Existence of multiple positive solutions of n th -order m -point boundary value problems

Sihua Liang; Jihui Zhang

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 1, page 15-28
  • ISSN: 0862-7959

Abstract

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The paper deals with the existence of multiple positive solutions for the boundary value problem ( ϕ ( p ( t ) u ( n - 1 ) ) ( t ) ) ' + a ( t ) f ( t , u ( t ) , u ' ( t ) , ... , u ( n - 2 ) ( t ) ) = 0 , 0 < t < 1 , u ( i ) ( 0 ) = 0 , i = 0 , 1 , ... , n - 3 , u ( n - 2 ) ( 0 ) = ∑ i = 1 m - 2 α i u ( n - 2 ) ( ξ i ) , u ( n - 1 ) ( 1 ) = 0 , where ϕ : ℝ → ℝ is an increasing homeomorphism and a positive homomorphism with ϕ ( 0 ) = 0 . Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.

How to cite

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Liang, Sihua, and Zhang, Jihui. "Existence of multiple positive solutions of $n{\rm th}$-order $m$-point boundary value problems." Mathematica Bohemica 135.1 (2010): 15-28. <http://eudml.org/doc/38107>.

@article{Liang2010,
abstract = {The paper deals with the existence of multiple positive solutions for the boundary value problem \[ \{\left\lbrace \begin\{array\}\{ll\} (\varphi (p(t)u^\{(n-1)\})(t))^\{\prime \} + a(t)f(t, u(t), u^\{\prime \}(t), \ldots , u^\{(n-2)\}(t)) = 0, \quad \ 0 < t < 1, \\ u^\{(i)\}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^\{(n-2)\}(0) = \sum \_\{i=1\}^\{m-2\}\alpha \_iu^\{(n-2)\}(\xi \_i),\quad u^\{(n-1)\}(1) = 0, \end\{array\}\right.\} \] where $\varphi \colon \mathbb \{R\} \rightarrow \mathbb \{R\}$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.},
author = {Liang, Sihua, Zhang, Jihui},
journal = {Mathematica Bohemica},
keywords = {boundary-value problems; positive solutions; fixed-point theorem; cone; boundary-value problem; positive solution; fixed-point theorem; cone},
language = {eng},
number = {1},
pages = {15-28},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of multiple positive solutions of $n\{\rm th\}$-order $m$-point boundary value problems},
url = {http://eudml.org/doc/38107},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Liang, Sihua
AU - Zhang, Jihui
TI - Existence of multiple positive solutions of $n{\rm th}$-order $m$-point boundary value problems
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 1
SP - 15
EP - 28
AB - The paper deals with the existence of multiple positive solutions for the boundary value problem \[ {\left\lbrace \begin{array}{ll} (\varphi (p(t)u^{(n-1)})(t))^{\prime } + a(t)f(t, u(t), u^{\prime }(t), \ldots , u^{(n-2)}(t)) = 0, \quad \ 0 < t < 1, \\ u^{(i)}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^{(n-2)}(0) = \sum _{i=1}^{m-2}\alpha _iu^{(n-2)}(\xi _i),\quad u^{(n-1)}(1) = 0, \end{array}\right.} \] where $\varphi \colon \mathbb {R} \rightarrow \mathbb {R}$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
LA - eng
KW - boundary-value problems; positive solutions; fixed-point theorem; cone; boundary-value problem; positive solution; fixed-point theorem; cone
UR - http://eudml.org/doc/38107
ER -

References

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