Nodal solutions for a second-order m -point boundary value problem

Ruyun Ma

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 4, page 1243-1263
  • ISSN: 0011-4642

Abstract

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We study the existence of nodal solutions of the m -point boundary value problem u ' ' + f ( u ) = 0 , 0 < t < 1 , u ' ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m - 2 α i u ( η i ) where η i ∈ ℚ ( i = 1 , 2 , ⋯ , m - 2 ) with 0 < η 1 < η 2 < ⋯ < η m - 2 < 1 , and α i ∈ ℝ ( i = 1 , 2 , ⋯ , m - 2 ) with α i > 0 and 0 < ∑ i = 1 m - 2 α i < 1 . We give conditions on the ratio f ( s ) / s at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.

How to cite

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Ma, Ruyun. "Nodal solutions for a second-order $m$-point boundary value problem." Czechoslovak Mathematical Journal 56.4 (2006): 1243-1263. <http://eudml.org/doc/31103>.

@article{Ma2006,
abstract = {We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^\{\prime \prime \}+ f(u)=0, \quad 0<t<1, u^\{\prime \}(0)=0, \quad u(1)=\sum ^\{m-2\}\_\{i=1\} \alpha \_i u(\eta \_i) \] where $\eta _i\in \mathbb \{Q\}$$(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _\{m-2\}<1$, and $\alpha _i\in \mathbb \{R\}$$(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^\{m-2\}_\{i=1\} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.},
author = {Ma, Ruyun},
journal = {Czechoslovak Mathematical Journal},
keywords = {multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems; multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems},
language = {eng},
number = {4},
pages = {1243-1263},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nodal solutions for a second-order $m$-point boundary value problem},
url = {http://eudml.org/doc/31103},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ma, Ruyun
TI - Nodal solutions for a second-order $m$-point boundary value problem
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 4
SP - 1243
EP - 1263
AB - We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 0<t<1, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb {Q}$$(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _{m-2}<1$, and $\alpha _i\in \mathbb {R}$$(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^{m-2}_{i=1} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.
LA - eng
KW - multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems; multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems
UR - http://eudml.org/doc/31103
ER -

References

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