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

Wojciech Mydlarczyk

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 1995-2002
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Wojciech Mydlarczyk. "Uniqueness of solutions to an Abel type nonlinear integral equation on the half line." Open Mathematics 10.6 (2012): 1995-2002. <http://eudml.org/doc/269188>.

@article{WojciechMydlarczyk2012,
abstract = {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.},
author = {Wojciech Mydlarczyk},
journal = {Open Mathematics},
keywords = {Nonlinear Volterra integral equations; Integral inequalities; Nontrivial solutions; Uniqueness of solution; Generalized Osgood condition; nonlinear Volterra integral equations; integral inequalities; nontrivial solutions; uniqueness of solution; generalized Osgood condition},
language = {eng},
number = {6},
pages = {1995-2002},
title = {Uniqueness of solutions to an Abel type nonlinear integral equation on the half line},
url = {http://eudml.org/doc/269188},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Wojciech Mydlarczyk
TI - Uniqueness of solutions to an Abel type nonlinear integral equation on the half line
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 1995
EP - 2002
AB - 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.
LA - eng
KW - Nonlinear Volterra integral equations; Integral inequalities; Nontrivial solutions; Uniqueness of solution; Generalized Osgood condition; nonlinear Volterra integral equations; integral inequalities; nontrivial solutions; uniqueness of solution; generalized Osgood condition
UR - http://eudml.org/doc/269188
ER -

References

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  7. [7] Mydlarczyk W., A condition for finite blow-up time for a Volterra integral equation, J. Math. Anal. Appl., 1994, 181(1), 248–253 http://dx.doi.org/10.1006/jmaa.1994.1018[Crossref] 
  8. [8] Mydlarczyk W., The existence problem for a nonlinear Abel equation on the half-line, Nonlinear Anal., 2010, 73(7), 2022–2026 http://dx.doi.org/10.1016/j.na.2010.05.031[Crossref] Zbl1204.45007
  9. [9] Mydlarczyk W., Okrasinski W., Positive solutions to a nonlinear Abel type integral equation on the whole line, Comput. Math. Appl., 2001, 41(7-8), 835–842 http://dx.doi.org/10.1016/S0898-1221(00)00323-0[Crossref] Zbl0986.45002
  10. [10] Nussbaum R.D., Hilbert’s Projective Metric and Iterated Nonlinear Maps, Mem. Amer. Math. Soc., 75(391), American Mathematical Society, Providence, 1988 Zbl0666.47028
  11. [11] Olmstead W.E., Ignition of a combustible half space, SIAM J. Appl. Math., 1983, 43(1), 1–15 http://dx.doi.org/10.1137/0143001[Crossref] Zbl0546.76131
  12. [12] Roberts C.A., Lasseigne D.G., Olmstead W.E., Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 1993, 5(4), 531–546 http://dx.doi.org/10.1216/jiea/1181075776[Crossref] Zbl0804.45002

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