Solvability of a nonlinear second order m-point boundary value problem with p-Laplacian at resonance

Abstract

Abstract We study the existence of solutions of the nonlinear second order m-point boundary value problem with p-Laplacian at resonance { ( ϕ p ( x ′ ) ) ′ = f ( t , x , x ′ ) , t ∈ [ 0 , 1 ] , x ′ ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m − 2 a i x ( ξ i ) , $$ \textstyle\begin{cases} (\phi _{p}(x'))'=f(t,x,x'),\quad t\in [0,1],\\ x'(0)=0, \qquad x(1)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \end{cases} $$ where ϕ p ( s ) = | s | p − 2 s $\phi _{p}(s)=|s|^{p-2}s$ , p > 1 $p>1$ , f : [ 0 , 1 ] × R 2 → R $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ is a continuous function, a i > 0 $a_{i}>0$ ( i = 1 , 2 , … , m − 2 $i=1,2,\ldots ,m-2$ ) with ∑ i = 1 m − 2 a i = 1 $\sum_{i=1}^{m-2}a_{i}=1$ , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m-2}<1$ . Based on the topological transversality method together with the barrier strip technique and the cut-off technique, we obtain new existence results of solutions of the above problem. Meanwhile some examples are also given to illustrate our main results.

Keywords

• m-point boundary value problem
• p-Laplacian
• Resonance
• Topological transversality method
• Barrier strip technique
• Cut-off technique