Existence theory for first order discontinuous functional differential equations

We prove the existence of extremal solutions for a first order functional differential equation subject to nonlinear boundary conditions of functional type. Moreover, the functions that define our problem are allowed to be discontinuous. The proof of our main result is based on a generalized iterative technique.

     

Existence and approximation of solutions for discontinuous functional differential equations

In this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly. By applying the method of upper and lower solutions, which may be discontinuous, some existence results of extremal solutions are obtained. Furthermore, we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences.     

Banach空间不连续的脉冲微分-积分方程的解与迭代解

本文通过建立一个比较结果,应用不动点定理与上、下解方法,讨论了Ｂａｎａｃｈ空间含间断项的一阶混合型脉冲微分－积分方程初值问题的最大解和最小解,并在非线性项满足Ｃａｒａｔｈｅｏｄｏｒｙ条件时获得了解的迭代,推广改进了某些文献中的相应结果．     

一类不连续时滞系统的一致最终有界性

主要讨论不连续的时滞自治系统,在Filippov解意义下的一致最终有界性问题．基于Lyapunov-Krasovskii泛函给出了全局强一致最终有界的Lyapunov定理,并将其应用到一类带有不连续摩擦项的时滞力学系统．     

一类不连续非自治系统的一致最终有界性

主要讨论右端不连续的非自治系统在Filippov解意义下的一致最终有界性问题.首先给出不连续系统全局强一致最终有界的定义,并得到了不连续系统全局一致强最终有界的Lya- punov定理.最后给出了在一类带有不连续摩擦项的力学系统中的应用.     

四阶方程两点边值问题变号解的存在性

利用一个已有的抽象结论,证明了一类非线性四阶方程两点边值问题变号解的存在性.     

Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.     

Nonlinear boundary value problems for discontinuous delayed differential equations

In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.     

DISCONTINUOUS SOLUTIONS IN L^∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed toexamine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.     

Monotonic solutions of a perturbed quadratic fractional integral equation

We present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. The concept of a measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof. Finally, we give an example for indicating the natural realizations of our abstract result presented in the paper.     

Existence and comparison results for quasilinear parrabolic inclusions woth state-dependent subdifferentials ∗

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by nonmonotone operators of Leray-Lions type and state-dependent subdifferentials. Our main goal is to prove the existence of solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques     

Non-linear functional boundary value problem for discontinuous functional differential equations

This paper is devoted to investigating the existence of solutionsfor discontinuous functional differential equations with non-linearfunctional boundary conditions. Some existence results of solutionsare obtained by employing a generalized method of upper andlower solutions, which may be discontinuous.     

Construction of strong solutions of SDE's via Malliavin calculus

In this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients. We illustrate this approach by studying stochastic differential equations driven by the Wiener process. Using Malliavin calculus we derive the result of A.K. Zvonkin (1974) [31] for bounded and measurable drift coefficients as a special case of our analysis of SDE's. Moreover, our approach yields the important insight that the solutions obtained by Zvonkin are even Malliavin differentiable. The latter indicates that the “nature” of strong solutions of SDE's is tightly linked to the property of Malliavin differentiability. We also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions.     

Regularity of solutions for a third order differential equation in Hilbert spaces

We study regularity of mild and strong solutions of an abstract mathematical model of a flexible space structure under appropriate initial conditions. We apply our results showing qualitative properties of the trajectories in the case of the negative Laplacian operator.     

Banach空间中含间断项Sturm—Liouville问题的解

利用拟上下解方法和混合单调迭代法,研究了Banach空间中含间断项的非线性Sturm-liouville问题解的存在唯一性,并给出逼近解迭代序列的误差估计．     

GLOBAL STABILITY OF SOLUTIONS WITH DISCONTINUOUS INITIAL DATA CONTAINING VACUUM STATES FOR THE RELATIVISTIC EULER EQUATIONS

The global stability of Lipschitz continuous solutions with discontinuous initial data for the relativistic Euler equations is established in a broad class of entropy solutions in L∞containing vacuum states. As a corollary, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in     

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Almost periodic invariant tori for the NLS on the circle

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.     

Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games

In this paper, we study the Hamilton-Jacobi-Isaacs equation of zerosum differential games with discontinuous running cost. For such class of equations, the uniqueness of the solutions is not guaranteed in general. We prove principles of optimality for viscosity solutions where one of the players can play either causal strategies or only a subset of continuous strategies. This allows us to obtain nonstandard representation formulas for the minimal and maximal viscosity solutions and prove that a weak form of the existence of value is always satisfied. We state also an explicit uniqueness result for the HJI equations for piecewise continuous coefficients, in which case the usual statement on the existence of value holds.