Conditions For Existence Of Solutions Research Articles

Resolution of the Transport equation subject to constraint

We present a method for solving the Transport equation when its solution has to belong to a constrained set which is not required to be convex. An autonomous formulation of the characteristics method allows us to use the tangency condition which has been introduced for ordinary differential equations. Thus we obtain a sufficient condition for existence of solutions, which shows the interplay between the geometry of the constraints set K and the velocity field β . A numerical method is proposed for solving the problem when the sufficient condition is not satisfied. A numerical experiment is presented showing the efficiency of the algorithm proposed.

Sufficient condition for existence of solutions for higher-order resonance boundary value problem with one-dimensional p-Laplacian

By using coincidence degree theory of Mawhin, existence results for some higher order resonance multi- point boundary value problems with one dimensional p-Laplacian operator are obtained.

非線形ディスクリプタシステムに対する安定条件

Descriptor equations have much more flexibility in describing nonlinear systems than state equations. In this paper, based on the Lyapunov's direct method, stability conditions are developed for nonlinear descriptor systems. An existence condition of solutions of descriptor equations is also derived. The results remove the differentiability assumptions on nonlinearities in descriptor equations, which are required in the existing works. As an application of the conditions, stability analysis of a Lur'e-type feedback system whose linear part has a direct path is considered by using a descriptor expression.

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.

Global solutions of inhomogeneous Hamilton-Jacobi equations

We consider the viscous Hamilton-Jacobi (VHJ) equationu t-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas considerable effort has been devoted to study the large time behavior of solutions of the equationu t-Δu=g(x,u), whereamplitude blow-up may occur if for instanceg(x,u)≈u p asu→∞ andp>1, relatively little is known in the case of (VHJ). The aim of this paper is to investigate this question. More precisely, we study the relations between and we obtain a precise description of the global dynamics for (VHJ). Namely, we show that (i) implies (ii) and that in this case, all global solutions converge uniformly to the (unique) stationary solution. In the radial case, we prove that, conversely, (ii) implies (i). Moreover, for certain (smooth) functionsh, we obtain the existence of global classical solutions with gradient blowing up in infinite time. For 1p-2 or for the Cauchy problem, all solutions are global, but we establish similar relations between the existence of bounded or locally bounded solutions and the existence of stationary solutions. Our proofs depend on some new gradient estimates of solutions, local and global in space, obtained by Bernstein type arguments. As another consequence of these estimates we prove a parabolic Liouville-type theorem for solutions ofu t\t-Δu=│Δu│p in ℝNx(\t-\t8,0). Various other results are obtained, including universal bounds for global solutions.

Theory of differential equations in discontinuous piecewise‐defined functions

AbstractA truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Explorando sistemas hamiltonianos: estudo analítico

Um pêndulo duplo submetido a torques externos é usado para introduzir alguns fundamentos da teoria de Sistemas Dinâmicos para os cursos de graduação em Física, logo após o aluno ter cursado a disciplina de mecânica analítica. Este sistema mostra-se um bom exemplo para a introdução de tais técnicas. As equações de movimento de Hamilton indicam a existência de soluções estacionárias (pontos de equilíbrio) no espaço de fase do modelo. A identificação da natureza desses pontos de equilíbrio permite a descrição da dinâmica do sistema na vizinhança linear destes. Além disso, os resultados obtidos na vizinhança linear de um ponto de equilíbrio não são alterados pela introdução de torques externos constantes não-nulos. Neste trabalho enfatiza-se a análise da vizinhança linear dos pontos de equilíbrio.

Some nonexistence results of stationary solutions for the Gray–Scott model

We study some stationary problems for the Gray–Scott model. The existence of stationary solutions has been established by Hale et al. [J.K. Hale, L.A. Peletier, W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000) 102–130]. In this paper we will show some sufficient conditions about the nonexistence of nontrivial stationary solutions. To prove the result, we need some a priori estimates derived by using the strong maximum principle.

On the existence of stationary optimal policies for partially observed MDPs under the long-run average cost criterion

This paper studies the problem of the existence of stationary optimal policies for finite state controlled Markov chains, with compact action space and imperfect observations, under the long-run average cost criterion. It presents sufficient conditions for existence of solutions to the associated dynamic programming equation, that strengthen past results. There is a detailed discussion comparing the different assumptions commonly found in the literature.

Large Eddy Simulation. Existence of stationary solutions to a dynamical model

We consider the existence of stationary solutions to the Germano Model - equations describing turbulent flow of fluids. The model comes from Large Eddy Simulation techniques yielding modified Navier-Stokes Equations with an additional nonlocal term. On one hand this nonlocalness disturbs monotonicity, but on the other hand it is helpful for compactness arguments. Thus we combine the methods of monotone operators and smoothing properties of convolutions in passing to the limit in the approximate problem.

Set-relations and optimality conditions in set-valued maps

Problems in set-valued optimization can be solved via set-optimization. In this paper properties of set-optimization are investigated. Conditions for existence of solutions are established. Directional derivatives are studied for set-valued mappings. Necessary and sufficient conditions in the existence of solutions are showed with directional derivatives.

STATIONARY SOLUTIONS OF A SELECTION MUTATION MODEL: THE PURE MUTATION CASE

A selection mutation equations model for the distribution of individuals with respect to the age at maturity is considered. In this model we assume that a mutation, perhaps very small, occurs in every reproduction where the noncompactness of the domain of the structuring variable and the two-dimensionality of the environment are the main features. Existence of stationary solutions is proved using the theory of positive semigroups and the infinite-dimensional version in Banach lattices of the Perron Frobenius theorem. The behavior of these stationary solutions when the mutation is small is studied.

Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.

Existence conditions in general quasimonotone variational inequalities

In this paper we study a general variational inequality model with set-valued quasimonotone operators, a model which includes several variational inequalities and equilibrium problems. We establish unifying conditions for existence of solutions in a topological vector space setting. Applications to parametric equilibrium models and to a contact problem are given.

Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl

We look for necessary and sufficient conditions for the existence of solutions to the minimisation problem <p align="center"> $ (P) \qquad\qquad\qquad $ inf $\int_\Omega f $ (curl $u(x)) dx : u \in u_{\xi_0} + W^{1,\infty}_0(\Omega;\mathbb R^3)$ <p align="left" class="times"> where the boundary data $u_{\xi_0}$ satisfies curl$u_{\xi_0}(x)= \xi_{0}$, for $\xi_0$ a given vector in $\mathbb R^3$.

Functional Equations in Asymptotical Problems of Queueing Theory

The investigation of communication systems with a large number of devices sometimes leads to initial-boundary problems for functional equations. In this work we consider several classes of such problems for differential-difference and integral-differential equations and for partial differential equations. We are interested in the global existence of solutions in the quarter-plane x > 0, t > 0; in the existence of stationary solutions, in their stability, and in their behavior as x → ∞.

Singular solutions of protter's problem for the 3 + 1;-D wave equation

For the wave equation we study boundary value problems, which are four-dimensional analogues of Darboux problems on the plane. It is shown that for n in ℕ there exists a right hand side smooth function from C n , for which the corresponding unique generalized solution has a strong power-type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone and does not propagate along the cone. The present article describes the exact behavior of the singular solutions at the point O. We give some necessary and sufficient conditions for existence of solutions with fixed order of singularity. It states some exact a priori estimates for the singular solutions

Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

Stability and instability of equilibria of an equation of size structured population dynamics

In this paper we consider a quasilinear equation with a nonlinear boundary condition modelling the dynamics of a biological population structured by size. We suppose vital rates depending on the total population. This hypothesis introduces some nonlinearities on the equation and on the boundary condition. We study the existence and uniqueness of solution of the initial value problem and the existence of stationary solutions. After we calculate the spectrum of the linearization at an equilibrium and we study its (local) stability.

On the Existence of Stationary Solutions of a Generalized Mullins Equation

We derive a nonlinear evolution equation which describes the development of a solid surface due to the mechanism of evaporation–condensation under a temperature gradient. The existence of stationary grain boundary grooves is shown and the stationary shapes are evaluated.