On some fixed point results for (\alpha, \beta)-Berinde-\varphi-Contraction mapppings with applications

The aim of this paper is to introduce a new class of mappings called (\alpha, \beta)-Berinde-\varphi-contraction mappings and to establish some fixed point results for this class of mappings in the frame work of metric spaces. Furthermore, we applied our results to the existence of solution of second order differential equations and the existence of a solution for the following nonlinear integral equation: \begin{align*}x(t)=g(t)+\int_a^bM(t,s)K(t,x(s))ds,\end{align*}where M:[a,b]\times [a,b]\to\mathbb{R}^+, K:[a,b]\times \mathbb{R}\to \mathbb{R} and g:[a,b]\to \mathbb{R} are continuous functions. Our results improve, extend and generalize some known results in the literature. In particular, our main result is a generalization of the fixed point result of Pant \cite{ran}.