By Józef Banaś, Mohammad Mursaleen

This ebook bargains with the learn of series areas, matrix adjustments, measures of noncompactness and their quite a few purposes. The idea of degree of noncompactness is likely one of the most valuable ones to be had and has many purposes. The publication discusses many of the life effects for numerous sorts of differential and quintessential equations with assistance from measures of noncompactness; particularly, the Hausdorff degree of noncompactness has been utilized to procure worthy and adequate stipulations for matrix operators among BK areas to be compact operators.

The ebook includes 8 self-contained chapters. bankruptcy 1 discusses the speculation of FK areas and bankruptcy 2 numerous duals of series areas, that are used to symbolize the matrix periods among those series areas (FK and BK areas) in Chapters three and four. bankruptcy five experiences the thought of a degree of noncompactness and its houses. The concepts linked to measures of noncompactness are utilized to symbolize the compact matrix operators in Chapters 6. In Chapters 7 and eight, the various lifestyles effects are mentioned for numerous different types of differential and quintessential equations, that are bought with assistance from argumentations in keeping with compactness conditions.

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Institut SANU (Beograd) 9(17), 143–234 (2000) 8. : Functional Analysis with Applications. Wiley, New Delhi (1989) 9. : On absolute almost convergence. J. Math. Anal. Appl. 164, 381–398 (1992) 10. : Infinite Matrices and Sequence Spaces. , New York (1955) 11. : Summability Theory and its Applications. Bentham Science Publishers, Istanbul (2011). (e-books, Monographs) 12. : Classical and Modern Methods in Summability. Oxford University Press, New York (2000) 13. : Theory and Application of Infinite Series.

15, p. 108) Let X ∗ α be B K space. Then X f f ∗ cl X (α). Hence, if X has AD, then X ⊂ X f f . Proof First we have to show α ⊂ X f in order for X f f to be meaningful. 16 (i)). 17 (b), we assume that X has AD and have to show X ⊂ X f f . Let x ∞ X be given. We define the functional f : X ≤ → C by f (ψ) = ψ(x) for all ψ ∞ X ≤ . Then we have | f (ψ) |=| ψ(x) |≥→ ψ →→ x →, and consequently f ∞ X ≤≤ . 16 (ii), since X has AK . Thus the inverse map q −1 : X f → X ≤ exists. We define the map g: X f → C by g(b) = ψ(x) (b ∞ X f ) where x = q −1 (b).

Then x (m) = m k=1 | x k | e this norm satisfies condition (i) but not (ii). It is not a monotone norm. (b) The spaces ∞ , c, c0 ,cs,bs have monotone norms. (c) Let A = (C, 1). Then c A has monotone norm. 6]) Let X be BK space with monotone norm T a triangle and B a positive triangle. Then X [B] is a BK space with x B(| x |) for all x ∈ X [B] . Proof For typographical convenience, we shall write . for x . is a norm on X [B] . Further, since X is a BK space, x (m) − x X [B] = B(| x (m) − x |) → 0 (m → ∞) implies Bn (| x (m) − x |) = n bnk | x (m) − x |→ 0 (m → ∞) k=0 for all n.

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