On the stability of gradient polynomial systems at infinity

We first define the notion of the infimum at infinity of a polynomial function and the notion of stability at infinity near the fiber of the gradient descent system. Then we prove that the gradient descent system is stable at infinity near the fiber of the infimum value at infinity.     

Multiple Orthogonal Polynomials on the Unit Circle

We introduce multiple orthogonal polynomials on the unit circle. We show how this is related to simultaneous rational approximation to Caratheodory functions (two-point Hermite-Pade approximation near zero and near infinity). We give a Riemann-Hilbert problem for which the solution is in terms of type I and type II multiple orthogonal polynomials on the unit circle, and recurrence relations are obtained from this Riemann-Hilbert problem. Some examples are given to give an idea of the behavior of the zeros of type II multiple orthogonal polynomials.     

Total energy decay for the wave equation in exterior domains with a dissipation near infinity

We show that the energy of solutions to the initial boundary value problem for the wave equation in exterior domains with a dissipation which is localized only near infinity tends to zero as the time goes to infinity. We do not make any geometrical condition like star-shapedness on the boundary.     

Multiple solutions for a class of elliptic equations with jumping nonlinearities

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.     

Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions

In the first part of this work, we recall variational methods related to invariant sets in ${C^1_0}$ . In the second part of the work, we consider an elliptic Dirichlet problem in a situation where the origin is a solution around which the nonlinearity has a slope between two consecutive eigenvalues of order larger than 2 and near + infinity the slope of the nonlinearity is smaller than the first eigenvalue. Then we discuss the conditions needed near - infinity in order to ensure the existence of a positive solution and two sign-changing solutions.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

Near minimally normed spline quasi-interpolants on uniform partitions

Spline quasi-interpolants (QIs) are local approximating operators for functions or discrete data. We consider the construction of discrete and integral spline QIs on uniform partitions of the real line having small infinity norms. We call them near minimally normed QIs: they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms. We give precise results for cubic and quintic QIs. Also the QI error is considered, as well as the advantage that these QIs present when approximating functions with isolated discontinuities.     

On a class of nonlinear integral equations with a noncompact operator

The paper is devoted to investigation of the solvability of a class of nonlinear integral equations on the semiaxis in a critical case, which possess a noncompact operator of almost Hammerstein type. Under some conditions on the equation kernel, the existence of a bounded, monotone increasing, positive solution is proved. The asymptotic behavior of the solution near infinity is studied.     

A characterization of a limit solution for finite horizon bargaining problems

We investigate a two-person random proposer bargaining game with a deadline. A bounded time interval is divided into bargaining periods of equal length and we study the limit of the subgame perfect equilibrium outcomes as the number of bargaining periods goes to infinity while the deadline is kept fixed. This limit is close to the discrete Raiffa solution when the time horizon is very short. If the deadline goes to infinity the limit outcome converges to the time preference Nash solution. Regarding this limit as a bargaining solution under deadline, we provide an axiomatic characterization.     

On the existence of extremal functions for a weighted Sobolev embedding with critical exponent

We investigate the existence of ground state solutions to the Dirichlet problem in , u = 0 on , where , and is a domain in . In particular we prove that a non negative ground state solution exists when the domain is a cone, including the case . Moroever, we study the case of arbitrary domains, showing how the geometry of the domain near the origin and at infinity affects the existence or non existence of ground state solutions. Received April 16, 1998 / Accepted June 24, 1998     

One-parameter Family of Positive Solutions for a Nonlinear Integral Equation Arising in Physical Kinetics

The paper is devoted to the question of solvability of a Urysohn type nonlinear integral equation. This equation has an application in the kinetic theory of gases and can be derived from Boltzmann model equation. We prove an existence theorem of one-parameter family of positive solutions in the space of functions possessing linear growth at infinity. Moreover, for each member of this family we find an exact asymptotic formula at infinity. We obtain two-sided estimates for solution, as well as describe an iterative method for construction of solution.We conclude the paper by giving examples of functions that describe nonlinearity and satisfy the conditions of the main theorem.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification

Critical points at infinity for autonomous differential systems are defined and used as an essential tool. Rn is mapped onto the unit ball by various mappings and the boundary points of the ball are used to distinguish between different directions at infinity. These mappings are special cases of compactifications. It is proved that the definition of the critical points at infinity is independent of the choice of the mapping to the unit ball.We study the rate of blow up of solutions in autonomous polynomial differential systems of equations via compactification methods. To this end we represent each solution as a quotient of a vector valued function (which is a solution of an associated autonomous system) by a scalar function (which is a solution of a related scalar equation).     

Higher-order parabolic variational inequality in unbounded domains

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 949–968, July, 2008.     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

About Asymptotic Behavior for a Transmission Problem in Hyperbolic Thermoelasticity

We consider a transmission problem in thermoelasticity with memory. We show the exponential decay of the solution in case of radially symmetric situations, as time goes to infinity.      

Radial symmetry of standing waves for nonlinear fractional Hardy–Schrödinger equation

In this paper, by applying the method of moving planes, we conclude the conclusions for the radial symmetry of standing waves for a nonlinear Schrödinger equation involving the fractional Laplacian and Hardy potential. First, we prove the radial symmetry of solution under the condition of decay near infinity. Based upon that, under the condition of no decay, by the Kelvin transform, we establish the results for the non-existence and radial symmetry of solution.     

Behavior of the solution of a random semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state.     

On the time-periodic problem for the Stokes system in domains with cylindrical outlets to infinity

We study the time-periodic Stokes problem in the domain with cylindrical outlets to infinity in weighted function spaces. We prove that there exists a unique solution with prescribed fluxes over the sections of outlets to infinity and that, in each outlet, this solution tends to the corresponding time-periodic Poiseuille flow. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 177–195, April–June, 2007.