Permutable entire functions satisfying algebraic differential equations

In this paper we characterize the relation between two entire functions which are permutable and satisfy certain algebraic differential equations.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

代数微分方程的代数解

Using value distribution theory and techniques,the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that the results are sharp.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

On the hyperbolicity of rapidly oscillating periodic solutions of functional differential equations

We consider periodic solutions of nonlinear functional differential equations with rational periods less than 2. We study the spectral properties of monodromy operators and state a hyperbolicity criterion for such solutions.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 82–85, 2005Original Russian Text Copyright © by A. L. Skubachevskii and H.-O. WaltherThe first authors research was supported by the Mercator-Programm of the Deutsche Forschungsgemeinschaft, RFBR grant No. 04-01-00256, and Russian Ministry of Education and Science grant No. E02-1.0-131.Translated by A. L. Skubachevskii and H.-O. Walther     

On the growth of meromorphic and entire solutions of second-order algebraic differential equations

In this paper, the growth of the meromorphic solutions of the equation

Factorization of entire solutions of some algebraic differential equations

We investigate the factorization of entire solutions of the following algebraic differential equations:

bn(z)finjn(f^′)+bn−1(z)fin−1jn−1(f^′)+?+b₀(z)fi₀j₀(f^′)=b(z),     

On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2.     

On solutions of linear differential equations with entire coefficients

The first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions. Subsequent results involve the separation of the zeros of a Bank-Laine function.     

Entire solutions of differential equations with finite order transcendental entire coefficients

In this paper, we investigate the complex oscillation of the differential equation

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:

fⁿ(z)+P_n−3(f)=p₁e^α₁z+p₂e^α₂z

On global continuability of solutions of infinite-dimensional ordinary differential equations

We prove necessary and sufficient conditions for global in time existence of solutions of ordinary differential equations on infinite dimensional Banach spaces and manifolds under some natural additional hypotheses, in particular, for equations with right-hand sides, given on everywhere dense subsets of phase spaces.     

Fatou sets of entire solutions of linear differential equations

This paper is devoted to studying the dynamical properties of solutions of f^″+A(z)f=0$f^{″}+A\left(z\right)f=0$, where A(z)$A\left(z\right)$ is an entire function of finite order. With some added conditions on A(z)$A\left(z\right)$, we prove that all the Fatou components of E(z)$E\left(z\right)$ are simply-connected provided that E=_f1f2$E=f_1{f}_2$ and _f1,_f2$f_1,f_2$ are two linearly independent solutions of such equations.     

Polynomial solutions of certain differential equations arising in physics

Conditions for the existence of polynomial solutions of certain second‐order differential equations have recently been investigated by several authors. In this paper, a new algorithmic procedure is given to determine necessary and sufficient conditions for a differential equation with polynomial coefficients containing parameters to admit polynomial solutions and to compute these solutions. The effectiveness of this approach is illustrated by applying it to determine new solutions of several differential equations of current interest. A comparative analysis is given to demonstrate the advantage of this algorithmic procedure over existing software. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation