Some fixed point results of Kannan maps on the Nakano sequence space (Q2046611)

Summary: In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence space \((\ell_{r (.)})_{\psi}\), where \(\psi(v)= \sum_{a = 0}^{\infty}( 1 / r_a) | v_a|^{r_a}\) and \(r_a\geq1\). They depended on their proof that the modular \(\psi\) has the Fatou property. Here we explain that this result is incorrect. Hence, in this paper, the concept of the premodular, which generalizes the modular, on the Nakano sequence space such as its variable exponent in \((1,\infty)\) and the operator ideal constructed by this sequence space and \(s\)-numbers, is introduced. We show the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. Several numerical experiments are presented to illustrate our results. Additionally, some applications to the existence of solutions of summable equations are presented. The novelty lies in the fact that our main results improve some well-known theorems which concern the variable exponent in the aforementioned space.