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

It is known that it a very restrictive condition for frame $\{f_{k}\}^{\infty }_{k=1}$ to have representation $ \{T^n \varphi \}_{n=0}^\infty as the orbit of bounded operator $T$ under single generator $\varphi \in \mathcal H.$ We prove that,