Multi-point boundary value problems for one-dimensionalp-Laplacian at resonance

Abstract

In this paper, we consider the multi-point boundary value problems for one-dimensional p-Laplacian at resonance:$(\phi _p (x'(t)))' = f(t,x(t),x'(t))$ subject to the boundary value conditions:$(\phi _p (x'(0)) = \sum\limits_{i = 1}^{n - 2} {\alpha _i \phi _p (x'(\xi _i ))}$,$(\phi _p (x'(1)) = \sum\limits_{j = 1}^{m - 2} {\beta _j \phi _p (x'(\eta _i ))}$ where ϕ _p (s)=|s|^p-2 s, p>1,α_i(1≤i≤n-2)∈R,β_{jit}(1≤j≤m-2)∈R, 0<ξ₁<ξ₂<...<ξ_n-2<1, 0<η₁<η₂<...<η_m-2<1, By applying the extension of Mawhin’s continuation theorem, we prove the existence of at least one solution. Our result is new.