Geodesic
The Wiener--Hopf Integral Equation in the Supercritical Case
Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 11-19.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the scalar homogeneous equation $$ S(x)=\int_0^\infty K(x-t)S(t)\,dt, \qquad x\in\mathbb R^+\equiv(0,\infty), $$ with symmetric kernel $K$: $K(-x)=K(x)$, $x\in\mathbb R_1$ satisfying the conditions $$ 0\leqslant K\in L_1(\mathbb R^+)\cap C^{(2)}(\mathbb R^+), \qquad \int_0^\infty K(t)\,dt>\frac12, $$ $K'\leqslant 0$ and $0\leqslant K''\downarrow$ on $\mathbb R^+$. We prove the existence of a real solution $S$ of the equation given above with asymptotic behavior $S(x)=O(x)$ as $x\to+\infty$. 
@article{MZM_2004_76_1_a1,
     author = {L. G. Arabadzhyan},
     title = {The {Wiener--Hopf} {Integral} {Equation} in the {Supercritical} {Case}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {11--19},
     publisher = {mathdoc},
     volume = {76},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a1/}
}
TY  - JOUR
AU  - L. G. Arabadzhyan
TI  - The Wiener--Hopf Integral Equation in the Supercritical Case
JO  - Matematičeskie zametki
PY  - 2004
SP  - 11
EP  - 19
VL  - 76
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a1/
LA  - ru
ID  - MZM_2004_76_1_a1
ER  - 
%0 Journal Article
%A L. G. Arabadzhyan
%T The Wiener--Hopf Integral Equation in the Supercritical Case
%J Matematičeskie zametki
%D 2004
%P 11-19
%V 76
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a1/
%G ru
%F MZM_2004_76_1_a1
L. G. Arabadzhyan. The Wiener--Hopf Integral Equation in the Supercritical Case. Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 11-19. http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a1/

[1] Wiener N., Hopf E., “Über eine Klasse singulaer Integralgleichungen”, Sitzung Preuss. Akad. Wiss., 1931, 696–706 | Zbl

[2] Krein M. G., “Integralnye uravneniya na polupryamoi s yadrom, zavisyaschim ot raznosti argumentov”, UMN, 13:5 (1958), 3–120 | MR | Zbl

[3] Gokhberg I. Ts., Krein M. G., “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, UMN, 13:2 (1958), 3–72 | MR

[4] Presdorf Z., Nekotorye klassy singulyarnykh uravnenii, Nauka, M., 1979

[5] Krein S. G., Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971

[6] Ambartsumyan V. A., Nauchnye trudy, Erevan, 1960

[7] Sobolev V. V., Kurs teoreticheskoi astrofiziki, Nauka, M., 1967

[8] Borovkov A. A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Nauka, M., 1972

[9] Cherchinyani K., Teoriya i prilozheniya uravneniya Boltsmana, Mir, M., 1978

[10] Engibaryan N. B., Arutyunyan A. A., “Integralnye uravneniya na polupryamoi s raznostnymi yadrami i nelineinye funktsionalnye uravneniya”, Matem. sb., 97:5 (1975), 35–76

[11] Arabadzhyan L. G., Engibaryan N. B., “Uravneniya v svertkakh i nelineinye funktsionalnye uravneniya”, Itogi nauki i tekhniki. Matem. analiz, 22, VINITI, M., 1984, 175–244 | MR

[12] Krein M. G., Shmulyan Yu. L., “Uravneniya Vinera–Khopfa, yadra kotorykh dopuskayut integralnoe predstavlenie cherez eksponenty”, Izv. AN ArmSSR. Ser. matem., 17:4 (1982), 307–327 | MR | Zbl

[13] Engibaryan N. B., Engibaryan B. N., “Integralnye uravneniya svertki na polupryamoi so vpolne monotonnymi yadrami”, Matem. sb., 187:10 (1996), 53–72 | MR | Zbl

[14] Akhiezer N. I., Klassicheskaya problema momentov, GIFML, M., 1961

[15] Grigoryan G. A., “Uravneniya Vinera–Khopfa v zakriticheskom sluchae”, Izv. NAN Armenii. Ser. matem., 32:1 (1997) | MR

[16] Kolmogorov A. L., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981

[17] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, T. 3, Nauka, M., 1973