Cyclic Fisher \((\psi ,{\mathcal {R}})\)-quasi-contractions in metric spaces equipped with a semi-partial order relation

A new class of mappings, called cyclic Fisher \((\psi ,{\mathcal {R}})\) $(\psi ,\mathcal{R})$ -quasi-contractions, is introduced and used to survey the existence and convergence of best proximity points in the framework of metric spaces equipped with a relation \({\mathcal {R}}\) $\mathcal{R}$ . In special case, by considering the class of cyclic \(\psi \) $\psi$ -quasi-contractions in reflexive Banach spaces, we extend and improve the main conclusions of Al-Thagafi and Shahzad (Nonlinear Anal 70:3665–3671, 2009).