On the study of a class of variational inequalities via Leray-Schauder degree
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We present existence results for general variational inequalities without monotonicity or coercivity assumptions. It relies on a Leray-Schauder degree approach and provides additional information about the location of solutions. 1. Introduction The study of variational inequalities is very important from a theoretic point of view in mathematics as well as for its various and significant applications in different fields, for instance, in what is called nonsmooth mechanics [1, 3, 10]. Comprehensive treatment of different problems related to variational inequalities and their applications can be found in the monographs [2, 5, 6, 7, 8]. A basic assumption in the results studying the variational inequalities on a Hilbert space is the monotonicity condition, in particular, the ellipticity (or coercivity) hypothesis on the (possibly nonlinear) operator entering the problem. The interest to relax this condition, by imposing other type of assumptions, is a real challenge in the recent developments. The present paper is devoted to this topic, where in place of monotonicity there are supposed suitable assumptions allowing the application of topological degree arguments. Our approach permits to encompass the solvability of cases that were not covered by the previous known results. We describe the functional setting of the paper. Let H be a real Hilbert space endowed with the scalar product ·, · and the associated norm · . Consider the following general assumptions on the data in our variational inequality formulation (see problem (1.3)): (H1) Φ : H → H is a compact mapping, that is, Φ is continuous and maps the bounded sets onto relatively compact sets; (H2) ϕ : H → R is a convex and continuous function which is bounded from above on the bounded subsets of H. Since a convex and lower semicontinuous function on H is bounded from below by an affine function, it is bounded from below on the bounded subsets of H. Hypothesis (H2) ensures thus that the function ϕ is bounded on the bounded subsets of H. We stress that Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:4 (2004) 261–271 2000 Mathematics Subject Classification: 49J40, 35K85, 47H11 URL: http://dx.doi.org/10.1155/S1687182004407074
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Variational inequalities via Leray-Schauder degree
the property of the function ϕ : H → R to be bounded from above on the bounded subsets of H as assumed in (H2) is not satisfied, in general, by a convex and continuous function ϕ on H. We provide an example in this direction based on private communication with J. Saint Raymond (2004). Example 1.1. Consider the Hilbert space 2 and the function f : 2 → R defined by
f (x) = sup 2nxn − n n≥0
∀x ∈ 2 ,
(1.1)
where xn are the components of x. The function f is convex, continuous, and not bounded on the bounded sets. Indeed, f is defined on 2 because for any x ∈ 2 the set
n : 2nxn − n ≥ 0 = n : xn ≥
1 2
(1.2)
is finite. The function f is convex, since it is the upper hull of the convex functions fn o
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