A penalty/Newton/conjugate gradient method for the solution of obstacle problems

Motivated by the search for non-negative solutions of a system of Eikonal equations with Dirichlet boundary conditions, we discuss in this Note a method for the numerical solution of parabolic variational inequality problems for convex sets such as $\text{K={v∣v∈H}{}_0{}^1\text{(Ω)}$, v?ψ a.e. on $\text{Ω}.}$ The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm requiring at each iteration the solution of a linear problem for a discrete analogue of the elliptic operator I?μΔ. Numerical experiments show that the resulting method has good convergence properties, even for small values of the penalty parameter. To cite this article: R. Glowinski et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On a class of nonlinear integral equations of Urysohn's type

In this paper we study an interesting class of nonlinear integral equations of Urysohn's type, namely,

$\text{u(x) +}\text{∑}\text{j=l}\text{n}\text{∫}{}_{\mathrm{\Omega }}\text{k}{}_{\mathrm{j}}\text{(x,y)f}{}_{\mathrm{j}}\text{(y, u(y)) dy= v(x)}\text{(x ∈ Ω)}$

. It is shown that such an equation can be considered as a nonlinear operator equation of Hammerstein type in an appropriate Banach space. One can in this way extend the theory of nonlinear operator equations of Hammerstein type (except for the part which uses variational methods) to this class of equations.     

Valeurs propres relatives à un cône convexe : caractérisation et résultats de cardinalité

Let A be an n×n real matrix, and $\text{K?}\text{R}{}^{\mathrm{n}}$ be a closed convex cone. The spectrum of A relative to K, denoted by σ(A,K), is the set of all $\text{λ∈}\text{R}$ for which the linear complementarity problem

$\text{x∈K,}\text{Ax?λx∈K}{}^{\mathrm{+}}\text{,}\text{〈x,Ax?λx〉=0}$

admits a nonzero solution $\text{x∈}\text{R}{}^{\mathrm{n}}$. The aim of this Note is to study the main properties of the set-valued mapping $\text{σ(·}\text{,K)}$, and discuss some structural differences existing between the polyhedral case (i.e., K is finitely generated) and the non-polyhedral case. To cite this article: A. Seeger, M. Torki, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

Random fixed points of K-set- and pseudo-contractive random maps

Let $\text{(Ω,Σ)}$ be a measurable space, X and Y separable Banach spaces, and C a weakly compact subset of X. Let $\text{f}\text{:}\text{Ω×C→Y}$ and $\text{T}\text{:}\text{Ω×C→Y}$ be continuous random operators. Then the deterministic solvability of the equation$\text{f(ω,x)−T(ω,x)=0}\text{(ω∈Ω,x∈C)}$implies the stochastic solvability of it provided that (f−T)(ω,.) is demiclosed at zero and T(ω,C) is bounded for each $\text{ω∈Ω}$. As applications, random fixed points of various types of pseudo-contractive and k-set-contractive random operators are obtained.     

Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Then we show that the iterative sequence converges strongly to a common element of the three sets. In this paper, we also give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping in Banach space $l^2$.     

Existence of solutions for nonlinear elliptic degenerate equations

In this paper, we study the problem$\text{−div}\text{a(x,u,}\text{∇}\text{u)−div}\text{φ(u)+g(x,u)=f}\text{in}\text{Ω}$in the setting of the weighted sobolev space $\text{W}{}_0{}^{\mathrm{1,p}}\text{(Ω,ν)}$. The main novelty of our work is L^∞ estimates on the solutions, and the existence of a weak and renormalized solution.     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

Variational inequalities with one-sided irregular obstacles

The authors show that the Hölder continuity of the solutionu∈K?{v∈H _o ¹ (Ω) | v≤ψ in Ω} of the variational inequality $$(\triangledown u,\triangledown u - \triangledown v) \leqslant (f,u - v),v\varepsilon \mathbb{K},$$ also holds under a one-sided Hölder condition on the obstacle ψ. This class of obstacles ψ contains the implicit obstacles of the quasivariational inequalities occuring in stochastic impulse control.     

Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations

Consider the following nonlinear Dirichlet boundary value problems:

$\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+f(u(t,x)),}\text{t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.1)

$\text{D}{}_{\mathrm{t}}\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+}\text{α}\text{L(D}{}_{\mathrm{t}}\text{u(t,x))+f(u(t,x)),}\text{α}\text{>0,t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.2)

$\text{Lu(x)+f(u(x)),}\text{x∈Ω}\text{u(x)=0,}\text{x∈?Ω}$

. (1.3) In all of these equations, f: R → R is a locally Lipschitzian asymptotically linear function with positive asymptotic slope, f(0) = 0, and L is a self-adjoint, negativedefinite and strongly elliptic second-order differential operator on a smooth domain Ω in Rⁿ. The solutions of (1.1) and (1.2) generate semiflows which are not pointdissipative and whose equilibria are determined by solutions of (1.3). In this paper, using an extension (due to the present author) of Conley's Morse index theory to noncompact spaces, we prove not only the existence of positive solutions of (1.3) (a result shown earlier by Peitgen and Schmitt using different methods), but also show the existence of (nonconstant) heteroclinic orbits of (1.1) and (1.2) joining two sets of equilibria.     

A comparison result related to Gauss measureUn résultat de comparaison relatif à la mesure de Gauss

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

where $\text{Ω}$ is an open set of $\text{R}{}^{\mathrm{n}}$ (n?2), ?(x)=(2π)^?n/2exp(?|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j??(x)|ξ|² a.e. $\text{x∈Ω}$, $\text{ξ∈}\text{R}{}^{\mathrm{n}}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\text{u∈}\text{H}{}^1{}_0\text{(?,Ω)}$ of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.     

A hybrid approximation method for equilibrium,variational inequality and fixed point problems

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the fixed points of $?$-asymptotically nonexpansive mapping, the set of solutions of the equilibrium problem and the set of solutions of the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces. We show that the iterative scheme converges strongly to a common element of the above three sets under appropriate conditions.