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

Abstract

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.

Keywords

• φ-fixed point
• φ-fixed disc
• ( γ , ψ , φ , ϕ ) $(\gamma ,\psi ,\varphi ,\phi )$ contraction
• ( ψ , φ , ϕ ) x 0 $(\psi ,\varphi ,\phi )_{x_{0}}$ contraction
• Nonlinear integral equations