Systems of Wiener–Hopf integral equations, and nonlinear factorization equations

N. B. Engibaryan   ; L. G. Arabadzhyan

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 181-208

Résumé

Systems of Wiener–Hopf integral equations \begin{equation} f(x)=g(x)+\int_0^\infty T(x-t)f(t)\,dt \end{equation} and corresponding nonlinear factorization equations \begin{align} U(x)&=T(x)+\int_0^\infty V(t)U(x+t)\,dt, \nonumber \\ V(x)&=T(-x)+\int_0^\infty V(x+t)U(t)\,dt,\qquad x>0, \end{align} are studied. It is assumed that $T$ is a matrix-valued function with nonnegative components from $L_1(-\infty,\infty)$, with $\mu=r(A)\leqslant1$, where $\displaystyle A=\int_{-\infty}^\infty T(x)\,dx$, and $r(A)$ is the spectral radius of the matrix $A$. The conservative case $\mu=1$, to which major attention is given, falls outside the general theory of Wiener–Hopf integral equations, since the symbol of equation (1) degenerates. A number of results have been obtained about the properties of the solution of the factorization equation (2), and about the existence, asymptotics and other properties of the solution of the homogeneous and nonhomogeneous conservative equation (1). Bibliography: 21 titles.

Bibliographie

[1] Hopf E., Mathematical problems of radiative equilibrium, No 31, Cambridge, 1934

[2] Wiener N., Hopf E., “Über eine Klasse singulärer Integraleichungen”, Sitz. Akad. Wiss., 1931, 696–706 | Zbl

[3] Ambartsumyan V. A., Nauchnye trudy, t. I, Erevan, 1960

[4] Sobolev V. V., Perenos luchistoi energii v atmosferakh zvezd i planet, Gostekhizdat, M., 1956 | MR

[5] Fok V. A., “O nekotorykh integralnykh uravneniyakh matematicheskoi fiziki”, Matem. sb., 14(56) (1944), 3–50

[6] Lindlay D. V., “The theory of queries with a singi sever”, Proc. Camb. Phil. Soc., 48 (1952), 277–289 | DOI

[7] Spitzer F., “Wiener–Hopf equation whose kernel is probability density”, Duke Math. J., 24:3 (1957), 327–343 | DOI | MR | Zbl

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

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

[10] Prësdorf Z., Nekotorye klassy singulyarnykh uravnenii, Mir, M., 1979 | MR

[11] Feller F., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 2, Mir, M., 1967

[12] Engibaryan N. B., Arutyunyan A. A., “Integralnye uravneniya na polupryamoi s raznostnymi yadrami i nelineinye funktsionalnye uravneniya”, Matem. sb., 97(139) (1975), 35–58 | Zbl

[13] Engibaryan N. B., “Faktorizatsiya matrits-funktsii i nelineinye integralnye uravneniya”, Izv. AN Arm. SSR. Seriya matem., 15:3 (1980), 233–244 | MR | Zbl

[14] Arabadzhyan L. G., “O konservativnom uravnenii Vinera–Khopfa”, Izv. AN Arm. SSR, 16:1 (1981), 65–80 | MR

[15] Arabadzhyan L. G., “O sistemakh integralnykh uravnenii vosstanovleniya”, Diff. uravneniya, 20 (1984) | MR | Zbl

[16] Arutyunyan A. A., “O nekotorykh svoistvakh nelineinykh operatorov, svyazannykh s faktorizatsiei operatora Vinera–Khopfa”, Matematicheskii analiz k ego prilozheniya, Erevan, 1980, 37–44

[17] Engibaryan N. B., Arabadzhyan L. G., O nelineinykh uravneniyakh faktorizatsii operatorov Vinera–Khopfa, Preprint NII FKS, 79–1, ErGU, Erevan, 1979

[18] Lankaster P., Teoriya matrits, Nauka, M., 1978 | MR

[19] Bellman R., Kuk K., Differentsialno-raznostnye uravneniya, Mir, M., 1967 | MR | Zbl

[20] Krasnoselskii M. A., Polozhitelnye resheniya operatornykh uravnenii, GIFML, M., 1962 | MR

[21] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR