The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this...

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u-\mathrm{div}𝐚(u,Du)=v$, where $v\in L^1$, $𝐚(z,\xi )={\nabla }_{\xi }f(z,\xi )$, and $f$ is a convex function of $\xi$ with linear growth as $\parallel \xi \parallel \to \infty$, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi )=\varphi \left(z\right)\psi \left(\xi \right)$, $\varphi \>0$, $\psi$ being a convex function with linear growth as $\parallel \xi \parallel \to \infty$. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the...

In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of second order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.

This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=f(t,x\left(t\right),x\left({\alpha }_1\left(t\right)\right),\cdots ,x\left({\alpha }_n\left(t\right)\right))\text{for}\text{a.e.}t\in [0,T],\Delta x\left(t_k\right)=I_k\left(x\left(t_k\right)\right),k=1,\cdots ,m,x\left(0\right)=x\left(T\right).\hfill \end{array}\right.$$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.

Utilizing the theory of fixed point index for compact maps, we establish new results on the existence of positive solutions for a certain third order boundary value problem. The boundary conditions that we study are of nonlocal type, involve Stieltjes integrals and are allowed to be nonlinear.

On donne dans cet exposé des bornes inférieures universelles, en limite semiclassique, de la hauteur des résonances de forme associées aux opérateurs de Schrödinger à l’extérieur d’obstacles avec des conditions au bord de Dirichlet ou de Neumann et des potentiels analytiquement dilatables et tendant vers $0$ à l’infini. Ces bornes inférieures sont exponentiellement petites par rapport à la constante de Planck.

A Cauchy problem for an abstract nonlinear Volterra integrodifferential equation is considered. Existence and uniqueness results are shown for any given time interval under weak time regularity assumptions on the kernel. Some applications to the heat flow with memory are presented.