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DISCONTINUOUS GENERALIZED QUASI-VARIATIONAL INEQUALITIES WITH APPLICATION TO FIXED POINTS

Paolo Cubiotti and Jen-Chih Yao
Taiwanese Journal of Mathematics
Vol. 15, No. 5 (October 2011), pp. 2059-2080
Stable URL: http://www.jstor.org/stable/taiwjmath.15.5.2059
Page Count: 22
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DISCONTINUOUS GENERALIZED QUASI-VARIATIONAL INEQUALITIES WITH APPLICATION TO FIXED POINTS
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Abstract

Abstract We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed space X with topological dual X*, two sets C,D ⊆ X and two multifunctions S : C → 2D and T : C → 2X*, find (x^, φ^)∈C×X* such that x^∈S(x^), φ^∈T(x^) and 〈φ^,x^−y〉≤0 for all y∈S(x^). We prove an existence theorem where T is not assumed to have any continuity or monotonicity property, improving some aspects of the main result of [7]. In particular, the coercivity assumption is meaningfully weakened. As an application, we prove a theorem of the alternative for the fixed points of a Hausdorff lower semicontinuous multifunction. In particular, we obtain sufficient conditions for the existence of a fixed point which belongs to the relative boundary of the corresponding value. 2010 Mathematics Subject Classification: 90C29, 49J40. Key words and phrases: Generalized quasi-variational inequalities, Affine hull, Lower semicontinuity, Hausdorff lower semicontinuity, Fixed points, Relative interior, Relative boundary.

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