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Applications of global bifurcation to existence theorems for Sturm-Liouville problems

100%
Gulgowski J.
Annales Polonici Mathematici
|
2004
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tom 83
|
nr 3
221-229
EN
We prove an existence theorem for Sturm–Liouville problems ⎧u''(t) + φ(t,u(t),u'(t)) = 0 for a.e. t ∈ (a,b), (∗) ⎨ ⎩l(u) = 0, where $φ: [a,b] × ℝ^{k} × ℝ^{k} → ℝ^[k}$ is a Carathéodory map. We assume that φ(t,x,y) = m₁φ₀(t,x,y) + o(|x|+|y|) as |x|+|y| → 0 and φ(t,x,y) = m₂φ₀(t,x,y) + o(|x|+|y|) as |x|+|y| → ∞, where m₁,m₂ are positive constants and φ₀ belongs to a class of nonlinear maps. The proof bases on global bifurcation results. We define a map $f: (0,∞) × C¹([a,b],ℝ^{k}) → C¹([a,b],ℝ^{k}) such that if f(1,u) = 0, then u is a solution of (∗). Then we show that there exists a connected set C of nontrivial zeroes of f such that there exist (λ₁,u₁),(λ₂,u₂)∈C with λ₁ < 1 < λ₂. In the last section we give examples of maps φ₀ leading to specific existence theorems.
2
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Approximation of solutions to second order nonlinear Picard problems with Carathéodory right-hand side

100%
Gulgowski J.
Open Mathematics
|
2014
|
tom 12
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nr 1
155-166
EN
We present an approximation method for Picard second order boundary value problems with Carathéodory righthand side. The method is based on the idea of replacing a measurable function in the right-hand side of the problem with its Kantorovich polynomial. We will show that this approximation scheme recovers essential solutions to the original BVP. We also consider the corresponding finite dimensional problem. We suggest a suitable mapping of solutions to finite dimensional problems to piecewise constant functions so that the later approximate a solution to the original BVP. That is why the presented idea may be used in numerical computations.
3
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Numerical detection of bifurcation point in the curve

100%
Gulgowski J.
Mathematica Applicanda
|
2015
|
tom 43
|
nr 1
EN
We are presenting a numerical method which detects the presence and position of a  bifurcation simplex, the regular $(k+1)$-dimensional simplex, which may be considered as "fat bifurcation point", in the curve of zeroes of the $C^1$ map $f:{\mathbb R}^{k+1}\to{\mathbb R}^k$. On the other hand the bifurcation simplex appears in the neighbourhood of the bifurcation point, meaning that we have the method to locate the bifurcation point as well. The method does not require any estimation of the derivative of the function $f$ and refers to the values of the map $f$ only in the vertices of certain triangulation. The bifurcation simplex is detected by change of the Brouwer degree value of the restriction of the map $f$ to the appropriate $k$-simplex.This publication is co-financed by the European Union as part of the European Social Fund within the project Center for Applications of Mathematics.
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