The existence of nonzero nonnegative solutions are established for semilinear equations at resonance with the zero solution and possessing at most linear growth. Applications are given to nonlinear boundary value problems of ordinary differential equations.

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance $$x^{\left(n\right)}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right),\cdots ,x^{(n-1)}\left(t\right)),t\in (0,1),x\left(0\right)=\sum _{i=1}^m{\alpha }_{i}x\left({\xi }_i\right),x^{\text{'}}\left(0\right)=\cdots =x^{(n-2)}\left(0\right)=0,x^{(n-1)}\left(1\right)=\sum _{j=1}^l{\beta }_j{x}^{(n-1)}\left({\eta }_j\right),$$ where $f:[0,1]\times {ℝ}^n\to ℝ$ is a Carathéodory function, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_m<1$, ${\alpha }_i\in ℝ$, $i=1,2,\cdots ,m$, $m\ge 2$ and $0<{\eta }_1<\cdots <{\eta }_l<1$, ${\beta }_j\in ℝ$, $j=1,\cdots ,l$, $l\ge 1$. In this paper, two of the boundary value conditions are responsible for resonance.

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $$\frac{d}{dt}\left\{r_i\left(t\right)\frac{d}{dt}\left[{\lambda }_i\left(t\right)\frac{d}{dt}\left(x_i\left(t\right)-f_i(t,x_1(t-{\sigma }_{i1}),x_2(t-{\sigma }_{i2}),x_3(t-{\sigma }_{i3}))\right)\right]\right\}+\frac{d}{dt}\left[r_i\left(t\right)\frac{d}{dt}{g}_i(t,x_1\left(p_{i1}\left(t\right)\right),x_2\left(p_{i2}\left(t\right)\right),x_3\left(p_{i3}\left(t\right)\right))\right]+\frac{d}{dt}{h}_i(t,x_1\left(q_{i1}\left(t\right)\right),x_2\left(q_{i2}\left(t\right)\right),x_3\left(q_{i3}\left(t\right)\right))=l_i(t,x_1\left({\eta }_{i1}\left(t\right)\right),x_2\left({\eta }_{i2}\left(t\right)\right),x_3\left({\eta }_{i3}\left(t\right)\right)),t\ge t_0,i\in \{1,2,3\}$$ in the following bounded closed and convex set $$\Omega (a,b)=\left\{x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right))\in C([t_0,+\infty ),{ℝ}^3):a\left(t\right)\le x_i\left(t\right)\le b\left(t\right),\forall t\ge t_0,i\in \{1,2,3\}\right\},$$ where ${\sigma }_{ij}>0$, $r_i,{\lambda }_i,a,b\in C([t_0,+\infty ),{ℝ}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times {ℝ}^3,ℝ)$, $p_{ij},q_{ij},{\eta }_{ij}\in C([t_0,+\infty ),ℝ)$ for $i,j\in \{1,2,3\}$. By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.