Some aspects of 𝜆-weak convergence using difference operator

In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>ı</m:mi> <m:mi>ȷ</m:mi> </m:msubsup> </m:math> Δ^{}_{} and the de la Vallée–Poussin mean, denoted as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">V</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>w</m:mi> </m:msub> <m:mo>,</m:mo> <m:msubsup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>ı</m:mi> <m:mi>ȷ</m:mi> </m:msubsup> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mi>m</m:mi> </m:msub> </m:math> [(V,λ)w,Δ^{}_{}]ₘ for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> m=0 , 1, and ∞. Further, we explore some algebraic and topological properties of these spaces, including their nature as linear, normed, Banach, and BK spaces. Additionally, we examine properties such as solidity, symmetry, and monotonicity. Finally, we define and establish some inclusion relations among generalized difference weak statistical convergence, generalized difference weak 𝜆-statistical convergence, and generalized difference weak <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi mathvariant="script">V</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> [V,λ] -convergence.