Showing papers in "Nonlinear Analysis-theory Methods & Applications in 1986"

Remarks on sublinear elliptic equations

Abstract: On considere le probleme (1): −Δu=f(x,u) sur Ω, u≥0, u±0 sur Ω, u=0 sur ∂Ω, ou Ω⊂R N est un domaine borne a frontiere lisse et f[x,u):Ω×[0,∞)→R. On demontre que (1) a au plus une solution. De plus, une solution de (1) existe si et seulement si λ 1 (−Δ−a 0 (x)) 0. (a 0 (x)=lim u→0 f(x,u)/u; a ∞ (x)=lim u→∞ f(x,u)/u.)

Cooperative systems of differential equations with concave nonlinearities

Abstract: On considere le systeme dynamique discret engendre par une application monotone, concave T:R + n →R + n . On etudie, ensuite, des systemes d'equations differentielles cooperatifs periodiques x'=F(t,x) a non-linearites concaves

Nontrivial solutions of operator equations and Morse indices of critical points of min-max type

Abstract: On montre que si les hypotheses du theoreme de point selle sont satisfaites, si dim Y≥2, et si f −1 ({c}) contient seulement des points critiques non degeneres, alors f −1 ({c}) contient au moins un point critique u 0 tel que 2≤indice de Morse u 0 ≤dim Y

Stabilization of Hamiltonian systems

Abstract: 1. HAMILTONIAN SYSTEMS IN THIS paper we will be concerned with the stabilization by feedback of Hamiltonian systems. In order to facilitate our discussions (especially when applying Lyapunov’s second method) we will restrict ourselves to a particular, although natural, subclass of Hamiltonian systems given in the following way [l]. Let Q be an n-dimensional smooth manifold, denoting the configuration space, and let T*Q be the cotangent bundle, denoting the phase space or srufe space. Furthermore there is a smooth m-dimensional output manifold Y (m < n) and a smooth output map C : Q- Y. (Smooth will mean C” or Ck, with k sufficiently big, although we shall restrict ourselves in the second part of Section 2 to analytic data.) For simplicity we take C to be submersive. so rank dC(q) = m. We assume that the system on T*Q has an internal energy which is the sum of a kinetic energy K and a porentiul energy V. This means that there exists a Riemannian metric (,) on Q, in local coordinates (q,, . . . , q,J for Q given by

On existence and uniqueness of solutions of Hamilton-Jacobi equations

Abstract: : The theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years. This development occurred because the authors established uniqueness criteria for generalized solutions - called viscosity solutions - which correctly identify the solutions sought in areas of application, including control theory, differential games and the calculus of variations. The concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept. A feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results. The current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs. It is also felt that the presentation of the comparison results, which imply uniqueness, continuous dependence, and are used to estimate moduli of continuity, has something to offer over earlier presentations in special cases.(Author)

The drop theorem,the petal theorem and Ekeland's variational principle

Abstract: On montre que le principe variationnel d'Ekeland, ou plus precisement, une forme legerement modifiee, est une consequence d'un resultat geometrique connu sous le terme de theoreme de la goutte

Emden-Fowler equations involving critical exponents

Abstract: On considere le probleme y″+t −k (λy 2 +y 2k−3 )=0 t 0 pour t 2, 0 0. Soit Tλ(γ)=inf{t>0:yλ(•,γ)>0 sur (t,∞)}, alors Tλ(γ)>0 ∀γ>0. On etudie comment la position Tλ(γ) du premier zero de yλ(t,γ) varie avec γ

Some backward uniqueness results

Abstract: Soit #7B-U un operateur non borne lineaire ou non lineaire sur un espace de Hilbert H, tel que #7B-U(0)=0. Soit φ une fonction, solution de l'equation d'evolution dφ/dt+#7B-U(φ)=0 et telle que φ(T)=0 pour un T>0. On etudie la propriete d'unicite retrograde sur [0,T] pour des inegalites ∥dφ(t)/dt+νA(t)φ(t)∥ H ≤n(t)∥φ(t)∥ D (A 1/2 (t)) , ou {A(t)} t ≥ 0 est une famille d'operateurs lineaires non bornes autoadjoint sur H et u∈L 2 (0,T)

Local soltutions for a nonlinear degenerate hyperbolic equation

Abstract: On etudie l'existence et l'unicite des solutions locales pour le probleme mixte associe a l'equation hyperbolique degenere du type ∂ 2 u/∂Γ 2 −M(∫|⊇u(x,t)| 2 dx)Δu=0, ou Ω est un ouvert borne de R n a frontiere lisse Γ et x est un vecteur de R n

Generalization of Fredholm alternative for nonlinear differential operators

Abstract: On considere le probleme aux valeurs limites ―(|u'(t)| p−2 u'(t))'=f(t,u(t))+g(t), t∈(0,π), u(0)=u(π)=0. On demontre que si lim u→±∞ inf f(t,u)/|u| p−2 u, lim u→±∞ sup f(t,u)/|u| p−2 u sont entre des valeurs propres du probleme aux valeurs propres ―(|u'| p−2 u')'=λ|u| p−2 u sur (0,π), u(0)=u(π)=0, alors le probleme aux limites est resoluble pour tout g∈L 1

A class of nonlinear elliptic-parabolic equations with time-dependent constraints

Abstract: On etudie les equations d'evolutions non lineaires de la forme f(t)∈du(t)/dt+∂φ t (v(t)), v(t)∈Bu(t), 0

Uniform decay rated for parabolic conservations laws

Abstract: On etablit des taux de decroissance uniformes optimaux pour des solutions du probleme de Cauchy avec donnees grandes pour des lois de conservation paraboliques scalaires a plusieurs dimensions d'espace

Remark on the decay for damped string and beam equations

Abstract: where A is an operator in a Hilbert space H, M is a real function. Our results improve those obtained by de Brito in a recent paper [l]. We briefly recall the notation and the assumptions in [l] referring to this paper for other information on physical meaning, existence and regularity theory, applications and bibliography. Let A be a self-adjoint positive operator in a (real) Hilbert space H with dense domain V. We assume that 5 > 0 is the least eigenvalue of A so a(u) = (Au, u) 3 51~1~ for u E V. Let M E Cl@+) be a real nondecreasing function such that M(s) 2 p + ks, k z 0, p-real. We have, with the notation

Numbers of zeros on invariant manifolds in reaction-diffusion equations

Abstract: On considere l'equation a une dimension u t =u xx +f(x,u), t>0, 0

A boundary value problem with a periodic nonlinearity

Abstract: Soit g:R→R une fonction continue et periodique de periode T>0 et soit h∈L 1 (0,Π). On considere le probleme d'existence pour le probleme aux limites de Picard−u''−u+g(u)=h, u(0)=0=u(Π)

On the peroidic BVP for the forced duffing equation with jumping nonlinearity

Abstract: On etudie le probleme suivant: u″+cu'+f(t,u)=e(t) p.p sur [0,T], u(T)−u(0)=0, u'(T)−u'(0)=0. C'est une fonction arbitraire dans L 1 (0,T;R), c∈R, et l'application non lineaire f est a saut

Time-dependent implicit evolution equations

Abstract: On presente des theoremes d'existence et d'unicite pour des problemes de Cauchy correspondant aux equations abstraites de la forme d(B(t)u(t))/dt+A(t,u(t))=f(t), ou B(t) est une famille dependante du temps d'operateurs lineaires

Global solutions for some nonlinear parabolic equations with nonmonotonic peturbations

Abstract: Publisher Summary This chapter discusses the existence, uniqueness, and decay property of the solutions of two problems. It also presents the Sobolev's Lemma and Gagliardo–Nirenberg inequality.

Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions

Abstract: On demontre des theoremes d'existence en utilisant des solutions superieures et inferieures

Necessary conditions, controllability and the value function for differential-difference inclusions

Abstract: On deduit des conditions necessaires pour la commande optimale d'un systeme contraint par des inclusions differentielles aux differences

Transverse homoclinic orbits in periodically perturbed parabolic equations

Abstract: On considere le systeme autonome x˙+Ax=g(x) ou A est un operateur sectoriel sur un espace de Banach X. On suppose que l'equation differentielle a une solution bornee ξ(t) telle que son equation variationnelle a une dichotomie exponentielle sur des demi-droites. On utilise la methode de Lyapunov-Schmidt pour obtenir la fonction de bifurcation pour des solutions bornees de l'equation perturbee non autonome

Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem

Abstract: On considere le systeme hamiltonien Jz=H'(z) ou J(x,y)=(−y,x)∀(x,y)∈R N ×R N . H est non convexe, a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l'exposant. Dans ce cas, on demontre que pour T>0, l'existence de solutions pour le systeme hamiltonien de periode minimale T