Some examples of singular fluid flows

We show, by an explicit construction, the existence of solutions to the Stokes system in dimension three that are singular on a fractal set. The singular points are understood in the sense given by Caffarelli, Kohn and Nirenberg [2]. As an application, we show the existence of suitable weak solutions to the Navier-Stokes equations, driven by rough forces, that are singular on a fractal set.     

Equilibrium price as a coincidence point of two mappings

The existence of an equilibrium price vector in a nonlinear market model is analyzed. In the model, the demand and supply functions are obtained by maximizing the producer utility and profit, respectively. Sufficient conditions for the existence of an equilibrium price vector and its stability with respect to small perturbations in the model are given. The results are consequences of theorems on the existence and stability of coincidence points in the theory of α-covering mappings.     

Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities*

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively.

We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties.     

Non-existence and existence of localized solitary waves for the two-dimensional long-wave–short-wave interaction equations

In this study, we establish the non-existence and existence results for the localized solitary waves of the two-dimensional long-wave–short-wave interaction equations. Both the non-existence and existence results are based on Pohozaev-type identities. We prove the existence of solitary waves by showing that the solitary waves are the minimizers of an associated variational problem.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

Existence and bifurcation of integral manifolds with applications

In this paper a bifurcation theorem on the existence of integral manifolds is obtained by using contracting principle. As an application, sufficient conditions for a higher dimensional system to have an integral manifold are given. Especially the existence and uniqueness of a 3-dimensional invariant torus appearing in a 4-dimensional autonomous system with singularity of codimension two are proved.     

Paréto equilibria in multicriteria metagames

Summary In this paper we study the existence of Paréto equilibria of a multicriteria metagame. A theorem on existence of a Paréto equilibrium and a theorem on existence of a Nash equilibrium with weights are presented, which improve and extend some known results in the theory of games with multiple payoffs. Also relations between a Paréto equilibrium and other solution concepts of an optimization problem with multiple criteria are discussed. Partially supported by NSFC, NSFIMTF and MADIS. The authors are grateful to the referee's valuable comments and suggestions.     

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.

     

Archimedeaness and additive utility on totally ordered semigroups

We study problems concerning the existence of additive utility funtions defined on totally ordered semigroups. The existence of an additive utility function on a semigroup is characterized by means of conditions that are similar, but not equivalent, to Archimedeaness. This fact is used to analyze the existence of utility representations (not necessarily additive) on totally ordered Abelian groups. In this direction, we show that the positive cone of a representable totally ordered Abelian group admits a countable partition into Archimedean semigroups. All the semigroups in that partition are representable by means of a utility function, but at most one is additively representable. Communicated by M. W. Mislove     

Envy-free division of discrete cakes

We address the problem of existence of an envy-free distribution of pieces among two or more players in the cake-cutting setting with the minimum number of cuts. Our cakes are discrete in the sense that the playersʼ valuations are concentrated on atoms. These atoms are placed on an interval and no two players give positive value to atoms placed at the same position. We prove the existence of an envy-free allocation for any discrete cake and any number of players by resorting to Spernerʼs Lemma, a suitable triangulation, and moving-knife arguments. Our results also apply to pies, which are defined over circumferences instead of intervals.     

The 3D Non-isentropic Compressible Euler Equations with Damping in a Bounded Domain

The authors investigate the global existence and asymptotic behavior of classical solutions to the 3D non-isentropic compressible Euler equations with damping on a bounded domain with slip boundary condition. The global existence and uniqueness of classical solutions are obtained when the initial data are near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.     

Banach空间非线性脉冲Volterra积分方程组的整体解

研究Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的非线性脉冲Volterra积分方程组解的存在性。给出了若干极值解的存在定理,改进了定义在有限区间上具有有限个脉冲点情形时该类方程的相应结果,并利用该结果讨论了一个无穷维积分方程组。     

On train algebras of rank 4

The existence of idempotent elements in train algebras of rank greater than 3 is an open question to be solved. Recent H. Guzzo results [7] on train algebras of rank 4 are based on the underlying assumption of the existence of an idempotent. In the present paper we establish the conditions that ensure the existence of such an idempotent. We also give additional properties on the Peirce decomposition which allow us to characterize some train algebras of rank 4. Finally, we give a characterization of the train algebras of rank 4 which are power-associative algebras or Jordan algebras.     

Mountain pass type solutions for quasilinear elliptic equations

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem where is a bounded domain in , , and the function is an increasing homeomorphism from onto . Under appropriate conditions on , , and the Orlicz-Sobolev conjugate of (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type. Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

On the existence of oscillating solutions in non-monotone Mean-Field Games

For non-monotone single and two-populations time-dependent Mean-Field Game systems we obtain the existence of an infinite number of branches of non-trivial solutions. These non-trivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutions predicted by bifurcation theory, and to study further properties of branches far away from bifurcation points.     

Global Existence and Asymptotic Behavior for the 3D Compressible Non–isentropic Euler Equations with Damping

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

关于一般化凸乘积空间上的极大元和平衡点

利用一般化凸乘积空间上的Fan-Browder型不动点定理给出了新的极大元存在定理,然后定义了两个概念：“类Uθ”和“类V”,并讨论了在抽象经济中平衡点的存在性问题．文中所得结论改进和推广了文献中的相应结果．     

Weak and classical solutions to predator-prey system with cross-diffusion

In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator-prey system with cross-diffusion in heterogeneous habitats. Predators are assumed to feed on preys with a Holling type II functional response to prey density and preys are assumed to follow a logistic growth in the absence of predation. The mobility of each classes is assumed to be influenced by the gradient of other classes. The existence result is proved by means of an approximation system, the Faedo-Galerkin method, and the compactness method. The global existence of classical solutions is proved under certain restrictions on the coefficients.     

Generalized bi-Quasi-Variational Inequalities for Upper Hemi-Continuous Operators in non-Compact Settings

By using the concept of escaping sequences introduced by Border [2], existence theorems on generalized bi-quasi-variational inequalities for F-semi-monotone (re- spectively, bi-quasi-semi-monotone) operators are obtained in non-compact set- tings. As application, an existence theorem on generalized bi-complementarity problem for F-semi-monotone (respectively, bi-quasi-semi-monotone) operator is given in non-compact setting.