Fixed point theorems for discontinuous mapping

Cromme, Ludwig J.; Diener, Immo

doi:10.1007/bf01586937

Cited by 29 publications

(16 citation statements)

References 7 publications

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“…Note that Cromme and Diener (1991) considered case dim X < and already proved (4.3). Corresponding to T , they obtained ∀ > 0 ∃x * ∈ M x * − Tx * ≤ T + In Phu and Truong (2000), we showed the existence of better invariant points, namely: If X is an n-dimensional normed space or the n-dimensional Euclidean space, then for all > 0 there exists x * ∈ M such that x * − Tx * ≤ n n + 1 T + or x * − Tx * ≤ n 2 n + 1 T + respectively.…”

Section: Ordermentioning

confidence: 67%

“…Downloaded by [Umeå University Library] Cromme (1997) and Cromme and Diener (1991) used two other measures of discontinuity, namely Downloaded by [Umeå University Library] Cromme (1997) and Cromme and Diener (1991) used two other measures of discontinuity, namely…”

Section: Discontinuous Mappingsmentioning

confidence: 99%

“…By applying Carathéodory's theorem (1911), Cromme and Diener (1991) showed the upper semi-continuity of T * in finite-dimensional space X. By applying Carathéodory's theorem (1911), Cromme and Diener (1991) showed the upper semi-continuity of T * in finite-dimensional space X.…”

Section: Discontinuous Mappingsmentioning

confidence: 99%

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Approximate Fixed-Point Theorems for Discontinuous Mappings

Phú

2004

Numerical Functional Analysis and Optimization

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For a given r ≥ 0, a mapping T M → M on some convex subset M of a normed linear space X is said to be around r-continuous if for all x ∈ M and > 0 there exists > 0 such that Ty − Tz < r + holds whenever y z ∈ M, y − x < , and z − x < . If does not depend on x then T is called uniformly r-continuous. By using the self-Jung constant J s X ∈ 1 2 , we state some theorems on approximate fixed points of such mappings. For instance, if M is compact and T is around r-continuous then, for all > 0, there exists x * ∈ M satisfying x * − Tx * ≤ 1 2 J s X r + , where can be replaced by zero under some additional assumptions. This property remains true if M is only relatively compact but T is uniformly r-continuous, or if the relative compactness of M is replaced by the relative compactness of T M .

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Section: Ordermentioning

confidence: 67%

Section: Discontinuous Mappingsmentioning

confidence: 99%

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Approximate Fixed-Point Theorems for Discontinuous Mappings

Phú

2004

Numerical Functional Analysis and Optimization

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“…Here, function

q_{n}^{*}

a discontinuous step function. By using the fixed point theorem for discontinuous mappings (Cromme and Diener, 1991), there exists a point

{\overset{β}{true}}^{*} \in B

, such that…”

Section: Full Likelihood-based Estimation Proceduresmentioning

confidence: 99%

Score Estimating Equations from Embedded Likelihood Functions Under Accelerated Failure Time Model

Ning¹,

Qin²,

Shen³

2014

Journal of the American Statistical Association

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SUMMARY The semiparametric accelerated failure time (AFT) model is one of the most popular models for analyzing time-to-event outcomes. One appealing feature of the AFT model is that the observed failure time data can be transformed to identically independent distributed random variables without covariate effects. We describe a class of estimating equations based on the score functions for the transformed data, which are derived from the full likelihood function under commonly used semiparametric models such as the proportional hazards or proportional odds model. The methods of estimating regression parameters under the AFT model can be applied to traditional right-censored survival data as well as more complex time-to-event data subject to length-biased sampling. We establish the asymptotic properties and evaluate the small sample performance of the proposed estimators. We illustrate the proposed methods through applications in two examples.

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“…Several generalizations to multivalued maps, or correspondences, were also made by others, especially by Smithson (, ) . Other extensions were discussed by Klee (), and then by Cromme and Diener (), Bula () and Cromme (). These authors dealt with almost continuous mappings, and they have proved that approximate continuous mappings admit approximate fixed points.…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

Fujimoto

2013

Metroeconomica

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This paper introduces economists to some fixed point theorems for discontinuous mappings with non‐convex images on a non‐convex domain. These theorems have recently been developed based on a new approach by mathematical economists and mathematicians. The new method of proof is first transformed into a sort of metatheorem, which is then used to obtain a set of necessary and sufficient conditions for a map to have a fixed point. Some fixed point theorems for discontinuous maps are then explained in more concrete cases. The formulations are intended for easier applications towards economic models involving discontinuity as well as non‐convexity.

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Fixed point theorems for discontinuous mapping

Cited by 29 publications

References 7 publications

Approximate Fixed-Point Theorems for Discontinuous Mappings

Approximate Fixed-Point Theorems for Discontinuous Mappings

Score Estimating Equations from Embedded Likelihood Functions Under Accelerated Failure Time Model

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

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