On some boundary value problems for second order nonlinear differential equations

Zuzana Došlá; Mauro Marini; Serena Matucci

Mathematica Bohemica (2012)

  • Volume: 137, Issue: 2, page 113-122
  • ISSN: 0862-7959

Abstract

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We investigate two boundary value problems for the second order differential equation with p -Laplacian ( a ( t ) Φ p ( x ' ) ) ' = b ( t ) F ( x ) , t ∈ I = [ 0 , ∞ ) , where a , b are continuous positive functions on I . We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: i ) x ( 0 ) = c > 0 , lim t → ∞ x ( t ) = 0 ; ii ) x ' ( 0 ) = d < 0 , lim t → ∞ x ( t ) = 0 .

How to cite

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Došlá, Zuzana, Marini, Mauro, and Matucci, Serena. "On some boundary value problems for second order nonlinear differential equations." Mathematica Bohemica 137.2 (2012): 113-122. <http://eudml.org/doc/246467>.

@article{Došlá2012,
abstract = {We investigate two boundary value problems for the second order differential equation with $p$-Laplacian \[ (a(t)\Phi \_\{p\}(x^\{\prime \}))^\{\prime \}=b(t)F(x), \quad t\in I=[0,\infty ), \] where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: \[ \{\rm i)\}\ x(0)=c>0, \ \lim \_\{t\rightarrow \infty \}x(t)=0; \quad \{\rm ii)\}\ x^\{\prime \}(0)=d<0, \ \lim \_\{t\rightarrow \infty \}x(t)=0. \]},
author = {Došlá, Zuzana, Marini, Mauro, Matucci, Serena},
journal = {Mathematica Bohemica},
keywords = {boundary value problem; $p$-Laplacian; half-linear equation; positive solution; uniqueness; decaying solution; principal solution; -Laplacian; boundary value problem; positive solution; noncompact interval},
language = {eng},
number = {2},
pages = {113-122},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some boundary value problems for second order nonlinear differential equations},
url = {http://eudml.org/doc/246467},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Došlá, Zuzana
AU - Marini, Mauro
AU - Matucci, Serena
TI - On some boundary value problems for second order nonlinear differential equations
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 2
SP - 113
EP - 122
AB - We investigate two boundary value problems for the second order differential equation with $p$-Laplacian \[ (a(t)\Phi _{p}(x^{\prime }))^{\prime }=b(t)F(x), \quad t\in I=[0,\infty ), \] where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x^{\prime }(0)=d<0, \ \lim _{t\rightarrow \infty }x(t)=0. \]
LA - eng
KW - boundary value problem; $p$-Laplacian; half-linear equation; positive solution; uniqueness; decaying solution; principal solution; -Laplacian; boundary value problem; positive solution; noncompact interval
UR - http://eudml.org/doc/246467
ER -

References

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