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AbstractConsider positive solutions and multiple positive solutions for a discrete nonlinear third-order boundary value problem  {Δ3u(t−1)=a(t)f(t,u(t)),t∈[1,T−2]Z,Δu(0)=u(T)=0,Δ2u(η)−αΔu(T−1)=0,$$ \textstyle\begin{cases} \Delta ^{3}u(t-1)=a(t)f(t,u(t)), \quad t\in [1,T-2]_{\mathbb{Z}},\\ \Delta u(0)=u(T)=0,\qquad \Delta ^{2} u(\eta )-\alpha \Delta u(T-1)=0, \end{cases} $$ which has the sign-changing Green’s function. Here T>8$T>8$ is a positive integer, [1,T−1]Z={1,2,…,T−2}$[1,T-1]_{\mathbb{Z}}=\{1,2,\dots ,T-2\}$, α∈[0,1T−1)$\alpha \in [0, \frac{1}{T-1})$, a:[0,T−2]Z→(0,+∞)$a:[0,T-2]_{\mathbb{Z}}\to (0,+\infty )$, f:[1,T−2]Z×[0,∞)→[0,∞)$f:[1,T-2]_{ \mathbb{Z}}\times [0,\infty )\to [0,\infty )$ is continuous.

Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green’s function