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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0104-9,
author = {Wojciech Mydlarczyk},
title = {Uniqueness of solutions to an Abel type nonlinear integral equation on the half line},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1995-2002},
zbl = {1263.45003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0104-9}
}
Wojciech Mydlarczyk. Uniqueness of solutions to an Abel type nonlinear integral equation on the half line. Open Mathematics, Tome 10 (2012) pp. 1995-2002. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0104-9/
[1] Bushell P.J., On a class of Volterra and Fredholm non-linear integral equations, Math. Proc. Cambridge Philos. Soc., 1976, 79(2), 329–335 http://dx.doi.org/10.1017/S0305004100052324[Crossref] | Zbl 0316.45003
[2] Bushell P.J., Okrasinski W., Uniqueness of solutions for a class of nonlinear Volterra integral equations with convolution kernel, Math. Proc. Cambridge Philos. Soc., 1989, 106(3), 547–552 http://dx.doi.org/10.1017/S0305004100068262[Crossref] | Zbl 0689.45013
[3] Gripenberg G., Unique solutions of some Volterra integral equations, Math. Scand., 1981, 48(1), 59–67 | Zbl 0463.45002
[4] Gripenberg G., Londen S.-O., Staffans O., Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990 http://dx.doi.org/10.1017/CBO9780511662805[Crossref] | Zbl 0695.45002
[5] Lasseigne D.G., Olmstead W.E., Ignition or nonignition of a combustible solid with marginal heating, Quart. Appl. Math., 1991, 49(2), 309–312 | Zbl 0731.76039
[6] Mydlarczyk W., The existence of nontrivial solutions of Volterra equations, Math. Scand., 1991, 68(1), 83–88 | Zbl 0701.45002
[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] 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] | Zbl 1204.45007
[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] | Zbl 0986.45002
[10] Nussbaum R.D., Hilbert’s Projective Metric and Iterated Nonlinear Maps, Mem. Amer. Math. Soc., 75(391), American Mathematical Society, Providence, 1988 | Zbl 0666.47028
[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] | Zbl 0546.76131
[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] | Zbl 0804.45002