# Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

• Volume: 69, Issue: 3, page 265-270
• ISSN: 0066-2216

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## Abstract

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The fourth order periodic boundary value problem ${u}^{\left(4\right)}-m⁴u+F\left(t,u\right)=0$, 0 < t < 2π, with ${u}^{\left(i\right)}\left(0\right)={u}^{\left(i\right)}\left(2\pi \right)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $±{10}^{-7}$.

## How to cite

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Lingbin Kong, and Daqing Jiang. "Multiple positive solutions of a nonlinear fourth order periodic boundary value problem." Annales Polonici Mathematici 69.3 (1998): 265-270. <http://eudml.org/doc/270758>.

@article{LingbinKong1998,
abstract = {The fourth order periodic boundary value problem $u^\{(4)\} - m⁴u + F(t,u) = 0$, 0 < t < 2π, with $u^\{(i)\}(0) = u^\{(i)\}(2π)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $±10^\{-7\}$.},
author = {Lingbin Kong, Daqing Jiang},
journal = {Annales Polonici Mathematici},
keywords = {nonlinear periodic boundary value problem; multiple positive solutions; cone; fixed point index; fourth-order periodic boundary value problem; positive solutions},
language = {eng},
number = {3},
pages = {265-270},
title = {Multiple positive solutions of a nonlinear fourth order periodic boundary value problem},
url = {http://eudml.org/doc/270758},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Lingbin Kong
AU - Daqing Jiang
TI - Multiple positive solutions of a nonlinear fourth order periodic boundary value problem
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 3
SP - 265
EP - 270
AB - The fourth order periodic boundary value problem $u^{(4)} - m⁴u + F(t,u) = 0$, 0 < t < 2π, with $u^{(i)}(0) = u^{(i)}(2π)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $±10^{-7}$.
LA - eng
KW - nonlinear periodic boundary value problem; multiple positive solutions; cone; fixed point index; fourth-order periodic boundary value problem; positive solutions
UR - http://eudml.org/doc/270758
ER -

## References

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1. [1] A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl. 185 (1994), 302-320. Zbl0807.34023
2. [2] A. Cabada and J. J. Nieto, A generalization of the monotone iterative technique for nonlinear second-order periodic boundary value problems, J. Math. Anal. Appl. 151 (1990), 181-189. Zbl0719.34039
3. [3] L. H. Erbe, S. C. Hu and H. Y. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), 640-648. Zbl0805.34021
4. [4] W. J. Gao and J. Y. Wang, On a nonlinear second order periodic boundary value problem with Carathéodory functions, Ann. Polon. Math. 62 (1995), 283-291. Zbl0839.34031
5. [5] D. Q. Jiang and J. Y. Wang, A generalized periodic boundary value problem for the one-dimensional p-Laplacian, Ann. Polon. Math. 65 (1997), 265-270. Zbl0868.34015
6. [6] V. Šeda, J. J. Nieto and M. Gera, Periodic boundary value problems for nonlinear higher order ordinary differential equations, Appl. Math. Comput. 48 (1992), 71-82. Zbl0748.34014
7. [7] M. X. Wang, A. Cabada and J. J. Nieto, Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions, Ann. Polon. Math. 58 (1993), 221-235. Zbl0789.34027

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