameter, $g \in C ([0, 1], (0, + \infty))$ , $f \in C (ℝ, ℝ)$ with $s f (s) > 0$ for $s \neq 0$ . The proof of the main results is based upon bifurcation techniques.