[Paper Review] Some properties of Generalized Fibonacci difference bounded and $p$-absolutely convergent sequences

Anupam Das, Bipan Hazarika|arXiv (Cornell University)|Apr 1, 2016

Advanced Mathematical Theories and Applications6 references3 citations

TL;DR

This paper introduces a new sequence space $ l_p(\hat{F}(r,s)) $ using the generalized Fibonacci difference matrix $ \hat{F}(r,s) $, establishes its inclusion relations, computes its $ \alpha $-, $ \beta $-, and $ \gamma $-duals, characterizes matrix transformations, and proves that for $ 1 < p < \infty $, the space has the Banach-Saks type $ p $ and the weak fixed point property, leveraging its linear isomorphism to $ \ell_p $. The key contribution lies in the geometric and structural analysis of this novel sequence space derived from Fibonacci matrices.

ABSTRACT

The main objective of this paper is to introduced a new sequence space $l_{p}(\hat{F}(r,s)),$ $ 1\leq p \leq \infty$ by using the band matrix $\hat{F}(r,s).$ We also establish a few inclusion relations concerning this space and determine its $α-,β-,γ-$duals. We also characterize some matrix classes on the space $l_{p}(\hat{F}(r,s))$ and examine some geometric properties of this space.

Motivation & Objective

• To define a new sequence space $ l_p(\hat{F}(r,s)) $ using the band matrix $ \hat{F}(r,s) $ derived from generalized Fibonacci numbers.
• To establish inclusion relations among $ l_p(\hat{F}(r,s)) $ and classical sequence spaces.
• To compute the $ \alpha $-, $ \beta $-, and $ \gamma $-duals of $ l_p(\hat{F}(r,s)) $, enabling characterization of matrix transformations.
• To investigate geometric properties such as the Banach-Saks type $ p $ and weak fixed point property in $ l_p(\hat{F}(r,s)) $ for $ 1 < p < \infty $.

Proposed method

• Define the generalized Fibonacci difference matrix $ \hat{F}(r,s) $ using Fibonacci numbers $ f_n $, with entries $ f_{nk}(r,s) = s\frac{f_{n+1}}{f_n} $ for $ k=n $, $ r\frac{f_n}{f_{n-1}} $ for $ k=n-1 $, and zero otherwise.
• Introduce the sequence space $ l_p(\hat{F}(r,s)) $ as the matrix domain $ \ell_p^{\hat{F}(r,s)} = \{ x \in \omega : \hat{F}(r,s)x \in \ell_p \} $, ensuring completeness and normed structure.
• Use standard techniques from functional analysis to compute the $ \alpha $-, $ \beta $-, and $ \gamma $-duals via summability conditions and matrix transformation characterizations.
• Establish inclusion theorems by comparing $ l_p(\hat{F}(r,s)) $ with classical spaces like $ \ell_p $, $ c_0 $, and $ \ell_\infty $, using properties of the matrix $ \hat{F}(r,s) $.
• Construct a Schauder basis for $ l_p(\hat{F}(r,s)) $ using the standard basis $ \{e^{(n)}\} $, proving its completeness and Schauder basis property.
• Prove geometric properties using the linear isomorphism between $ l_p(\hat{F}(r,s)) $ and $ \ell_p $, and apply known results on Banach-Saks type $ p $ and weak fixed point property.

Experimental results

Research questions

• RQ1 What are the inclusion relations between the newly defined space $ l_p(\hat{F}(r,s)) $ and classical sequence spaces such as $ \ell_p $, $ c_0 $, and $ \ell_\infty $?
• RQ2 What are the $ \alpha $-, $ \beta $-, and $ \gamma $-duals of the space $ l_p(\hat{F}(r,s)) $?
• RQ3 Which matrix classes map $ l_p(\hat{F}(r,s)) $ into other sequence spaces, and how can they be characterized?
• RQ4 Does $ l_p(\hat{F}(r,s)) $ possess the Banach-Saks type $ p $ property for $ 1 < p < \infty $?
• RQ5 Does $ l_p(\hat{F}(r,s)) $ have the weak fixed point property, and what is the value of the García-Falset coefficient $ R(l_p(\hat{F}(r,s))) $?

Key findings

• The space $ l_p(\hat{F}(r,s)) $ is a Banach space with the norm induced by the matrix $ \hat{F}(r,s) $, and it is linearly isomorphic to $ \ell_p $, preserving key geometric properties.
• The $ \alpha $-, $ \beta $-, and $ \gamma $-duals of $ l_p(\hat{F}(r,s)) $ are fully characterized using summability conditions on the matrix entries.
• The space $ l_p(\hat{F}(r,s)) $ has the Banach-Saks type $ p $ for $ 1 < p < \infty $, meaning every weakly null sequence has a subsequence whose Cesàro means grow at most as $ (n+1)^{1/p} $.
• The space $ l_p(\hat{F}(r,s)) $ satisfies the weak fixed point property, as $ R(l_p(\hat{F}(r,s))) = 2^{1/p} < 2 $, which ensures the existence of fixed points for nonexpansive mappings.
• The Schauder basis of $ l_p(\hat{F}(r,s)) $ is the standard basis $ \{e^{(n)}\} $, and the space is separable and complete.
• The inclusion relations show that $ l_p(\hat{F}(r,s)) \subset \ell_p $ for $ 1 \leq p < \infty $, and $ \ell_p \subset l_p(\hat{F}(r,s)) $ under certain conditions on $ r $ and $ s $.