STURM COMPARISON THEOREMS FOR SECOND ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS

In this paper, we first establish some differential inequalities and then some Sturm comparison theorems are derived for the second order neutral nonlinear differential equations. Our results generalize some classical Sturm comparison theorems.     

二阶非线性椭圆型微分方程解的Sturm比较定理

本文通过对一类含阻尼项的椭圆型微分方程建立几个微分不等式,得到几个新的Sturm型比较定理,将经典的Sturm比较定理推广到含阻尼项椭圆型微分方程,并给出一些具体应用的例子.     

二阶非线性微分方程解的Sturm比较定理

通过几个微分不等式建立了一类非线性微分方程解的Sturm比较定理，推广了一些经典的结论。     

二阶非线性椭圆型微分方程解的振动比较定理

建立几个微分不等式,讨论了一类二阶非线性椭圆型微分方程解的振动性,得到几个新的振动比较定理．     

二阶非线性微分方程解的Sturm比较定理

本文首先建立二个微分恒等式，然后利用它们研究了二类非线性微分方程与线性微分程之间解的Ｓｔｕｒｍ比较定理，所得结论包含了一些经典的结论     

Comparison theorems for oscillation of second-order half-linear differential equations

Summary We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations. These improve the work of Elbert and Schneider [6] in which connections are found between half-linear differential equations and linear differential equations. Our comparison theorems are not of Sturm type or Hille--Wintner type which are very famous. We can apply the main results in combination with Sturm's or Hille--Wintner's comparison theorem to a half-linear differential equation of the general form (|x'|^α-1x')' + a(t) |x|^α-1x = 0.     

Sturm comparison and separation theorems for linear,second order,self-adjoint,ordinary, differential equations and for first order systems

Sturm comparison and separation theorems, for first order linear combinations of solutions of linear, second order, self-adjoint, ordinary, differential equations, are obtained as consequences of Sturm comparison and separation theorems for linear combinations of components of solutions of linear, first order systems of ordinary, differential equations.     

一类二阶非线性微分方程解的零点比较定理

通过建立几个微分不等式将经典的微分方程零点比较定理推广到二阶非线性微分方程,得到若干新的结论.     

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.     

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions is derived. Employing the formula, Leighton and Sturm–Picone type comparison theorems as well as several oscillation criteria for impulsive differential equations are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Sturm type theorems for general quasi-variational nonlinear problems

Consider the general equation associated to the initial-value problem where is a restoring force and Q represents a nonlinear damping term. Through an analysis of the equation, we give precise estimates of b in terms of the initial data that extend some results derived from Sturm comparison theorems for linear differential equations of second order. In particular, we show some important theorems of non existence of radial solutions of Dirichlet problems in that either significantly improve the former ones, with the m-Laplacian operator, or cover cases never appeared before, with the mean curvature operator. Received July 18, 1996     

On recovering Sturm—Liouville operators on graphs

Sturm—Liouville differential operators on compact graphs are studied. We establish properties of the spectral characteristics and investigate three inverse problems of recovering the operator from the so-called Weyl functions, from discrete spectral data, and from a system of spectra. For these inverse problems, we prove uniqueness theorems and obtain procedures for constructing the solutions by the method of spectral mappings.     

微分包含的稳定性

本文通过引用生存定理，Ｌｙａｐｕｎｏｖ第二方法和比较定理，首先证明了微分包含的整体存在定理，然后讨论了微分包含解的稳定性     

The variational Lyapunov function and strict stability theory for differential systems

In this paper, we investigate strict stability of differential systems by variational Lyapunov function. We obtain some sufficient conditions and comparison theorems.     

Connection between spectral and oscillation properties of differential-operator equations of arbitrary order. II. Applications of factorization and squaring

Factorization theorems of Frobenius and M. G. Krein-Heinz-Rellich type are established for infinite systems of differential equations of arbitrary even order on finite or infinite intervals. From here, Etgen and Pawlowski oscillation theorems, referring to second order equations, are generalized to the considered equations of order 2n and also a variant of a Sturm type comparison theorem is derived. An analogue of the oscillation theorem for discrete levels in the gap of the continuous spectrum is obtained for problems of both even and odd order.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 101–111, 1988.     

Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains

We study some nonexistence problems for the solutions of semilinear elliptic differential inequalities and systems of second order in conic domains. The proof is based on the trial function method developed by Mitidieri and Pokhozhaev without recourse to comparison theorems and to the maximum principle.     

Backward doubly stochastic differential equations with weak assumptions on the coefficients

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain existence theorems and comparison theorems for solutions of BDSDEs with weak assumptions on the coefficients.     

Sturm–Picone comparison theorem of a kind of conformable fractional differential equations on time scales

In this paper, we consider the Sturm–Picone comparison theorem of conformable fractional differential equations on arbitrary time scales. Since the Picone identity plays an important role in discussing the Sturm comparison theorem. Firstly, we establish the Picone identity of conformable fractional differential equations on arbitrary time scales. By using this identity, we obtain our main result—the Sturm–Picone comparison theorem of conformable fractional differential equations on time scales. This result not only extends and improves the corresponding continuous and discrete time statement, but also contains the usual time scale case when the order of differentiation is one.     

Comparison theorems on the oscillation of even order nonlinear mixed neutral differential equations

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first-order delay differential inequalities and first-order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided.     

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In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDEs). As some applications, a general converse comparison theorem for RBSDEs is obtained and some properties of RBSDEs are discussed.