The boundedness and convergence of solutions of second-order differential equations with applications

Summary For multidimensional second-order differential equations we present new conditions for the boundedness and convergence of all solutions and their derivatives, and the existence of periodic solutions. In particular, for scalar case we give the necessary and sufficient conditions for the boundedness and convergence of all solutions and their derivatives. These results are used to improve a result of Levin and Nohel for the reactor dynamics and a result of Holmes for the evolution equations derived from a conveying fluid and to give an answer to the conjecture on the continuous stirred tank proposed by Uppal etc.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

Qualitative behavior of global solutions to some nonlinear fourth order differential equations

We study global solutions to a fourth order semilinear ordinary differential equation. We determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions. Furthermore, we discuss their qualitative behaviors such as oscillations and boundedness.     

Local boundedness of solutions to quasilinear elliptic systems

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.     

Convergence,boundedness, and ergodicity of regime-switching diusion processes with infinite memory

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.     

Estimates and asymptotic properties of solutions and their derivatives for a weakly nonlinear third order ordinary differential equation

Sufficient conditions are established for the estimate, boundedness, power absolute integrability on the semiaxis, tending to zero including exponential and power laws for solutions and their first and second derivatives to a weakly nonlinear third order ordinary differential equation.     

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

In this paper, we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first study an ordinary integral equation, where the integral is defined in the Young sense, and we prove an existence result and the boundedness of the solutions. Then, we apply this result pathwise to solve the stochastic problem.     

Das Maximum-Prinzip vom Miranda-Agmon-Typ für Lösungen der biharmonischen Gleichung in einem Rechteck

Utilizing the quadratic functional criteria and positive functionals, the author develops a comparison theorem for oscillation of partial differential equations. This result parallels a comparison of the oscillation of scalar ordinary differential equations to oscillation of vector-matrix differential equations established by LEWIS and WRIGHT . We want to look at the matrix partial differential equation.     

Convergence of the newton process to multiple solutions

In 1870, E.Schröder showed that the convergence of the Newton process of successive approximations to a multiple solution of a scalar equation was geometric in character, and that quadratic convergence could be restored by multiplying the ordinary corrections by a constant. Here, this result is extended to finite systems, and it is shown that there exist various subspaces of the given space in which the convergence is geometric with a rate characteristic of the given subspace. Quadratic convergence may be restored by applying a given fixed linear operator to the ordinary corrections. The conditions under which these results apply to equations in infinite-dimensional Banach spaces are given. Numerical examples involving scalar equations and a simple 2 × 2 system are presented.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No. DA-11-022-ORD-2059.     

Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients

We obtain deterministic first-order linear differential equations with ordinary and variational derivatives and deterministic initial conditions for the expectation and the second moment function of the solution of an ordinary scalar first-order linear inhomogeneous differential equation whose coefficients are random processes. We derive existence conditions for mean periodic solutions. In particular, we consider Gaussian and uniformly distributed random coefficients.     

Stability in functional differential equations established using fixed point theory

In this paper we consider a nonlinear scalar delay differential equation with variable delays and give some new conditions for the boundedness and stability by means of Krasnoselskii’s fixed point theory. A stability theorem with a necessary and sufficient condition is proved. The results in [T.A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003) 15–32; T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11 (2002) 499–519; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63 (2005) e233–e242] are improved and generalized. Some examples are given to illustrate our theory.     

A second-order nonlinear problem with two-point and integral boundary conditions

The paper gives sufficient conditions for the existence and nonuniqueness of monotone solutions of a nonlinear ordinary differential equation of the second order subject to two nonlinear boundary conditions one of which is two-point and the other is integral. The proof is based on an existence result for a problem with functional boundary conditions obtained by the author in [6].     

A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a class of differential vector variational inequalities in finite dimensional Euclidean spaces. We establish a relationship between differential vector variational inequalities and differential scalar variational inequalities. Under various conditions, we obtain the existence and linear growth of solutions to the scalar variational inequalities. In particular we prove existence theorems for Carathéodory weak solutions of the differential vector variational inequalities. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the initial-value differential vector variational inequalities.     

On a theorem of Favard

We obtain sufficient conditions for the existence of almost periodic solutions of almost periodic linear differential equations thereby extending Favard's classical theorem. These results are meant to complement previous results of the authors who have shown by means of a counterexample that the boundedness of all solutions is not, by itself, sufficient to guarantee the existence of an almost periodic solution to a linear almost periodic differential equation.

     

Unified boundedness,periodicity, and stability in ordinary and functional differential equations

Summary We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.On leave from Anhui University, Hefei, Anhui, People's Republic of China     

Evaluating the Evans function: Order reduction in numerical methods

We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss-Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss-Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.

     

Asymptotic convergence of solutions of a scalar q-difference equation with double delays

We obtain sufficient conditions for the asymptotic convergence of all solutions of a scalar q-difference equation with double delays. Moreover, we prove that the limits of the solutions could be formulated in terms of the initial functions and the solution of a corresponding sum equation.     

关于一类五阶常微分方程解的渐近性质

给出了一类五阶常微分方程所有解一致有界和当t→∞时收敛于零的充分条件。得到的结果包含并改善了Abou-El-Ela和Sadek1999年关于非自治微分方程渐近解的结果。     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.