RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Ufimsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 1, Pages 118–136 (Mi ufa414)

On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

M. Saidani, B. Belaїdi

Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem-(Algeria)

Abstract: In this paper, by using the value distribution theory, we study the growth and the oscillation of meromorphic solutions of the linear differential equation
\begin{align*} f^{(k) }&+( A_{k-1,1}(z) e^{P_{k-1}(z) }+A_{k-1,2}(z) e^{Q_{k-1}(z) }) f^{( k-1) }
& +\cdots +( A_{0,1}(z) e^{P_{0}(z) }+A_{0,2}(z) e^{Q_{0}(z) }) f=F(z), \end{align*}
where $A_{j,i}(z) ( \not\equiv 0)$ $( j=0,\ldots,k-1),$ $F(z)$ are meromorphic functions of a finite order, and $P_{j}(z),Q_{j}(z)$ $(j=0,1,\ldots,k-1;i=1,2)$ are polynomials with degree $n\geqslant 1$. Under some conditions, we prove that as $F\equiv 0$, each meromorphic solution $f\not\equiv 0$ with poles of uniformly bounded multiplicity is of infinite order and satisfies $\rho _{2}(f)=n$ and as $F\not\equiv 0$, there exists at most one exceptional solution $f_{0}$ of a finite order, and all other transcendental meromorphic solutions $f$ with poles of uniformly bounded multiplicities satisfy ${\overline{\lambda }(f)=\lambda (f)=\rho ( f) =+\infty }$ and $\overline{\lambda }_{2}( f) =\lambda _{2}( f) =\rho _{2}( f) \leq \max \{ n,\rho ( F) \}.$ Our results extend the previous results due Zhan and Xiao [19].

Keywords: Order of growth, hyper-order, exponent of convergence of zero sequence, differential equation, meromorphic function.

Full text: PDF file (480 kB)
References: PDF file   HTML file

English version:
Ufa Mathematical Journal, 2018, 10:1, 115–134 (PDF, 453 kB); https://doi.org/10.13108/2018-10-1-115

Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: 34M10, 30D35
Language: English

Citation: M. Saidani, B. Belaïdi, “On the growth of solutions of some higher order linear differential equations with meromorphic coefficients”, Ufimsk. Mat. Zh., 10:1 (2018), 118–136; Ufa Math. J., 10:1 (2018), 115–134

Citation in format AMSBIB
\Bibitem{SaiBel18} \by M.~Saidani, B.~Bela\"{\i}di \paper On the growth of solutions of some higher order linear differential equations with meromorphic coefficients \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 1 \pages 118--136 \mathnet{http://mi.mathnet.ru/ufa414} \elib{http://elibrary.ru/item.asp?id=32705557} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 1 \pages 115--134 \crossref{https://doi.org/10.13108/2018-10-1-115} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000432413800009} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85044296162}