On an Ambrosetti–Prodi type problem for a class of fourth-order ODEs involving Dirac weights

The aim of this paper is to establish an Ambrosetti–Prodi type result involving Dirac weights \(\begin{aligned} \left\{ \begin{array}{ll} u''''(x)+q(x)u(x)=(c (x)+\sum \limits _{i=1}^{p}c_{i}\delta (x-x_i))(g(u(x))+f(x)),~~~~&{}x\in (0,1),\\ \ u(0)=u(1)=u''(0)=u''(1)=0,\\ \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}^{\prime \prime \prime \prime }(x)+q(x)u(x)=(c(x)+\sum _{i=1}^p{c}_i\delta (x-x_i))(g(u(x))+f(x)),\hfill & x\in (0,1),\hfill \\ u(0)=u(1)=u^{\prime \prime }(0)=u^{\prime \prime }(1)=0,\hfill \end{array}\right)\end{array}$$ where \(\delta (x-x_i)\) $\delta (x-x_i)$ is the canonical Dirac delta function at the point \(x_i\) $x_i$ , \(i=1,2,\ldots ,p\) $i=1,2,\dots ,p$ , \(p\in \mathbb {N}\) $p\in \mathbb{N}$ , \(0=x_0<x_1<\cdots<x_p<x_{p+1}=1\) $0=x_0<x_1<\cdots <x_p<x_{p+1}=1$ , \(q\in C([0,1],[0,+\infty ))\) $q\in C([0,1],[0,+\infty ))$ , \(f\in L^1([0,1],\mathbb {R})\) $f\in L^1([0,1],\mathbb{R})$ , \(g\in C^{1}(\mathbb {R},\mathbb {R})\) $g\in C^1(\mathbb{R},\mathbb{R})$ , \(c\in C([0,1],[0,+\infty ))\) $c\in C([0,1],[0,+\infty ))$ , \(c_i\in [0,+\infty )\) $c_i\in [0,+\infty )$ . The main tools used are the sub-super-solution method and Leray–Schauder topological degree theory.