Existence results for nonlinear problems with $\varphi$- Laplacian operators and nonlocal boundary conditions

Abstract

Using Leray-Schauder degree theory we study the existence of at least one solution for the boundary value problem of the type
\[
\left\{\begin{array}{lll}
(\varphi(u' ))' = f(t,u,u') & & \\
u'(0)=u(0), \ u'(T)= bu'(0), & & \quad \quad
\end{array}\right.
\]
where $\varphi: \mathbb{R}\rightarrow \mathbb{R}$ is a homeomorphism such that $\varphi(0)=0$, $f:\left[0, T\right]\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function, $T$ a positive real number, and $b$ some non zero real number.