One barrier reflected backward doubly stochastic differential equations with discontinuous monotone coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) monotone generator. An existence theorem and a comparison theorem for solutions of the class of RBDSDEs are established. Some known results are extended.     

Reflected backward doubly stochastic differential equations with discontinuous coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) generator. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.     

Reflected backward doubly stochastic differential equations driven by a Lévy process

We prove the existence and uniqueness of a solution for reflected backward doubly stochastic differential equations (RBDSDEs) driven by Teugels martingales associated with a Lévy process, in which the obstacle process is right continuous with left limits (càdlàg), via Snell envelope and the fixed point theorem.     

Existence d'une sélection continue pour les applications multivoques à images non nécessairement convexes

In this Note, we prove a continuous selection existence result for the multivalud maps with not necessarily convex values, but continuously contractible, a notion that we define below. To cite this article: Y. Askoura, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the existence of free topological groups

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U,V)-group over X if (a) X is a subspace of G, (b) G ϵ U, and (c) every continuous f: X → H with H ϵ V extends uniquely to a continuous homomorphism f̄: G→H. For certain classes U and V, we consider the question of the existence of free (U,V)- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then

• 1.(a) there is a free (PA,PA)-group over X iff; X=Ø and
• 2.(b) there is for each X a free (PA,CA)-group over X in which X is closed.

     

Sélections approchées dans les espaces métriques de dimension finie

We prove an existence result of approximate selections for multi-valued maps defined on finite dimensional metric spaces. We impose to our applications to be upper semi continuous and to have a values having contractible small neighborhoods. To cite this article: Y. Askoura, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A unification of the existence theorems of the nonlinear complementarity problem

The nonlinear complementarity problem is the problem of finding a point x in the n-dimensional Euclidean space,R ⁿ , such that x ? 0, f(x) ? 0 and 〈x,f(x)～ = 0, where f is a nonlinear continuous function fromR ⁿ into itself. Many existence theorems for the problem have been established in various ways. The aim of the present paper is to treat them in a unified manner. Eaves's basic theorem of complementarity is generalized, and the generalized theorem is used as a unified framework for several typical existence theorems.     

Hypercyclic semigroups and somewhere dense orbitsSemigroupes hypercycliques et orbites quelque part denses

We study hypercyclicity of linear strongly continuous semigroups. In the case of iterations of a single operator Bourdon and Feldman have recently proved that the existence of somewhere dense orbits implies hypercyclicity. We show the corresponding result for semigroups. As a consequence, a conjecture of Herrero concerning iterations of a single operator also holds for strongly continuous semigroups. To cite this article: G. Costakis, A. Peris, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 895–898.     

Notes on the Feynman integral,II

In the setting of Cameron and Storvick's recent theory, our main result establishes the existence of the analytic Feynman integral for functions on v-dimensional Wiener space of the form F(X) = exp {? ∝_a^b (A(s) X(s), X(s)) ds}. Here X is a $\text{R}$^v-valued continuous function on [a, b] such that X(a) = 0 and {A(s): a ? s ? b} is a commutative family of real, symmetric, positive definite matrices such that the square roots of the eigenvalues are functions of bounded variation on [a, b]. We obtain the existence theorem just referred to without having to construct special spaces, quadratic forms, etc., to fit the particular problem of interest.     

An existence result for optimal economic growth problems

An existence result for optimal control problems of Lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain. This subsumes the main existence result of R. F. Baum ¦J. Optim. Theory Appl.19 (1976), 89–116¦ and has the existence results for optimal economic growth problems of S.-I. Takekuma ¦J. Math. Econom.7 (1980), 193–208¦ and M. J. P. Magill ¦Econometrica49 (1981), 679–711; J. Math. Anal. Appl.82 (1981), 66–74¦ as simple corollaries. In addition, a new notion of uniform integrability is used, which coincides with the classical notion if the time domain is bounded.     

Cocycles and Mañe sequences with an application to ideal fluids

Exponential dichotomy of a strongly continuous cocycle Φ is proved to be equivalent to existence of a Mañe sequence either for Φ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.     

Some infinite classes of Hadamard matrices

Some results of Williamson [Duke Math. J., 11 1944, Bull. Amer. Math. Soc., 53 1947] and Wallis (J. Combinatorial Theory, 6 1969] are considerably improved to establish that in each case referred to, the same stated condition or conditions, which according to either of the authors give rise to one Hadamard matrix, actually imply the existence of an infinite series of Hadamard matrices. Also proved is the existence of some infinite series of Williamson's matrices, which coupled with the interesting findings of Turyn [J. Combinatorial Theory Ser. A, 16 1974] establish the existence of infinitely many more series of Hadamard matrices than those known so far.     

Invariant measures for dichotomous stochastic differential equations in Hilbert spaces

We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. We consider infinite dimensional noise that is white in time and colored in space and we assume that the nonlinearities are Lipschitz continuous. We show that if the equation is dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure. To cite this article: O. Van Gaans, S. Verduyn Lunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1083–1088.     

Sensibilité de l'équation de la chaleur aux sauts de conductivité

We investigate the differentiability of the solution of the heat equation with respect to the conductivity when this is piecewise continuous. We prove the existence of Lagrangian and punctual differentials and give their respective expressions. Finally, an application to the identification of a discontinuity is presented. Here, we propose an alternative method to the classical fast derivative method, which greatly simplifies the computations. To cite this article: O. Pantz, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Properties of the Principal Eigenvalues of a General Class of Non-classical Mixed Boundary Value Problems

In this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations5, 999-1030). In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations138, 86-132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations146, 336-374).     

Almost sure convergence of the kT-occupation time density estimator

In this Note, we introduce an extension of the k-nearest neighbor estimator in continuous time, the $k_T$-occupation time estimator, and we give sufficient conditions for its existence. Then, we show the almost sure convergence for α-mixing and bounded processes in two cases, the superoptimal case (when parametric rates are reached) and the optimal case (when i.i.d. rates of density estimation are reached). To cite this article: B. Labrador, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Random nonlinear Hammerstein equations

We prove existence theorems for random differential equations defined in a separable reflexive Banach space. These theorems are proved through the use of theory of random analysis established in [X. Z. Yuan, Random nonlinear mappings of monotone type, J. Math. Anal. Appl. 19] which differs from the other means, for example in [R. Kannan and H. Salehi, Random nonlinear equations and monotonic nonlinearities, J. Math. Anal. Appl. 57 (1977), 234–256; D. Kravvaritis, Existence theorems for nonlinear random equations and inequalities, J. Math. Anal. Appl. 86 (1982), 61–73; D. A. Kandilakis and N. S. Papageorgious, On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl. 126 (1987), 11–23].     

Periodic solutions for systems of forced coupled pendulum-like equations

The existence of periodic solutions for systems of forced pendulum-like equations was studied in the papers by J. A. Marlin (Internat. J. Nonlinear Mech.3 (1968), 439–447) and J. Mawhin (Internat. J. Nonlinear Mech.5 (1970), 335–339). In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N-coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M. Levi, F. C. Hoppensteadt, and W. L. Miranker (Quart. Appl. Math.36 (1978), 167–198).     

The periodic and eigen solutions of the equation governing a nonrotating viscoelastic earth model under transient force

The problem of existence of the periodic solution of the equation governing a nonrotating viscoelastic earth model under transient force is examined. By first formulating the governing equations, using the methods of Coleman and Noll (Rev. Modern Physics33 (2) (1961), 239–249), Dahlen and Smith (Philos. Trans. Roy. Soc. London A279 (1975), 583–624), and Biot (“Mechanics of Incremental Deformations,” Wiley, New York, 1965), these equations are subjected to oscillatory displacement resulting in an eigenvalue problem whose solutions are the viscoelastic-gravitational displacement eigenfunctions U(x) with associated eigenfrequencies ω. A theorem is then proved to show the existence of a periodic solution.     

On the totality of sequences in topological vector spaces

Price and Zink [Ann. of Math.82 (1965), 139–145] gave necessary and sufficient conditions for the existence of a multiplier m so that {mφ_n}₁^∞ is total; that is, the linear span is dense in L²[0, 1], thus answering a question raised by Boas and Pollard [Bull. Amer. Math. Soc.54 (1948), 512–522]. Using techniques similar to those of Price and Zink, it is shown that this result can be extended to more general spaces. Indeed, if X is either a separable Fréchet space or a complete separable p-normed space (0 < p ? 1), then the existence of a continuous linear operator A so that {Aφ_n}₁^∞ spans a dense subspace is implied by the existence of a nested, equicontinuous family of commuting projections which in addition has the properties that the union of their ranges is dense and that, for each projection, the projection of the original sequence is total in the projected space. Conversely, in a Banach space, it is shown that if such an operator exists and is 1-1 and scalar, then such a family of projections also exists. Further, it is shown that the above considerations extend the first half of the Price-Zink result to L^p[0, 1] (0 < p < ∞) and the other half to L^p[0, 1] (1 ? p < ∞).