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.

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.

The paper defines and studies the Drazin inverse for a closed linear operator $A$ in a Banach space $X$ in the case that $0$ belongs to a spectral set of the spectrum of $A$. Results are applied to extend a result of Krein on a nonhomogeneous second order differential equation in a Banach space.

We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw2(a, b) spaces.