Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
semimonotone operators; uniformly convex Banach spaces
semimonotone operators; uniformly convex Banach spaces
Summary:
The aim of this paper is to present an existence theorem for the operator equation of Hammerstein type $x+KF(x)=0$ with the discontinuous semimonotone operator $F$. Then the result is used to prove the existence of solution of the equations of Urysohn type. Some examples in the theory of nonlinear equations in $L_p(\Omega )$ are given for illustration.
References:
[1] Amann H.: Ein Existenz und Eindeutigkeitssatz fur die Hammersteinshe Gleichung in Banachraumen. Math. Z. 111 (1969), 175-190. MR 0254687
[2] Amann H.: Uber die naherungsweise Losung nichlinearer Integralgleichungen. Numer. Math. 19 (1972), 29-45. MR 0303772
[3] Brezis H., Browder F.: Nonlinear integral equations and systems of Hammerstein's type. Adv. Math. 10 (1975), 115-144. MR 0394341
[4] Brezis H., Sibony M.: Méthod d'approximation et d'itération pour les opérateurs monotones. Arch. Rat. Mech. Anal. 28 (1968), 1 59-82. MR 0220110
[5] Buong N.: On solution of the operator equation of Hammerstein type with semimonotone and discontinuous nonlinearity (in Vietnamese). J. Math. 12 (1984), 25-28.
[6] Buong N.: On approximate solutions of Hammerstein equation in Banach spaces (in Russian). Ukrainian Math. J. 8 (1985), 1256-1260.
[7] Gaidarov D.P., Raguimkhanov P.K.: On integral inclusion of Hammerstein (in Russian). Sibirian Math. J. 21 (1980), 2 19-24. MR 0569173
[8] Ganesh M., Joshi M.: Numerical solvability of Hammerstein equations of mixed type. IMA J. Numer. Anal. 11 (1991), 21-31. MR 1089546
[9] Gupta C.P.: Nonlinear equations of Urysohn's type in a Banach space. Comment. Math. Univ. Carolinae 16 (1975), 377-386. MR 0500344 | Zbl 0326.47058
[10] Emellianov C.V.: Systems of Automatic Control with Variable Structure (in Russian). Moscow, Nauka, 1967.
[11] Joshi M.: Existence theorem for a generalized Hammerstein type equation. Comment. Math. Univ. Carolinae 15 (1974), 283-291. MR 0348569 | Zbl 0291.47034
[12] Krasnoselskii A.M.: On elliptic equations with discontinuous nonlinearities (in Russian). Dokl. AN RSPP 342 (1995), 6 731-734. MR 1346871
[13] Pavlenko V.N.: Nonlinear equations with discontinuous operators in Banach space (in Russian). Ukrainian Math. J. 31 (1979), 5 569-572. MR 0552495
[14] Pavlenko V.N.: Existence of solution for nonlinear equations involving discontinuous semimonotone operators (in Russian). Ukrainian Math. J. 33 (1981), 4 547-551.
[15] Raguimkhanov: Concerning an existence problem for solution of Hammerstein equation with discontinuous mappings (in Russian). Izvestia Vukov, Math. (1975), 10 62-70.
[16] Vaclav D.: Monotone Operators and Applications in Control and Network Theory. Elsevier, Amsterdam-Oxford-New York, 1979. MR 0554232 | Zbl 0425.93002
[17] Vainberg M.M.: Variational Method and Method of Monotone Operators (in Russian). Moscow, Nauka, 1972. MR 0467427
Search
Partner of
