Correction to: M. Arif and M. Imdad, Fixed point results under nonlinear suzuki $(F,\mathcal{R}^{\neq})$-contractions with an application, Filomat 36:9 (2022), 3155–3165.

Abstract

{\bf Correction to: M. Arif and M. Imdad, Fixed point results under nonlinear Suzuki $(F,\mathcal{R}^{\neq})$-contractions with an application, Filomat 36:9 (2022), 3155–3165.} {\bf {https://doi.org/10.2298/FIL2209155A}}\\
Here, we have noticed an obvious flaw in the proof of the alternative part of the main results, especifically in Theorems 4.1 and 4.2, where the contraction condition can be applied if the Suzuki condition is met out in the presence of amorphous binary relation. We made an error in applying the contraction condition (denoted as $(e)$) as noted on page 3160, line 18 (from the top) of reference [1]. In our analysis, we used the condition
$\frac{1}{2}d(u,u_{n_k})<d(u,u_{n_k})$ to employ condition $(e)$ on the pair $(u,u_{n_k})$. However, condition $(e)$ is applicable only for those pairs $(u,v)\in \mathcal{R}$ for which either $\frac{1}{2}d(u,Tu)<d(u,v)$ or $\frac{1}{2}d(v,Tv)<d(u,v)$ holds. Therefore, in order to apply hypothesis $(e)$ to the pair $(u,u_{n_k})$, we need to establish either
$\frac{1}{2}d(u_{n_{k}},u_{n_{k}+1})\leq d(u_{n_{k}},u)$ or
$\frac{1}{2}d(u_{n_{k}+1},u_{n_{k}+2})\leq d(u_{n_{k}+1},u)$. However, the main results of \cite{F1} corresponding to $\mathcal{R}$-continuity of $T$ never demands any modification.