Equations with discontinuous nonlinear semimonotone operators

Nguyen Buong

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 1, page 7-12
  • ISSN: 0010-2628

Abstract

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The aim of this paper is to present an existence theorem for the operator equation of Hammerstein type x + K F ( 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 ( Ω ) are given for illustration.

How to cite

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Buong, Nguyen. "Equations with discontinuous nonlinear semimonotone operators." Commentationes Mathematicae Universitatis Carolinae 40.1 (1999): 7-12. <http://eudml.org/doc/248385>.

@article{Buong1999,
abstract = {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.},
author = {Buong, Nguyen},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semimonotone operators; uniformly convex Banach spaces; Hammerstein equation; nonlinear operator; semimonotone operator; discontinuous operator},
language = {eng},
number = {1},
pages = {7-12},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Equations with discontinuous nonlinear semimonotone operators},
url = {http://eudml.org/doc/248385},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Buong, Nguyen
TI - Equations with discontinuous nonlinear semimonotone operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 1
SP - 7
EP - 12
AB - 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.
LA - eng
KW - semimonotone operators; uniformly convex Banach spaces; Hammerstein equation; nonlinear operator; semimonotone operator; discontinuous operator
UR - http://eudml.org/doc/248385
ER -

References

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  1. Amann H., Ein Existenz und Eindeutigkeitssatz fur die Hammersteinshe Gleichung in Banachraumen, Math. Z. 111 (1969), 175-190. (1969) MR0254687
  2. Amann H., Uber die naherungsweise Losung nichlinearer Integralgleichungen, Numer. Math. 19 (1972), 29-45. (1972) MR0303772
  3. Brezis H., Browder F., Nonlinear integral equations and systems of Hammerstein's type, Adv. Math. 10 (1975), 115-144. (1975) MR0394341
  4. Brezis H., Sibony M., Méthod d'approximation et d'itération pour les opérateurs monotones, Arch. Rat. Mech. Anal. 28 (1968), 1 59-82. (1968) MR0220110
  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. (1984) 
  6. Buong N., On approximate solutions of Hammerstein equation in Banach spaces (in Russian), Ukrainian Math. J. 8 (1985), 1256-1260. (1985) 
  7. Gaidarov D.P., Raguimkhanov P.K., On integral inclusion of Hammerstein (in Russian), Sibirian Math. J. 21 (1980), 2 19-24. (1980) MR0569173
  8. Ganesh M., Joshi M., Numerical solvability of Hammerstein equations of mixed type, IMA J. Numer. Anal. 11 (1991), 21-31. (1991) MR1089546
  9. Gupta C.P., Nonlinear equations of Urysohn's type in a Banach space, Comment. Math. Univ. Carolinae 16 (1975), 377-386. (1975) Zbl0326.47058MR0500344
  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. (1974) Zbl0291.47034MR0348569
  12. Krasnoselskii A.M., On elliptic equations with discontinuous nonlinearities (in Russian), Dokl. AN RSPP 342 (1995), 6 731-734. (1995) MR1346871
  13. Pavlenko V.N., Nonlinear equations with discontinuous operators in Banach space (in Russian), Ukrainian Math. J. 31 (1979), 5 569-572. (1979) MR0552495
  14. Pavlenko V.N., Existence of solution for nonlinear equations involving discontinuous semimonotone operators (in Russian), Ukrainian Math. J. 33 (1981), 4 547-551. (1981) 
  15. Raguimkhanov, Concerning an existence problem for solution of Hammerstein equation with discontinuous mappings (in Russian), Izvestia Vukov, Math. (1975), 10 62-70. (1975) 
  16. Vaclav D., Monotone Operators and Applications in Control and Network Theory, Elsevier, Amsterdam-Oxford-New York, 1979. Zbl0425.93002MR0554232
  17. Vainberg M.M., Variational Method and Method of Monotone Operators (in Russian), Moscow, Nauka, 1972. MR0467427

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