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1
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Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

100%
Mydlarczyk W.
Open Mathematics
|
2012
|
tom 10
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nr 6
1995-2002
EN
We consider a convolution-type integral equation u = k ⋆ g(u) on the half line (−∞; a), a ∈ ℝ, with kernel k(x) = x α−1, 0 < α, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if α ∈ (1, 4), then for any two nontrivial solutions u 1, u 2 there exists a constant c ∈ ℝ such that u 2(x) = u 1(x +c), −∞ < x. The results are obtained by applying Hilbert projective metrics.
2
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A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$

100%
Mydlarczyk W.
Annales Polonici Mathematici
|
1998
|
tom 68
|
nr 2
177-189
EN
We consider the problem of the existence of positive solutions u to the problem $u^{(n)}(x) = g(u(x))$, $u(0) = u'(0) = ... = u^{(n-1)}(0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $∫₀^δ 1/s [s/g(s)]^{1/n} ds < ∞$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.
3
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A singular initial value problem for second and third order differential equations

100%
Mydlarczyk W.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
249-257
4
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An initial value problem fora third order differential equation

100%
Mydlarczyk W.
Annales Polonici Mathematici
|
1994
|
tom 59
|
nr 3
215-223
EN
For an initial value problem u'''(x) = g(u(x)), u(0) = u'(0) = u''(0) = 0, x > 0, some theorems on existence and uniqueness of solutions are established.
5
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The blow-up solutions of integral equations

100%
Mydlarczyk W.
Colloquium Mathematicum
|
1999
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tom 79
|
nr 1
147-156
6
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The existence of solutions to a Volterra integral equation

100%
Mydlarczyk W.
Annales Polonici Mathematici
|
1996
|
tom 64
|
nr 2
175-182
EN
We study the equation u = k∗g(u) with k such that ln k is convex or concave and g is monotonic. Some necessary and sufficient conditions for the existence of nontrivial continuous solutions u of this equation are given.
7
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Galerkin methods for multidimensional nonlinear Volterra type equations

50%
Mydlarczyk W.
Applicationes Mathematicae
|
1991
|
tom 21
|
nr 1
113-125
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