On the growth of solutions of some higher order linear differential equations

Abdallah El Farissi; Benharrat Belaidi

Applications of Mathematics (2012)

  • Volume: 57, Issue: 4, page 377-390
  • ISSN: 0862-7940

Abstract

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In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation f ( k ) + A k - 1 f ( k - 1 ) + ⋯ + A 2 f ' ' + ( D 1 ( z ) + A 1 ( z ) e a z ) f ' + ( D 0 ( z ) + A 0 ( z ) e b z ) f = F ( k ≥ 2 ) , where a , b are complex constants that satisfy a b ( a - b ) ≠ 0 and A j ( z ) ( j = 0 , 1 , ⋯ , k - 1 ) , D j ( z ) ( j = 0 , 1 ) , F ( z ) are entire functions with max { ρ ( A j ) ( j = 0 , 1 , ⋯ , k - 1 ) , ρ ( D j ) ( j = 0 , 1 ) } < 1 . We also investigate the relationship between small functions and the solutions of the above equation.

How to cite

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El Farissi, Abdallah, and Belaidi, Benharrat. "On the growth of solutions of some higher order linear differential equations." Applications of Mathematics 57.4 (2012): 377-390. <http://eudml.org/doc/246386>.

@article{ElFarissi2012,
abstract = {In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation \begin\{align\} &f^\{(k)\}+A\_\{k-1\}f^\{(k-1)\}+\dots +A\_\{2\}f^\{\prime \prime \}+(D\_\{1\}(z) +A\_\{1\}(z) \{\rm e\}^\{az\})f^\{\prime \}\\ &\hfill +( D\_\{0\}(z)+A\_\{0\}(z) \{\rm e\}^\{bz\}) f=F\quad (k\ge 2), \end\{align\} where $a$, $b$ are complex constants that satisfy $ab(a-b) \ne 0 $ and $A_\{j\}(z)$$(j=0,1,\dots ,k-1)$, $D_\{j\}(z) $$(j=0,1)$, $F(z) $ are entire functions with $\max \lbrace \rho (A_\{j\}) \ (j=0,1,\dots ,k-1), \ \rho (D_\{j\})$$(j=0,1)\rbrace <1$. We also investigate the relationship between small functions and the solutions of the above equation.},
author = {El Farissi, Abdallah, Belaidi, Benharrat},
journal = {Applications of Mathematics},
keywords = {linear differential equations; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros; linear differential equation; entire solution; exponent of convergence of zeros},
language = {eng},
number = {4},
pages = {377-390},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the growth of solutions of some higher order linear differential equations},
url = {http://eudml.org/doc/246386},
volume = {57},
year = {2012},
}

TY - JOUR
AU - El Farissi, Abdallah
AU - Belaidi, Benharrat
TI - On the growth of solutions of some higher order linear differential equations
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 4
SP - 377
EP - 390
AB - In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation \begin{align} &f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_{2}f^{\prime \prime }+(D_{1}(z) +A_{1}(z) {\rm e}^{az})f^{\prime }\\ &\hfill +( D_{0}(z)+A_{0}(z) {\rm e}^{bz}) f=F\quad (k\ge 2), \end{align} where $a$, $b$ are complex constants that satisfy $ab(a-b) \ne 0 $ and $A_{j}(z)$$(j=0,1,\dots ,k-1)$, $D_{j}(z) $$(j=0,1)$, $F(z) $ are entire functions with $\max \lbrace \rho (A_{j}) \ (j=0,1,\dots ,k-1), \ \rho (D_{j})$$(j=0,1)\rbrace <1$. We also investigate the relationship between small functions and the solutions of the above equation.
LA - eng
KW - linear differential equations; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros; linear differential equation; entire solution; exponent of convergence of zeros
UR - http://eudml.org/doc/246386
ER -

References

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