Monotone and cash-invariant convex functions and hulls

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function f, we find the greatest closed convex monotone and cash-invariant function majorized by f. We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.     

Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L²(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.     

On the local existence of solutions to a class of nonconvex evolution inclusions

We prove the local existence of solutions to the Cauchy problemx'∈-?_F V(x)+F(x+f(t,x),x(0)=x ₀, where? _FV is the Fréchet subdifferential of a functionV with aψ-monotone subdifferential of order 2,F is an upper semicontinuous set-valued map contained in the Fréchet subdifferential of aφ- convex function of order two andf is a Carathéodory mapping.     

On the Carathéodory superposition of multifunctions and an existence theorem

Let I ? ? be an interval and Y a reflexive Banach space. We introduce the (H) property of a multifunction F from I × Y to Y and prove that the Carathéodory superposition of F with each continuous function f from I to Y is a derivative provided that F has the (H) property. Some application of this theorem to the existence of solutions of differential inclusions f′(x) ∈ F(x, f(x)) is given.     

Perturbed nonlinear differential equations

The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t, y) + F(t, y). The unperturbed system x = f(t, x) has a dichotomy in which some solutions exists and are well-behaved as t increases to ∞ and some solution exists and are well-behaved as t decreases to ? ∞. A similar study is made for a perturbed nonlinear differential equation defined on a half line, say, R⁺, and the existence of a family of solutions with special boundedness properties is established. Finally, the ideas are applied to the study of integral manifolds. Examples are given.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

The Poisson equation on complete manifolds with positive spectrum and applications

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function f has decay f=O(r−1−ε) for some ε>0, where r is the distance function to a fixed point, then the Poisson equation Δu=f has a solution u with at most exponential growth.We apply this result on the Poisson equation to study the existence of harmonic maps between complete manifolds and also existence of Hermitian-Einstein metrics on holomorphic vector bundles over complete manifolds, thus extending some results of Li-Tam and Ni.Assuming moreover that the manifold is simply connected and of Ricci curvature between two negative constants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result to the Ricci flow on complete surfaces.     

Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on ${\bf R}^n$

The existence/nonexistence question is studied for the inhomogeneous elliptic equation in . In particular, we establish that the above equation possesses infinitely many positive entire solutions for small provided that , p is large enough, and the locally H?lder continuous function f satisfies suitable decay conditions at . Received March 23, 2000 / Accepted September 21, 2000 / Published online March 12, 2001     

Global tolerances in the problems of combinatorial optimization with an additive objective function

It is known that by means of minimal values of tolerances one can obtain necessary and sufficient conditions for the uniqueness of the optimal solution of a combinatorial optimization problem (COP) with an additive objective function and the set of nonembedded feasible solutions. Moreover, the notion of a tolerance is defined locally, i.e., with respect to a chosen optimal solution. In this paper we introduce the notion of a global tolerance with respect to the whole set of optimal solutions and prove that the nonembeddedness assumption on the set of feasible solutions of the COP can be relaxed, which generalizes the well known relations for the extremal values of the tolerances. In particular, we formulate a new criterion for the uniqueness of the optimal solution of the COP with an additive objective function, which is based on certain equalities between locally and globally defined tolerances.     

Minimum norm solutions of single stiff linear analytic differential equations

The $\text{l}$₂-norm of the infinite vector of the terms of the Taylor series of an analytic function is used to measure the “unsmoothness” of the function. The sets of solutions to the scalar differential equations y′(t) = λy(t) + f(t) and y′(t) = q(t)y(t) + f(t) are analyzed with respect to this norm. A number of results on the particular solution with minimum norm are given.     

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G_w for w∈D of a nondegenerate interpolation problem of the Nevanlinna-Pick type with a class of particular solutions F_w of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna-Pick type interpolation problem by using the known assertions corresponding to F_w. It turns out that G_w(w∈D) is exactly or provides us with the unique solution to these two extremal problems in a manner.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

A counter-example to the characterization of the discontinuous value function of control problems with reflectionUn contre-exemple à la caractérisation de la fonction-valeur discontinue d'un problème de contrôle réfléchi

We consider a finite horizon deterministic optimal control problem with reflection. The final cost is assumed to be merely a locally bounded function which leads to a discontinuous value function. We address the question of the characterization of the value function as the unique solution of an Hamilton–Jacobi equation with Neumann boundary conditions. We follow the discontinuous approach developed by Barles and Perthame for problems set in the whole space. We prove that the minimal and maximal discontinuous viscosity solutions of the associated Hamilton–Jacobi can be written in terms of value functions of control problems with reflection. Nethertheless, we construct a counter-example showing that the value function is not the unique solution of the equation. To cite this article: O. Ley, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 469–473.     

Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case

We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations f _t + cf _x + F(t, x, c)f _c = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.     

Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line

In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm-Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing.     

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

We introduce the notion of ?-unique bounded solution to the nonlinear differential equation x′ = f(x) ? h(t), where f: ? → ? is a continuous function and h(t) is an arbitrary continuous function bounded on ?. We derive necessary and sufficient conditions for the existence and ?-uniqueness of bounded solutions to this equation.     

A Comparison Theorem for Stochastic Differential Equations Under the Novikov Condition

We consider a system of stochastic differential equations driven by a standard n-dimensional Brownian motion where the drift function b is bounded and the diffusion coefficient is the identity matrix. We define via a duality relation a vector Z (which depends on b) of square integrable stochastic processes which is shown to coincide with the unique strong solution of the previously mentioned equation. We show that the process Z is well defined independently of the boundedness of b and that it makes sense under the more general Novikov condition, which is known to guarantee only the existence of a weak solution. We then prove that under this mild assumption the process Z solves in the strong sense a related stochastic differential inequality. This fact together with an additional assumption will provide a comparison result similar to well known theorems obtained in the presence of strong solutions. Our framework is also suitable to treat path-dependent stochastic differential equations and an application to the famous Tsirelson equation is presented.     

Existence of solutions for a two point boundary value problem

We consider the two point boundary value problemF[y]=y″?f(x,y,y′)=0,a≤x≤b, y(a)=A, y(b)=B. Assuming thatf satisfies certain differential inequalities associated with the existence ofF-subfunctions andF-superfunctions, and thatf also satisfies a suitable growth condition with respect toy′, we prove that the two point boundary value problem has a solutiony with (x, y(x), y′(x)) in a specified region; indeed we show that the problem has a maximal and a minimal solution in this region. Our results unify and generalize earlier results of K. Ako, L. K. Jackson, M. Nagumo, and others.     

Dirichlet problem on locally finite graphs

In this paper, we study the existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let B be a subset of the vertices of a graph G. The Dirichlet problem is to find a function whose discrete Laplacian on G?B and its values on B are given. Each infinite connected component of G?B is called an end of G relative to B. If there are no ends, then there is a unique solution to the Dirichlet problem. Such a solution can be obtained as a limit of an averaging process or as a minimizer of a certain functional or as a limit-solution of the heat equation on the graph. On the other hand, we show that if G is a locally finite graph with l ends, then the set of solutions of any Dirichlet problem, if non-empty, is at least l-dimensional.     

Second order optimality conditions for the extremal problem under inclusion constraints

In this paper, we establish second order optimality conditions for the problem of minimizing a function f on the solution set of an inclusion 0∈F(x), where f and the support function of set valued map F have compact second order approximations at x?.