A Modern Theory of Integration

Bartle, Robert G.

doi:10.1090/gsm/032

Cited by 163 publications

(160 citation statements)

References 0 publications

Supporting

Mentioning

151

Contrasting

Unclassified

Order By: Relevance

“…In Section 4 we present our main results: the existence of a positive solution to BVP (1.1), (1.2) (Theorem 4.1) and to BVP (1.3), (1.2) (Corollary 4.2). In limiting processes we use the Vitali's convergence theorem (see, for example, [3], [6]) since it is impossible to find a Lebesgue integrable majorant function for the sequence {f n (t, g −1 (x n (t)), x n (t))} which is necessary for applying the Lebesgue dominated convergence theorem. We include also two examples (Examples 4.3 and 4.4) to illustrate our theory.…”

mentioning

confidence: 99%

Positive Solutions of Nonlocal Singular Boundary Value Problems

2004

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 99%

Positive Solutions of Nonlocal Singular Boundary Value Problems

2004

View full text Add to dashboard Cite

show abstract

“…Proof is based on the Arzelà-Ascoli theorem and the Vitali's convergence theorem, see, e.g. [5], [6], [11]. Section 4 present an example to illustrate our main result.…”

Section: Definition 11mentioning

confidence: 96%

Existence of Solutions for Nonlocal Boundary Value Problem with Singularity in Phase Variables

Tian¹,

2006

Journal of Applied Analysis

View full text Add to dashboard Cite

show abstract

“…The book of Lee and Výborný serves well as an introduction and reference for anyone interested in this topic. Other good sources are Gordon [7], which covers much of the same material from a somewhat different perspective, and the forthcoming book of Bartle [3].…”

Section: Dt Does Not Exist As a Lebesgue Integral)mentioning

confidence: 99%

The Integral: An Easy Approach after Kurzweil and Henstock

Alewine¹,

Schechter²,

Yee³

et al. 2001

The American Mathematical Monthly

View full text Add to dashboard Cite

95 softcover.Reviewed by J. Alan Alewine and Eric Schechter.A simple definition. Riemann's integral of 1867 can be summarized asThis summary conceals some of the complexity-for example, the limit is of a net, not a sequence-but it displays what we wish to emphasize: The integral is formed by combining the values f (τ i ) in a very direct fashion. The values of f are used less directly in Lebesgue's integral (1902), which can be described as lim n→∞ b a g n (t)dt. The approximating functions g n must be chosen carefully, using deep, abstract notions of measure theory. Simpler definitions are possible-for example, functional analysts might consider the metric completion of C[0, 1] using the L 1 norm-but such a definition does not give us easy access to the Lebesgue integral's simple and powerful properties such as the Monotone Convergence Theorem. We generally think in terms of those simple properties, rather than the various complicated definitions, when we actually use the Lebesgue integral.The KH integral (also known as the gauge integral, the generalized Riemann integral, etc.) was discovered or invented independently by Kurzweil and Henstock in the 1950's; it has attracted growing interest in recent years. It offers the best of both worlds: a powerful integral with a simple definition. In fact, its definition is nearly identical to that of the Riemann integral, as we now show. For any tagged partition

show abstract

A Modern Theory of Integration

Cited by 163 publications

References 0 publications

Positive Solutions of Nonlocal Singular Boundary Value Problems

Positive Solutions of Nonlocal Singular Boundary Value Problems

Existence of Solutions for Nonlocal Boundary Value Problem with Singularity in Phase Variables

The Integral: An Easy Approach after Kurzweil and Henstock

Contact Info

Product

Resources

About