An asymptotic theorem for a class of nonlinear neutral differential equations

The neutral differential equation is considered under the following conditions: n 2, > 0, = ±1, F(t, u) is nonnegative on [t ₀, ) × (0, ) and is nondecreasing in u (0, infin;), and lim g(t) = as t . It is shown that equation (1.1) has a solution x(t) such that 0}}{\text{.}} \hfill \\ \end{gathered} $$ " align="middle" border="0"> Here, k is an integer with 0 k n–1. To prove the existence of a solution x(t) satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.     

A stability theorem of backward stochastic differential equations and its application

In this Note, we establish a stability theorem for backward stochastic differential equations, and we apply this theorem to study the homogenization of systems of semilinear parabolic partial differential equations.     

An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique 2π$2\pi$-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2k$2k$th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2k$2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for 2k$2k$th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2π$2\pi$-periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.     

A theorem on non-associative algebras and its application to differential equations

For a given field F, the set of F-algebras (resp. commutative F-algebras) of arity n≥2 and F-dimension m can be identified with the mⁿ⁺¹ (resp. m(^m+n?1 _n)) dimensional F-affine space S of structure coefficients. We show: If F is algebraically closed, then there exists an affine subvariety A of S with A≠S, which is defined over the prime field of F, such that all F-algebras corresponding to the points of S-A posses precisely n^m?1 idempotent elements ≠0 and fail to have nil potent elements ≠0. This implies for a system of ordinary differential equations $$\left( * \right)\dot X_i = D_i \left( {X_l ,..,X_m } \right),i = l,..,m,$$ with D_i(X_i,...,X_m)∈?[X₁,...,X_m] homogeneous polynomials of degree n: If the coefficients of the polynomials D_i, i=1,...,m, are algebraically independent over the field of rationals, then (*) possesses precisely n^m?1 ray solutions and fails to have a critical point other than the origin.     

A oscillation theorem for a class of even order neutral differential equations

The even order neutral differential equation

(1)     

An anti-periodic LaSalle oscillation theorem for a class of functional differential equations

In this paper, we establish a result on the existence of anti-periodic solution for a class of functional differential equations, which substantially extends and improves some important results in the literature.     

Comparison theory for Riccati equations

We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B ²+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations

Dedicated to the memory of Professor David Vernon Widder (1898–1990)     

Global bifurcation theorem for a class of boundary conditions for ordinary differential equations of second order

In this paper we deal with boundary value problems where l : C¹([a, b], ?^k) → ?^k × ?^k is continuous, μ ≤ 0 and φ is a Caratheodory map. We define the class S of maps l, for which a global bifurcation theorem holds for the problem (+), with φ(t, x, y, λ) = λ(|x₁|, …, |x^k|) + o(|x| + |y|). We show that the class S contains Sturm‐Liouville boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison methods for a class of function valued stochastic partial differential equations

Summary A comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDE's) with Lipschitz coefficients driven by cylindrical and regular Hilbert space valued Brownian motions. Moreover, we obtain necessary and sufficient conditions for the positivity of the mild solutions of the SPDE's where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDE's, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDE's on the cone of positive functions with non-Lipschitz drift parts including the case of arbitrary polynomialsR(x) withR(O)O and leading negative coefficient.This research was supported by a grant from ONR     

An embedding theorem for differential equations

The present paper establishes a general embedding theorem for differential equations, which is useful in establishing the maximum principle for optimal control problems.The preparation of this paper was sponsored by the Office of Naval Research and the US Army Research Office.     

Min-max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω⊂Rⁿ, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min-max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.     

Riccati scheme for integrating nonlinear systems of differential equations

A modification of the Riccati system that makes it possible to reduce to linear problems the initial-value problem for systems of ordinary differential equations with bilinear nonlinearity is discussed. It is shown that from the algebraic point of view it is natural, in the framework of the scheme, to consider functions that take values in an algebra with two multiplications related by a condition of the type of associativity.State University, St. Petersburg. Translated from Teoretcheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 352–363, March, 1995.     

On a class of Riccati equations in a Hilbert space

We establish existence and uniqueness of solutions of a class of Riccati equations in Hilbert space ocurring in filtering problems for distributed parameter systems using point sensors.     

Adaptive high-order splitting schemes for large-scale differential Riccati equations

We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in an efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used, e.g., for time step adaptivity, and our numerical experiments in this direction show promising results.