DISCONTINUOUS GENERALIZED QUASI-VARIATIONAL INEQUALITIES WITH APPLICATION TO FIXED POINTS

• Published in: Taiwanese journal of mathematics Vol. 15; no. 5; pp. 2059 - 2080
• Main Authors: Cubiotti, Paolo, Yao, Jen-Chih
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.10.2011
• Subjects: Banach space; Coercivity; Convexity; Existence theorems; Function discontinuity; Mathematical discontinuity; Mathematical inequalities; Topological spaces; Topological theorems; Variational inequalities
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/twjm/1500406423

We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed spaceXwith topological dualX*, two setsC,D⊆Xand two multifunctionsS:C→2D andT : 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 whereTis 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. 2010Mathematics 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.