Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations

Luisa Malaguti; Valentina Taddei

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2005)

  • Volume: 44, Issue: 1, page 97-113
  • ISSN: 0231-9721

Abstract

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The paper deals with the quasi-linear ordinary differential equation ( r ( t ) ϕ ( u ' ) ) ' + g ( t , u ) = 0 with t ∈ [ 0 , ∞ ) . We treat the case when g is not necessarily monotone in its second argument and assume usual conditions on r ( t ) and ϕ ( u ) . We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta Math. Hung. 1990]. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem associated to the equation when ϕ ( u ) = u . Several examples are included.

How to cite

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Malaguti, Luisa, and Taddei, Valentina. "Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 44.1 (2005): 97-113. <http://eudml.org/doc/32454>.

@article{Malaguti2005,
abstract = {The paper deals with the quasi-linear ordinary differential equation $(r(t)\varphi (u^\{\prime \}))^\{\prime \}+g(t,u)=0$ with $t \in [0, \infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r(t)$ and $\varphi (u)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta Math. Hung. 1990]. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem associated to the equation when $\varphi (u)=u$. Several examples are included.},
author = {Malaguti, Luisa, Taddei, Valentina},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques; quasi-linear differential equation; oscillatory solution},
language = {eng},
number = {1},
pages = {97-113},
publisher = {Palacký University Olomouc},
title = {Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations},
url = {http://eudml.org/doc/32454},
volume = {44},
year = {2005},
}

TY - JOUR
AU - Malaguti, Luisa
AU - Taddei, Valentina
TI - Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2005
PB - Palacký University Olomouc
VL - 44
IS - 1
SP - 97
EP - 113
AB - The paper deals with the quasi-linear ordinary differential equation $(r(t)\varphi (u^{\prime }))^{\prime }+g(t,u)=0$ with $t \in [0, \infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r(t)$ and $\varphi (u)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta Math. Hung. 1990]. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem associated to the equation when $\varphi (u)=u$. Several examples are included.
LA - eng
KW - quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques; quasi-linear differential equation; oscillatory solution
UR - http://eudml.org/doc/32454
ER -

References

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