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 prove the existence of a sequence $\left(x_k\right)$ satisfying $0\in f\left(x_k\right)+{\sum }_{i=1}^M{a}_i\nabla f(x_k+{\beta }_i(x_{k+1}-x_k))(x_{k+1}-x_k)+F\left(x_{k+1}\right)$, where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.

Given a Hilbert space $H$ with a Borel probability measure $\nu$, we prove the $m$-dissipativity in $L^1(H,\nu )$ of a Kolmogorov operator $K$ that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator $K$. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un $C_0$-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case $\parallel K\parallel \ge 1$. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the...

∗ Cette recherche a été partiellement subventionnée, en ce qui concerne le premier et le dernier auteur, par la bourse OTAN CRG 960360 et pour le second auteur par l’Action Intégrée 95/0849 entre les universités de Marrakech, Rabat et Montpellier.The primary goal of this paper is to shed some light on the maximality of the pointwise sum of two maximal monotone operators. The interesting purpose is to extend some recent results of Attouch, Moudafi and Riahi on the graph-convergence of maximal monotone...

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.