Generalized variational inequalities and fixed point theorems. (English) Zbl 0912.49006

In this paper, the authors deduce, from a well-known fixed point theorem of Idzik, new variational inequalities for generalized upper hemicontinuous multimaps. Consequently, all of the results of W. K. Kim and K.-K. Tan [Bull. Aust. Math. Soc. 46, No. 1, 139-148 (1992; Zbl 0747.47037)] and X. Ding [J. Suchuan Norm. Univ. 17, No. 6, 10-16 (1994; MR 96b:47064)] and some others are substantially extended and improved. Moreover, they establish new fixed point theorems on generalized upper hemicontinuous multimaps including a large number of historically well-known extensions of the Brouwer or Kakutani theorems. The fixed point theorems established by the authors are useful tools in nonlinear analysis; they have many interesting applications.


49J40 Variational inequalities
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
58E35 Variational inequalities (global problems) in infinite-dimensional spaces


Zbl 0747.47037
Full Text: DOI


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