CONSTRUCTION OF A TOPOLOGICAL DEGREE THEORY IN GENERALIZED SOBOLEV SPACES

Year 2018, Volume: 1 Issue: 2, 116 - 129, 31.07.2018

Mustapha Ait Hammou Elhoussine Azroul

Abstract

In this paper, we construct an integer-valued degree function in a suitable classes of mappings of monotone type, using a complementary system formed of Generalized Sobolev Spaces in which the variable exponent p in P(log)(Omega) satisfy 1 < p'-  < p'+ < + ifinity, where  Omega is in RN is open and bounded.
This kind of spaces are not refexives

Keywords

Primary Topological degree, Generalized Sobolev spaces, Secondary Complementary system

References

• Berkovits, J.: On the degree theory for nonlinear mappings of monotone type. -Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 58,1986.
• Berkovits, J., and V. Mustonen: On topological degree for mappings of monotone type. Nonlinear Anal. 10,1986,1373-1383.
• Berkovits, J., and V. Mustonen: Nonlinear mappings of monotone type I. Classification and degree theory. Preprint No 2/88, Mathematics, University of Oulu.
• Brouwer, L. E. J: Uber Abbildung von Mannigfaltigkeiten. - Math. Ann. 71, 1912 ,97-115.
• F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983) 139.
• Browder, F E: Degree of mapping for nonlinear mappings of monotone type. Proc. Natl. Acad. Sci. USA 80, 1771-1773 (1983).
• Deimling, K: Nonlinar functional analysis. Springer, Berlin (1985).
• L. Dingien, P. Harjulehto, P. Hasto, M. Ruzicka: Lebesgue and Sobolev Spaces with Variable Exponents, Springer (2011).
• L. Fuhrer, Ein elementarer analytischer Beweis zur Eindeutigkeit des A bbildungsgrades im Rn, Math. Nachr. 54 (1972), 259-267.
• J. P. Gossez; Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Am. Math. Soc. 190 (1974), 163-205.
• O. Kovacik and J. Rakosnik: On spaces Lp(x) and W1;p(x), Czechoslovak Math. J. 41 (1991), 592-618.
• Leray, J, Schauder, J: Topologie et equationes fonctionnelles. Ann. Sci. Ec. Norm. Super. 51, 45-78 (1934).
• Landes, R., and V. Mustonen: Pseudo-monotne mappings in Orlicz-Sobolev spaces and nonlinear boundary value problems on unbounded domains. J. Math. Anal. 88,1982,25-36.
• Narici, L., and E. Beckenstein: Topological vector spaces. -Marcel Dekker, Inc., New York and Basel, 1985.
• Skrypnik, I V. : Nonlinear higher order elliptic equations. Naukova Dumka, Kiev (1973)(in Russian).
• Skrypnik,IV: Methods for analysis of nonlinear elliptic bondary value problems. Amer. Math. Soc. Transl., Ser. II, vol. 139. AMS, Providence(1994).
• H. Amann and S. Weiss, On the uniqueness of the topological degree, Math. Z. 130 (1973), 39-5.
• Zeidler, E: Nonlinear functional analysis and its applications I: Fixed-Point-Theorems.Springer, New York (1985).