On some fixed point principles for cones in linear normed spaces

Fournier, Gilles; Peitgen, Heinz‐Otto

doi:10.1007/bf01425238

Cited by 10 publications

(5 citation statements)

References 15 publications

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“…Krasnosel'skiȋ's idea of compressions and expansions of cones gave rise to some generalizations in asymptotic fixed point theory: The first of these results is due to Nussbaum [61], other contributions [33,68] were motivated by the concept of ejectivity. For applications, cf.…”

Section: Historymentioning

confidence: 99%

70 Years of asymptotic fixed point theory

Steinlein

2015

J. Fixed Point Theory Appl.

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In this expository article we aim to illustrate asymptotic fixed point theory from the viewpoints of its history, the main motivation from applications, the two basic techniques for proofs and a survey on the major results. The bibliography should cover most aspects but is, for sure, not complete. We confine ourselves to results from topological fixed point theory, i.e., metric fixed point theory is not at all touched.

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Section: Historymentioning

confidence: 99%

70 Years of asymptotic fixed point theory

Steinlein

2015

J. Fixed Point Theory Appl.

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“…Our results although similar to those of ( [11]), are independent of these last results. However, they will generalize Theorems 2.5 and 2.6 of [6].…”

Section: Applicationsmentioning

confidence: 89%

“…Since P\B r is contractible (cf. [6]), P\B r is acyclic. Furthermore g(P\B r ) C/(5V) which is relatively compact.…”

Section: Applicationsmentioning

confidence: 98%

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Fixed Point Principles for Cones of a Linear Normed Space

Fournier

1980

Can. j. math.

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In [8] and [9], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [11], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to condensing maps and he applied this theorem to prove the existence of periodic solutions of nonlinear autonomous functional differential equations.Nussbaum's proof makes an essential use of the difficult Zabreiko and Krasnosel'skiĭ, and Steinlein (mod p)-theorem for the fixed point index [13 -16]. In [6], Fournier and Peitgen proved two different versions of this theorem for completely continuous maps each one being sufficient for Nussbaum's applications. The proofs of these two theorems are much less involved and, although they are different, they make use of the same easier generalized Lefschetz number calculations (see [12] for (mod p) and [5] for compact attractor).

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“…Remark 2.22. If E is a normed linear space, C ⊆ E a cone and U 0 an open ball in E, then [5] guarantees that (2.4) is true.…”

Section: Definition 22 We Saymentioning

confidence: 99%

Continuation theorems for weakly inward Kakutani and Strongly inward acyclic maps

O'Regan¹

2022

J. Nonlinear Sci. Appl.

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In this paper we begin by presenting a general Leray-Schauder alternative and a topological transversality theorem for Kakutani (upper semicontinuous maps with nonempty convex compact values) compact weakly inward maps. Then with some observations and extra assumptions we present a Leray-Schauder alternative and a topological transversality theorem for acyclic (upper semicontinuous maps with nonempty acyclic compact values) compact strongly inward maps.

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On some fixed point principles for cones in linear normed spaces

Cited by 10 publications

References 15 publications

70 Years of asymptotic fixed point theory

70 Years of asymptotic fixed point theory

Fixed Point Principles for Cones of a Linear Normed Space

Continuation theorems for weakly inward Kakutani and Strongly inward acyclic maps

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