Some Fixed Point Results of Kannan Maps on the Nakano Sequence Space

Abstract

In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence space ℓ r . ψ , where ψ v = ∑ a = 0 ∞ 1 / r a v a r a and r a ≥ 1 . They depended on their proof that the modular ψ has the Fatou property. But we have explained 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 , ∞ and the operator ideal constructed by this sequence space and s -numbers is introduced. We construct the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. It is interesting that several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of summable equations are introduced. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.

Decision-making on the solution of non-linear dynamical systems of Kannan non-expansive type in Nakano sequence space of fuzzy numbers

“…In Reference [11], the only attempt was made to explain Kannan operators in modular vector spaces. For more details on Kannan's fixed point theorems see [2,[4][5][6][7]25]. Given that the demonstration of many fixed point theorems in a given space entails either expanding the space itself or its self-mapping, both of these alternatives are possible.…”

Kannan Nonexpansive Operators on Variable Exponent Cesàro Sequence Space of Fuzzy Functions

“…In Reference [14], the only attempt was made to explain Kannan operators in modular vector spaces. For more details on Kannan's fixed point theorems, see [15][16][17][18][19][20]. Given that the proof of many fixed point theorems in a given space requires either growing the space itself or expanding the self-mapping that acts on it, both options are viable.…”