We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right),u^{\text{'}\text{'}\text{'}}\left(t\right)\right),\text{a.e.}t\in [0,1],\hfill \\ u\left(0\right)=a,u^{\text{'}}\left(0\right)=b,u\left(1\right)=c,u^{\text{'}\text{'}}\left(1\right)=d,\hfill \end{array}\right.$$ where the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ is a strong Carathéodory function. By constructing suitable height functions of the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.

Let $g$: $𝐑\to 𝐑$ be a continuous function, $e$: $[0,1]\to 𝐑$ a function in $L^2[0,1]$ and let $c\in 𝐑$, $c\ne 0$ be given. It is proved that Duffing’s equation $u^{\text{'}\text{'}}+c{u}^{\text{'}}+g\left(u\right)=e\left(x\right)$, $0<x<1$, $u\left(0\right)=u\left(1\right)$, $u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)$ in the presence of the damping term has at least one solution provided there exists an $𝐑>0$ such that $g\left(u\right)u\ge 0$ for $\left|u\right|\ge 𝐑$ and ${\int }_0^{1}e\left(x\right)dx=0$. It is further proved that if $g$ is strictly increasing on $𝐑$ with ${lim}_{u\to -\infty }g\left(u\right)=-\infty$, ${lim}_{u\to \infty }g\left(u\right)=\infty$ and it Lipschitz continuous with Lipschitz constant $\alpha <4{\pi }^2+c^2$, then Duffing’s equation given above has exactly one solution for every $e\in L^2[0,1]$.

This paper deals with the generalized nonlinear third-order left focal problem at resonance $$\left\{\begin{array}{c}{\left(p\left(t\right)u^{\text{'}\text{'}}\left(t\right)\right)}^{\text{'}}-q\left(t\right)u\left(t\right)=f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),t\in ]t_0,T[,m(u\left(t_0\right),u^{\text{'}\text{'}}\left(t_0\right))=0,n(u\left(T\right),u^{\text{'}}\left(T\right))=0,l(u\left(\xi \right),u^{\text{'}}\left(\xi \right),u^{\text{'}\text{'}}\left(\xi \right))=0,\hfill \end{array}\right.$$ where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the...

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.