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1
Content available remote

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

100%
Kong L., Jiang D.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 3
265-270
EN
The fourth order periodic boundary value problem $u^{(4)} - m⁴u + F(t,u) = 0$, 0 < t < 2π, with $u^{(i)}(0) = u^{(i)}(2π)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $±10^{-7}$.
2
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Chaos in some planar nonautonomous polynomial differential equation

100%
Wójcik K.
Annales Polonici Mathematici
|
2000
|
tom 73
|
nr 2
159-168
EN
We show that under some assumptions on the function f the system $ż = z̅(f(z) e^{iϕt} + e^{i2ϕt})$ generates chaotic dynamics for sufficiently small parameter ϕ. We use the topological method based on the Lefschetz fixed point theorem and the Ważewski retract theorem.
3
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An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$

100%
Wójcik K.
Annales Polonici Mathematici
|
1996-1997
|
tom 65
|
nr 3
203-211
EN
We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n+1}$ for which $∂E = ℝ^n × {0}$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in E\∂E such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^n × [0,∞)$.
4
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Minimal surface maps, fixed point indices and a Jiang-Guo type inequality

100%
Kelly M. R.
Banach Center Publications
|
1999
|
tom 49
|
nr 1
227-233
5
Content available remote

The coincidence index for fundamentally contractible multivalued maps with nonconvex values

88%
Gabor D.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 2
143-166
EN
We study a coincidence problem of the form A(x) ∈ ϕ (x), where A is a linear Fredholm operator with nonnegative index between Banach spaces and ϕ is a multivalued A-fundamentally contractible map (in particular, it is not necessarily compact). The main tool is a coincidence index, which becomes the well known Leray-Schauder fixed point index when A=id and ϕ is a compact singlevalued map. An application to boundary value problems for differential equations in Banach spaces is given.
6
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Partially dissipative periodic processes

75%
Andres J., Górniewicz L., Lewicka M.
Banach Center Publications
|
1996
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tom 35
|
nr 1
109-118
EN
Further extension of the Levinson transformation theory is performed for partially dissipative periodic processes via the fixed point index. Thus, for example, the periodic problem for differential inclusions can be treated by means of the multivalued Poincaré translation operator. In a certain case, the well-known Ważewski principle can also be generalized in this way, because no transversality is required on the boundary.
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