TWIN POSITIVE SYMMETRIC SOLUTIONS FOR LIDSTONE BOUNDARY VALUE PROBLEMS

Abstract

In this paper, we consider the Lidstone boundary value problem $$(\Phi (y^{(2n-1)}))^{'}(t)=f(t,y(t),y^{''}(t),\cdots ,y^{(2(n-1))}(t)), ~~~0\leq t\leq 1,$$ $$y^{(2i)}(0)=y^{(2i)}(1)=0,~~~0\leq i\leq n-1,$$ where $f: [0,1]\times R^{n}\rightarrow R$ is continuous, $\Phi(v)=|v|^{p-2}v,~p\gt 1$. Growth conditions are imposed on $f$ which yield the existence of at least two symmetric positive solutions by using a fixed point theorem in cones.