Conditions For Existence Of Solutions Research Articles

Inverse spectral problems of the theory of identification of linear dynamic systems

The problem of restoration of the parameters of the linear operator from a finite set of eigenvalues was considered. A new scheme of its solution was proposed, both the necessary and sufficient conditions for existence of solutions were presented, and a procedure of their numerical construction was substantiated. Consideration was given to the applied problems of identification and controllability of the linear dynamic systems.

Existence of solutions in variational relation problems without convexity

Two main existence conditions for solutions of variational relation problems are established without convexity. The first one is based on a finite solvability property and the second one on generalized KKM mappings. These conditions unify and strengthen several existing results in the literature on the topic. A model of satisficing process by rejection is considered which gives an economic interpretation of the introduced concepts.

Existence of solutions of Minty type scalar and vector variational inequalities

Scalar and vector variational inequalities involving bifunctions are considered in this article. Necessary and sufficient conditions for existence of solutions are proposed, based on the finite intersection property of a suitable set-valued function. In the case of a properly quasi-monotone bifunction the relation to the KKM property is established. The result of John (A note on Minty variational inequality and generalized monotonicity. Proc. Generalized Convexity and Monotonicity, 6 (2001), pp. 240–246) concerning the characterization of properly quasi-monotone functions in terms of existence of solutions of variational inequalities (scalar case) is extended to bifunctions both in the scalar and vector cases. Existence of solutions for scalar equilibrium problems with bifunctions are considered Bianchi and Pini (A note on equilibrium problems with properly quasimonotone bifunctions. J. Global Optim. 20 (2001), pp. 67–76) and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present article also generalizes this result, even in the scalar case (here the bifunctions are not assumed quasi-convex). As for the vector case, it should be stressed that two types of variational inequalities are studied, and respectively, the quasi-monotonicity is understood in two different ways. Finally, as a particular case variational inequalities of differentiable type are discussed.

An Internal Model Principle for Consensus in Heterogeneous Linear Multi-Agent Systems

The problem of reaching consensus in a heterogeneous multi-agent system is considered. The agents are modeled as linear time-invariant systems with potentially different state dimension and different dynamics. The interconnection topology between the agents is modeled as a directed and weighted graph. We propose an internal model principle for consensus translating in necessary conditions for existence of solutions to the output and state consensus problem.

Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.

A note on a result of M. Grossi

The purpose of this note is to present a fact complementary to a result in a recent paper of M. Grossi. Making use of an energy balance identity, it is shown that the sufficient conditions for existence of solutions proved in Grossi's paper are also almost necessary.

R-Boundedness of Smooth Operator-Valued Functions

In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems.

Propriétés probabilistes des processus GARCH périodiques

This Note examines the probabilistic structure of a GARCH-type stochastic difference equation with periodically time-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions, geometrically ergodic (in the periodic sense) and having finite higher-order moments. To cite this article: A. Bibi, A. Aknouche, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A Link-Node Discrete-Time Dynamic Second Best Toll Pricing Model with a Relaxation Solution Algorithm

Dynamic congestion pricing has become an important research topic because of its practical implications. In this paper, we formulate dynamic second-best toll pricing (DSBTP) on general networks as a bilevel problem: the upper level is to minimize the total weighted system travel time and the lower level is to capture motorists’ route choice behavior. Different from most of existing DSBTP models, our formulation is in discrete-time, which has very distinct properties comparing with its continuous-time counterpart. Solution existence condition of the proposed model is established independent of the actual formulation of the underlying dynamic user equilibrium (DUE). To solve the bilevel DSBTP model, we adopt a relaxation scheme. For this purpose, we convert the bilevel formulation into a single level nonlinear programming problem by applying a link-node based nonlinear complementarity formulation for DUE. The single level problem is solved iteratively by first relaxing the strick complementarity by a relaxation parameter, which is then progressively reduced. Numerical results are also provided in this paper to illustrate the proposed model and algorithm. In particular, we show that by varying travel time weights on different links, DSBTP can help traffic management agencies better achieve certain system objectives. Examples are given on how changes of the weights impact the optimal tolls and associated objective function values.

Existence of solutions to first-order impulsive periodic boundary value problems

This paper is concerned with the solvability of periodic boundary value problems for first-order impulsive ordinary differential equations. By using Schaeffer’s theorem, a sufficient condition for existence of solution of the problem is obtained, which extends and improves some known results in the literature.

On a mixed problem for Navier-Stokes system in the unit cube

Mixed problems on the unit balls have special complexity. By using a new method we shall give some sufficient conditions for existence of solutions of the Fefferman's problem B ([4]). 2000 Mathematics Subject Classification. Primary: 35A99. .

Necessary and Sufficient conditions for Existence of Solutions of a Divergence-type Variational Problem

Necessary and Sufficient conditions for Existence of Solutions of a Divergence-type Variational Problem

Triple configurations of traveling shock waves in inviscid gas flows

Triple configurations of shock waves in supersonic inviscid flows of a perfect gas are considered. The basic parameters of triple configurations are determined, and the conditions of solution existence are analyzed.

A mixed problem for Navier-Stokes system

Mixed problems are changed to boundary value problems by making used of Laplace transform. In classical boundary value problems, boundary conditions are local, but boundary conditions could be global ([2, 3, 11]). Every boundary value problem depends on a potential method in mathematical-physics theory. Of course, one couldn't solve some many problems by potential methods. We shall give a method in which one could reduce every boundary value problem to the second kind Fredholm integral equations and then solve it. In fact, we could obtain solution of every local, non-local or global boundary value problem by this method. Finally, we shall give some sufficient conditions for existence of solutions of the Fefferman's problem A ([4]). .

Stationary Solutions of a Model for the Growth of Tumors and a Connection between the Nonnecrotic and Necrotic Phases

Results of existence of stationary solutions are proved for a problem modeling the growth of a spheroid tumor in absence of inhibitor agents, for both the nonnecrotic and necrotic cases. The results obtained for the nonnecrotic case are used to prove the existence of stationary solutions for the necrotic case, thus clarifying the connection between both cases. Some bounds for the inner and external radii of the necrotic tumor are given. We also discuss the critical nutrient concentration that determines the necrotic phase.

Table of solutions and coefficients for second-order differential equations and inverse problems

In the present paper, formulas for solutions and coefficients of second-order differential equations are presented. These formulas can be used for finding solutions of direct and inverse problems encountered in mathematical modeling of various evolutionary processes. Besides, new statements of inverse problems are considered, and results obtained through solution of these problems are reported. In particular, inverse mathematical-analysis problems are formulated pertaining to the Taylor formula and to the integration average theorem. Also, equations with superpositions, and inverse problems for evolutionary and functional equations with a parameter, are considered. Formulas for solutions of inverse problems are given, and existence conditions for solutions of equations with superpositions are formulated.

Asymptotic results for a class of fourth order quasilinear difference equations

The authors consider the fourth order quasilinear difference equation where α and β are positive constants and {p n } and {q n } are positive real sequences. They classify the nonoscillatory solutions according to their asymptotic behavior for large n and then give necessary and sufficient conditions for existence of solutions of these various types. The results are illustrated with examples.

Dynamic equilibria of group vaccination strategies in a heterogeneous population

In this paper we present an evolutionary variational inequality model of vaccination strategies games in a population with a known vaccine coverage profile over a certain time interval. The population is considered to be heterogeneous, namely its individuals are divided into a finite number of distinct population groups, where each group has different perceptions of vaccine and disease risks. Previous game theoretical analyses of vaccinating behaviour have studied the strategic interaction between individuals attempting to maximize their health states, in situations where an individual's health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here we extend such analyses by applying the theory of evolutionary variational inequalities (EVI) to a (one parameter) family of generalized vaccination games. An EVI is used to provide conditions for existence of solutions (generalized Nash equilibria) for the family of vaccination games, while a projected dynamical system is used to compute approximate solutions of the EVI problem. In particular we study a population model with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). The smaller group is considered much less vaccination inclined than the larger group. Under these hypotheses, considering that the vaccine coverage of the entire population is measured during a vaccine scare period, we find that our model reproduces a feature of real populations: the vaccine averse minority will react immediately to a vaccine scare by dropping their strategy to a nonvaccinator one; the vaccine inclined majority does not follow a nonvaccinator strategy during the scare, although vaccination in this group decreases as well. Moreover we find that there is a delay in the majority's reaction to the scare. This is the first time EVI problems are used in the context of mathematical epidemiology. The results presented emphasize the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by EVI in this area of research.

On a stochastic version of Prouse model in fluid dynamics

A stochastic version of modified Navier–Stokes equations (introduced by Prouse) is considered in a three-dimensional torus; its main feature is that instead of the linear term − ν △ u of the Navier–Stokes equations there is a nonlinear term − △ Φ ( u ) − ∇ div Φ ( u ) . First, for this equation we prove existence and uniqueness of martingale solutions; then existence of stationary solutions. In the last part of the paper a new model, obtained from Prouse model with the nonlinearity Φ ( u ) = ν | u | 4 u , is analysed; for the structure function of this model, some insights towards an expression similar to that obtained by the Kolmogorov 1941 theory of turbulence are presented.

On certain conditions for the existence of solutions of equilibrium problems

The main purpose of this paper is the study of sufficient and/or necessary conditions for existence of solutions of equilibrium problems. We discuss some of the assumptions of the problem, under which the introduced conditions are sufficient and/or necessary, and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem.