AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract

The oscillation theorem for two simultaneous Sturm-Liouville systems in two parameters is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theorem under another important definiteness condition.     

EXPANSIONS IN EIGENFUNCTIONS OF A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS. II

Abstract

In an earlier work we investigated the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order under left definiteness and semi-definiteness assumptions. In this paper we fix our attention upon the indefinite case.     

An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Sufficient Conditions for the Oscillation of Delay Difference Equations

The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [J. Appl. Math. Simulation 2 (1989), 101–111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations.     

Oscillatory applications of some fourth-order differential equations

Fourth-order advanced differential equations naturally appear in models concerning physical, biological, chemical phenomena applications. The aim of this paper is to study the oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operators. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve the well-known oscillation results in the literature. Some examples to illustrate the results are given.     

Difference Equations and Lettenmeyer's Theorem

Many classical results for ordinary differential equations have counterparts in the theory of difference equations, although, in general, the technical details for the difference versions are more involved than the corresponding ones for differential equations. This note surveys material related to a difference analogue of Lettenmeyer's theorem. The projection method of Harris et al. , developed to treat certain questions in the analytic theory of ordinary differential equations is used to obtain counterparts for linear difference equations as well as extensions to certain nonlinear differential and difference equations.     

On nonlinear oscillations in a suspension bridge system

In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.

     

Oscillation of partial population model with diffusion and delay

By using the upper- and lower-solution method of partial functional differential equations and the oscillation theory of functional differential equation, the oscillation of a population equation with diffusion and delay is studied and a sufficient condition for all positive solutions of the equation to oscillate about the positive equilibrium is obtained. Finally, a model arising from ecology is given to illustrate the obtained results.     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

Stability in Terms of Two Measures for Stochastic Differential Equations with Markovian Switching

Abstract

In this paper, we investigate the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Our new theory can not only be used to show a given system to be stochastically stable in the classical sense, but can also be used to deal with some situations where the classical stability theory is not applicable.     

Elliptic quadratic forms,focal points,and a generalized theory of oscillation

The purpose of this paper is to generalize the theory, methods, and results for oscillation of second-order normal ordinary differential equations. This purpose is obtained by use of a theory of quadratic forms on Hilbert spaces given by Hestenes and the author.In particular, the ideas of this paper may be applied to second-order abnormal problems of differential equations, higher-order control problems, integral and partial differential equations, abstract approximation problems, and to finite dimensional approximations which lead to meaningful computer algorithms.For expository purposes some examples are included. Finally we show that specific existence and comparison theorems for the second-order case may be generalized to the 2nth-order case.     

Quasi-periodic solutions of nonlinear elliptic partial differential equations

In a recent paper [9] the KAM theory has been extended to non-linear partial differential equations, to construct quasi-periodic solutions. In this article this theory is illustrated with three typical examples: an elliptic partial differential equation, an ordinary differential equation and a difference equation related to monotone twist mappings.     

Asymptotic Diagonalization and Normalization of Linear q -Difference Equations

Asymptotic diagonalizations of linear differential equations are studied by several authors. The problems for linear difference equations are investigated recently by Bodine and Sacker. In their work, the full spectrum condition plays essential role. Here we consider a related problem for q-difference equations, |q| < 1, which do not satisfy the full spectrum condition. Our tool is the Arnold normal form for matrix.     

Higher order curvature information and its application in a modified diagonal Secant method

ABSTRACT

A secant equation (quasi-Newton) has one of the most important rule to find an optimal solution in nonlinear optimization. Curvature information must satisfy the usual secant equation to ensure positive definiteness of the Hessian approximation. In this work, we present a new diagonal updating to improve the Hessian approximation with a modifying weak secant equation for the diagonal quasi-Newton (DQN) method. The gradient and function evaluation are utilized to obtain a new weak secant equation and achieve a higher order accuracy in curvature information in the proposed method. Modified DQN methods based on the modified weak secant equation are globally convergent. Extended numerical results indicate the advantages of modified DQN methods over the usual ones and some classical conjugate gradient methods.     

On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.     

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas. March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001.     

带次线性中立项的二阶广义Emden-Fowler微分方程的Philos型振动准则

研究一类带次线性中立项的二阶非线性广义Emden-Fowler时滞微分方程的振动性.利用Riccati变换和不等式技巧,在非正则条件下建立了该类方程多个简便的Philos型和Kamenev型新振动准则.所得定理也适应于包括经典Euler方程等线性非中立型方程,推广和改进了已有文献中的相应结果.最后还给出应用实例展示了所...     

The existence of solutions for an impulsive fractional coupled system of (p,q)‐Laplacian type without the Ambrosetti‐Rabinowitz condition

In this article, based on the variational approach, the existence of at least one nontrivial solution is studied for (p, q)‐Laplacian type impulsive fractional differential equations involving Riemann‐Liouville derivatives. Without the usual Ambrosetti‐Rabinowitz condition, the nonlinearity f in the paper is considered under some suitable assumptions.     

On the oscillation of some linear difference equations with periodic coefficients

Some linear difference equations with periodic coefficients (not necessarily nonnegative) are considered. Necessary conditions and sufficient conditions for the oscillation of the solutions are established. Conditions under which all nonoscillatory solutions tend to zero at ∞ are also presented. The results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients, which were given earlier by the second author [Oscillations of some delay differential equations with periodic coefficients, J. Math. Anal. Appl. 162 (1991) 452–475].