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