Existence theorems for second order multi-point boundary value problems

James S. W. Wong, The University of Hong Kong, City University of Hong Kong and Chinney Investment Ltd., Hong Kong

E. J. Qualitative Theory of Diff. Equ., No. 41. (2010), pp. 1-12.

Abstract: 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 $(t, y) \in (0, 1) \times \Bbb R$ 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 $\|\cdot \|_1$ 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 $t=0$.

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