The connection between spectral and oscillation properties of differential-operator equations of arbitrary order. I. Sturm type theorems

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 48, pp. 101–111, 1987.     

A generalization of the theorems of alternation and non-oscillation for differential-operator equations of arbitrary even order

For a differential equation of arbitrary even order with operator coefficients, a generalization of Arnold’s theorems of alternation and non-oscillation is obtained. The oscillation theorem for a problem with general boundary conditions is proved.     

高阶中立型方程的强迫振动

讨论了一类具有偏差变元的高阶中立型方程，给出了该类方程解振动的充分条件。     

Oscillations in integrodifferential equations of arbitrary order

Sufficient conditions are derived for all bounded solutions of a class of scalar linear integrodifferential equations of arbitrary order to be either oscillatory or convergent to zero asymptotically. When the equations are suitably specialised to be delay-differential equations, our sufficient conditions also become necessary conditions for all bounded solutions of such equations to be oscillatory.     

Opial's inequality and oscillation of 2nd order equations

For a second-order differential equation, we obtain from Opial's inequality lower bounds for the spacing between two zeros of a solution or between a zero of a solution and a zero of its derivative. These bounds are expressed in terms of antiderivatives of the potential, and in particular we derive some new Liapunov type inequalities from them.

     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

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.     

On oscillation properties of elliptic equations

For a second-order elliptic equation, we obtain oscillation and nonoscillation criteria of a new type in which, unlike well-known criteria, the majorant function is not a function of r = |x|.     

Forced oscillation of second order linear and half-linear difference equations

Oscillation properties of solutions of the forced second order linear difference equation

are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation

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are also given.

     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

On the oscillation of second order quasilinear elliptic equations

By using the Riccati technique, some oscillation criteria for the second order quasilinear elliptic equation $\sum _{i,j=1}^N{D}_i\left[\Phi \left(y\right)A_{ij}\left(x\right){\left|Dy\right|}^{p-2}{D}_{j}y\right]+p\left(x\right)f\left(y\right)=0$ are established. These results contain the known oscillation theorems from the literature as special cases.     

On the oscillation of third order nonlinear difference equations

Some new criteria for the oscillation of third order nonlinear difference equations $$\begin{array}{l}\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)+q(k)f(x[g(k)])=0\quad\mbox{and}\\[6pt]\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)=q(k)f(x[g(k)])+p(k)h(x[\sigma(k)])\end{array}$$ are established.     

New classes of second order difference equations solvable by factorization method

Under some assumptions we find a general solution of the factorization problem for a family of second order difference equations.     

The boundary-value problem for differential-operator equations of the first order in a locally convex space

In this paper we describe a method that allows one to study boundary-value problems for the first-order differential-operator equations in an arbitrary locally convex space.