Positive Solutions for a Class of Fourth-order Singular BVPs on the Positive Half-line

Smaıl Djebali, Ouiza Saifi

Abstract

In this work, we are concerned with the existence and multiplicity of positive solutions for the singular fourth-order boundary value problem on the half-line $x^{(4)}(t)-\eta x''(t)+\lambda x(t)=\phi(t)f(t,x(t),x'(t),x''(t),x'''(t)),\quad t\in I=(0,+\infty)$,$x(0)=x''(0)=0$, $x(+\infty)=x''(+\infty)=0$ where$f\in{C}(\mathbb{R}^{+}\times I\times\mathbb{R}^3, \mathbb{R}^{+})$and $\eta, \lambda$ are real positive constants such that $\eta^{2}>4\lambda$. By using the fixed point index theory on cones in appropriate Banach spaces, we obtained existence results of single and multiple positive solutions.