Generalized g-Fractional Vector Representation Formula And Bochner Integral Type Inequalities for Banach Space Valued Functions

Abstract Using partitions of the unity ((PU)-partition), a new definition of an integral is given for a function f : [a, b] → X, where X is a Banach space, and it is proved that this integral is equivalent to the Bochner integral.

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Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

Abstract and Applied Analysis ◽

10.1155/2012/352159 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 2

Author(s):

Yuji Liu

Keyword(s):

Banach Space ◽

Differential Equations ◽

Positive Solutions ◽

Sufficient Conditions ◽

Second Order ◽

Integral Type ◽

One Dimensional ◽

Second Order Differential Equations ◽

Type Boundary ◽

The One

This paper is concerned with the integral type boundary value problems of the second order differential equations with one-dimensionalp-Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensionalp-Laplacian term[ρ(t)Φ(x’(t))]’involved with the functionρ, which makes the solutions un-concave.

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Expectation in metric spaces and characterizations of Banach spaces

Demonstratio Mathematica ◽

10.1515/dema-2013-0205 ◽

2009 ◽

Vol 42 (4) ◽

Author(s):

Artur Bator ◽

Wiesław Zięba

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Metric Space ◽

Metric Spaces ◽

Bochner Integral ◽

Random Elements

AbstractWe consider different definitions of expectation of random elements taking values in metric spaces. All such definitions are valid also in Banach spaces and in this case the results coincide with the Bochner integral. There may exist an isometry between considered metric space and some Banach space and in this case one can use the Bochner integral instead of expectation in metric space. We give some conditions which ensure existence of such isometry, for two different definitions of expectation in metric space.

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On Measurability for Vector-Valued Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1963-062-4 ◽

1963 ◽

Vol 15 ◽

pp. 613-621 ◽

Cited By ~ 1

Author(s):

D. O. Snow

Keyword(s):

Banach Space ◽

Infinite Series ◽

Unconditional Convergence ◽

Integration Theory ◽

Bochner Integral ◽

Norm Topology ◽

Vector Valued Functions ◽

Vector Valued

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

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