Some new φ-fixed point and φ-fixed disc results via auxiliary functions

Abstract In this paper, we first introduce several new contractions by some auxiliary functions. Then establish φ -fixed point results for $(\gamma ,\psi ,\varphi ,\phi )$ ( γ , ψ φ ϕ ) contractions, rational- , \psi and almost- \varphi disc $(\psi )_{x_{0}}$ x 0 type 1 2 3 in metric spaces. We generalize existing circle the literature. Further, give examples to demonstrate usefulness of our investigate existence uniqueness solutions nonlinear integral equations.