Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 11 2 2020 12 01 On some fixed point results for $(\alpha, \beta)$-Berinde-$\varphi$-Contraction mapppings with applications 363 378 4491 10.22075/ijnaa.2020.20635.2183 EN Akindele Mebawondu School of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Durban, South Africa Chinedu Izuchukwu School of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Durban, South Africa Kazeem Aremu School of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Durban, South Africa Oluwatosin Temitope Mewomo School of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Durban, South Africa Journal Article 2020 06 15 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*}<br />x(t)=g(t)+\int_a^bM(t,s)K(t,x(s))ds,<br />\end{align*}<br />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}. https://ijnaa.semnan.ac.ir/article_4491_75a0c78d4122489ab92f56eb7b478bbf.pdf