Fixed Point Result of Weak Paired Contractions

Abstract

In this paper, we introduce a new class of mappings called paired weak contractions, which unify and extend two recent concepts in fixed point theory: weak contractions in Banach spaces (Berinde) and paired contractive mappings in metric spaces (Chand & Rohen). For a mapping \(T: X \to X\) on a metric space \(X\), we define the condition:\[d(Tx,Ty) + d(Ty,Tz) \leq \delta (d(x,y)+d(y,z))+L(d(y,Tx)+d(z,Ty)),\]for all \(x,y,z \in X\) which are pairwise distinct, where \(\delta \in (0,1)\) and $L\geq 0$. We establish existence and uniqueness of fixed points. We also apply our result to the Fredholm integral equation. Our results recover the main theorems of Chand & Rohen as special cases.