Variational inequalities with the duality operator in Banach spaces

Yıl 2020, Cilt: 3 Sayı: 2, 78 - 84, 30.06.2020

In-sook Kim

Öz

We study variational inequality by way of metric projection in Banach spaces. The main method is to use a topological degree theory
for the class of operators of monotone type in Banach spaces. More precisely, some variational inequality associated with the duality operator
is considered. As applications, the problem is discussed in the Lebesgue spaces $L^p$ and the Sobolev spaces $W^{1,2}$.

Anahtar Kelimeler

variational inequality, operators of monotone type, degree theory

Kaynakça

• [1] Y.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lect. Notes Pure Appl. Math., vol. 178, Marcel Dekker, New York, 1996, pp. 15–50.
• [2] J. Berkovits, On the degree theory for mappings of monotone type, Ann. Acad. Sci. Fenn. Ser. A1 Diss. 58 (1986) 1–58.
• [3] J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differ. Equ. 234 (2007) 289–310.
• [4] J. Berkovits, V. Mustonen, On the topological degree for mappings of monotone type, Nonlinear Anal. 10 (1986) 1373–1383.
• [5] F.E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983) 1–39.
• [6] F.E. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Natl. Acad. Sci. USA 80 (1983) 1771–1773.
• [7] F.E. Browder, B.A. Ton, Nonlinear functional equations in Banach spaces and elliptic super-regularization, Math. Z. 105 (1968) 177–195.
• [8] I.-S. Kim, S.-J. Hong, A topological degree for operators of generalized (S + ) type, Fixed Point Theory Appl. 2015 (2015), 16 pp.
• [9] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, 1990.