Boundary Value Problems (2017-10-01)

# The method of lower and upper solutions for fourth order equations with the Navier condition

• Ruyun Ma,
• Jinxiang Wang,
• Dongliang Yan

DOI
https://doi.org/10.1186/s13661-017-0887-5
Journal volume & issue
Vol. 2017, no. 1
pp. 1 – 9

## Abstract

Abstract The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form y ( 4 ) ( x ) + ( k 1 + k 2 ) y ″ ( x ) + k 1 k 2 y ( x ) = f ( x , y ( x ) ) , x ∈ ( 0 , 1 ) , $$y^{(4)}(x)+(k_{1}+k_{2}) y''(x)+k_{1}k_{2} y(x)=f\bigl(x,y(x)\bigr), \quad x\in(0,1),$$ with the Navier condition y ( 0 ) = y ( 1 ) = y ″ ( 0 ) = y ″ ( 1 ) = 0 $$y(0) = y(1) = y''(0) = y''(1) = 0$$ under the condition k 1 < 0 < k 2 < π 2 $k_{1}<0<k_{2}<\pi^{2}$ . The main tool is the Schauder fixed point theorem.