φ -fixed point results for nonlinear contractions with an application

In this paper, using the notion of w -distance in metric spaces, we introduce two types of nonlinear contractions, (ι ,ψ ,φ ,ϕ ,p) and rational- (ι ,ψ , φ ,ϕ ,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.