In this work, we define new sequence spaces by combining generalized weighted mean and difference operator. Afterward, we investigate topological structure which are completeness, AK-property, AD-property. Also, we compute the α− , β− and γ− duals, and obtain bases for these sequence spaces. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes (c (u, v, ∆) : ℓ∞) and (c (u, v, ∆) : c) are established.2000 Mathematics Subject Classification. 46A05, 46A35, 46A45. Key words and phrases. Difference sequence space, generalized weighted mean, AK and AD properties, the α−, β− and γ− duals and bases of sequences, matrix mappings.
In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature. MSC: Primary 47H10
The concept of αβ-statistical convergence was introduced and studied by Aktuglu (Korovkin type approximation theorems proved via αβ-statistical convergence, J Comput Appl Math 259: [174][175][176][177][178][179][180][181] 2014). In this work, we generalize the concept of αβ-statistical convergence and introduce the concept of weighted αβ-statistical convergence of order γ , weighted αβ-summability of order γ , and strongly weighted αβ-summable sequences of order γ . We also establish some inclusion relation, and some related results for these new summability methods. Furthermore, we prove Korovkin type approximation theorems through weighted αβ-statistical convergence and apply the classical Bernstein operator to construct an example in support of our result.
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