φ-fixed point results for nonlinear contractions with an application

Abstract In this paper, using the notion of w-distance in metric spaces, we introduce two types of nonlinear contractions, ( ι , ψ , φ , ϕ , p ) $(\iota ,\psi ,\varphi ,\phi ,p)$ and rational- ( ι , ψ , φ , ϕ , p ) $(\iota ,\psi , \varphi ,\phi ,p)$ contractions. Based on these contractions, we prove the existence and uniqueness of a φ-fixed point for the corresponding contraction. We also provide some examples to demonstrate the correctness and practicability of our results, along with a numerical experiment. Finally, we apply the obtained results to linear matrix equations and nonlinear Fredholm integral equations.