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Article

Keywords:
fixed point; Hilbert space; weak topology; discontinouous operator
Summary:
We construct a class of discontinuous operators in infinite-dimensional separable Hilbert spaces, answering a natural question which arises in comparing a fixed point theorem of Altman and Shinbrot ([1], [4]) with its improvement obtained by Ricceri ([2], [3]).
References:
[1] Altman M.: A fixed point theorem in Hilbert space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5 (1957), 19-22. MR 0087064 | Zbl 0077.31902
[2] Ricceri B.: Un théorème d'existence pour les inéquations variationnelles. C.R. Acad. Sci. Paris, Série I 301 (1985), 885-888. MR 0823146 | Zbl 0606.49006
[3] Ricceri B.: Existence theorems for nonlinear problems. Rend. Accad. Naz. Sci. XL 11 (1987), 77-99. MR 0930860 | Zbl 0643.47058
[4] Shinbrot M.: A fixed point theorem, and some applications. Arch. Rational Mech. Anal. 17 (1964), 255-271. MR 0169068 | Zbl 0156.38502
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