A degree theory for variational inequalities with sums of maximal monotone and (S$_+$) operators

In-Sook Kim, Martin Väth

DOI: http://dx.doi.org/10.12775/TMNA.2016.022

Abstract


We develop a degree theory for variational inequalities which contain multivalued (S$_+$)-perturbations of maximal monotone operators. The multivalued operators need not necessarily be convex-valued. The result is simultaneously an extension of a degree theory for variational inequalities (developed by Benedetti, Obukhovskii and Zecca) and of the Skrypnik-Browder degree and extensions thereof.

Keywords


Degree theory; maximal monotone operator; operator of type (S$_+$); multivalued map; variational inequality; Browder-Skrypnik degree

Full Text:

PREVIEW FULL TEXT

References


G.M. Asfaw and A.G. Kartsatos, A Browder topological degree theory for multi-valued pseudomonotone perturbations of maximal monotone operators, Adv. Math. Sci. Appl. 22 (2012), 91–148.

I. Benedetti and V.V. Obukhovskiı̆, On the index of solvability for variational inequalities in Banach spaces, Set-Valued Anal. 16 (2008), 67–92.

I. Benedetti and P. Zecca, Relative topological degree and variational inequalities, Mediter. J. Math. 3 (2006), 47–65.

K. Borsuk, Theory of Retracts, Polish Scientific Publ., Warszawa, 1967.

H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123–144.

F.E. Browder, Nonlinear maximal monotone mappings in Banach spaces, Math. Ann. 175 (1968), 81–113.

F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., vol. XVIII, Part 2, Amer. Math. Soc., Providence, R.I., 1976.

F.E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983), 1–39.

M.M. Day, Normed Linear Spaces, 3rd ed., Springer–Verlag, 1973.

R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Longman, 1999.

B.-T. Kien, M.-M. Wong and N.-C. Wong, On the degree theory for general mappings of monotone type, J. Math. Anal. Appl. 340 (2008), 707–720.

I.V. Skrypnik, Nonlinear Elliptic Boundary Value Problems, Teubner–Verlag, 1986.

M. Väth, Topological Analysis. From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions, de Gruyter-Verlag, 2012.

E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Springer–Verlag, 1990.

S.-S. Zhang and Y.-Q. Chen, Degree theory for multivalued (S)-type mappings and fixed point theorems, Appl. Math. Mech. 11 (1990), 441–454.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism