Existence theorems for second order multi-point boundary value problems - Wong, James S. W. (Electronic journal of qualitative theory of differential equations, 2010)
We are interested in the existence of nontrivial solutions for the second order nonlinear differential equation (E): $y'' (t) = f\big(t, y (t)\big) = 0, 0 < t < 1$ subject to multi-point boundary conditions at $t=1$ and either Dirichlet or Neumann conditions at $t=0$. Assume that $f(t, y)$ satisfies $|f(t, y)| \le k(t) |y| + h(t)$ for non-negative functions $k, h \in L^1 (0, 1)$ for all $(t, y) \in (0, 1) × \Bbb R$ and $f(t, 0) \not\equiv 0$ for $t\in (0, 1)$. We show without any additional assumption on $h(t)$ that if $\|k\|_1$ is sufficiently small where $\|· \|_1$ denotes the norm of $L^1(0, 1)$ then there exists at least one non-trivial solution for such boundary value problems. Our results reduce to that of Sun and Liu and Sun for the three point problem with Neumann boundary condition at $t=0$.