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

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.

Keywords

• elastic beam
• fourth order equations
• lower and upper solutions
• Green function