Abstract
A weighted Nakano sequence space and thes-numbers it contains are the subject of this article, which explains the concept of the pre-quasi-norm and its operator ideal. We show that both Kannan contraction and nonexpansive mappings acting on these spaces have a fixed point. A slew of numerical experiments back up our findings. The presence of summable equations’ solutions is shown to be useful in a number of ways. Weight and power of the weighted Nakano sequence space are used to define the parameters for this technique, resulting in customizable solutions.
Highlights
IntroductionThe spaces of all, bounded, r-absolutely summable, and null sequences of real numbers will be denoted throughout the article by RN0 , l∞, lr, and c0, respectively, where N0 is the set of nonnegative integers
The spaces of all, bounded, r-absolutely summable, and null sequences of real numbers will be denoted throughout the article by RN0, l∞, lr, and c0, respectively, where N0 is the set of nonnegative integers. ey = f0, 0, ⋯, 1, 0, 0, ⋯g, while 1 lies in the yth place, with y ∈ N0
A function s : LðE, H Þ ⟶ 1⁄20,∞ÞN0, where LðE, H Þ is the space of all bounded linear operators from a Banach space E into a Banach space H and if E = H, we write LðEÞ, which transforms every mapping J ∈ LðE, HÞ
Summary
The spaces of all, bounded, r-absolutely summable, and null sequences of real numbers will be denoted throughout the article by RN0 , l∞, lr, and c0, respectively, where N0 is the set of nonnegative integers. (c) Ideal property: syðWH JÞ ≤ kWksyðHÞ kJk, for every J ∈ LðE0, EÞ, H ∈ LðE, H Þ, and W ∈ LðH , H 0Þ, where E0 and H 0 are any two Banach spaces (d) If J ∈ LðE, H Þ and δ ∈ R, we have syðδJÞ = ∣δ ∣ syðJÞ (e) Rank property: if rank ðJÞ ≤ y, syðJÞ = 0, for all J ∈ LðE, H Þ (f) Norming property: sk≥yðIyÞ = 0 or sk
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