Existence Results for Nonlinear Boundary Value Problems

Abstract

In the present paper, we are concerned to prove under some hypothesis the existence of fixed points of
the operator $L$ defined on $C(I)$ by
\[
Lu(t)=\int_{0}^{w}G(t,s)h(s)f(u(s))ds,\;t\in I,\,\omega \in \{1,\infty\}
\]%
where the functions $f\in C([0,\infty );[0,\infty )),\,h\in C(I;[0,\infty ))$, \\$G\in
C(I\times I)$ and $\begin{cases}
& I=[0,1]\text{ if},\, \omega=1,\\
&I=[0,\infty)\text{ if},\,\omega=\infty.
\end{cases}$. By using Guo Krasnoselskii fixed point theorem, we establish the existence of at least
one fixed point of the operator $L$.