Abstract: We consider Hammerstein-Voltera type nonlinear system of integral equations in critical case. Above mentioned equations have applications in rediative transfer theory and kinetic theory of gases. Using special iteration methods and method of monotone operators theory we prove the exitstence of by component positive solutions in space of bounded and summerable functions with zero limit at infinity. Some examples of corresponding equations representing separate interest are also given.
Keywords: nonlinear system of integral equations, Hammerstein-Voltera type operator, iteration, monotonisity, primitive matrix, summerable solution
1. Ambartsumyan V. A. Nauchnie trudi [Sceientific works], vol. 1,
Yerevan, National Academy of Sciences, 1960, 431 p.
2. Engibaryan N. B. On a problem in nonlinear radiative transfer.
Astrophysics, 1966, vol. 2, Issue 1, pp. 12-14.
3. Arabadzhyan L.G. An integral equation of transport theory in an
inhomogeneous medium. Differentsialnye Uravneniya, 1987, vol. 23,
no. 9, pp. 1618-1622 [in Russian].
4. Arabadzhyan L. G., Engibaryan N. B. Convolution equations and
nonlinear functional equations, J. of Soviet Math., 1987, vol. 36,
Issue 6, pp. 745-791.
5. Arabajian L. G. Solvability of a Volterra type integral equation
on the half-line, J. Contemp. Math. Anal., 1999, 34 , no. 2, 75-79
(2000); transl. from Izv. Nats. Akad. Nauk Armenii Math., 1999, 34,
no. 2, pp. 80-83 (2000) [in Russian].
6. Zarebina M. A numerical Solution of Nonlinear Volterra-Fredholm
Integral Equations, J. of Appl. Anal. and Comp., 2013, vol. 3, no.
1, pp. 95-104.
7. Lauran M. Existence results for some Nonlinear Integral
Equations, Miskole Math. Notes, 2012, vol. 13, no. 1, pp. 67-74.
8. Karapetyants N. K., Kilbas A. A., Saigo M. On the Solutions of
Nonlinear Volterra Convolution Equation with power Nonlinearity. J.
of Integral Equations and Appl., 1996, vol. 8, no. 4, pp. 429-445.
9. Khachatryan Kh. A., Grigoryan S. A. On nontrivial solvability of
a nonlinear Hammerstein-Volterra type integral equation,
Vladikavkaz. Math. Zh., 2012, vol. 14, no. 2, pp. 57-66 [in
Russian].
10. Lancaster P. Theory of Matrices, Academic Press, NY, 1969, 316
p.
11. Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I.,
Sbolevskii P. E. Integral operators in spaces of summable functions
[Noordhoff International Publishing], Leyden, 520, 1976.
12. Kolmogorov, A. N. and Fomin, V. S., Elementy teorii funktsii i
funktsional’nogo analiza [Elements of the Theory of Functions and
Functional Analysis], Moscow: Nauka, 1981, 544 p.
13. Khachatryan Kh. A. Some classes of Urysohn's nonlinear integral
equations on the semi-axis, Doklady of the National Academy of
Sciences of Belarus, 2011, vol. 55, no. 1, pp. 5-9.
