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Quasidifferentiability and positive solutions of equations for set-valued mappings. (English) Zbl 0867.47037

From the text: “In this paper, the quasidifferentiability and the quasiderivative of set-valued mappings are introduced and applied to the study of problems concerning the positive solution of equations and the positive (relative) eigenvector and eigenvalue for the set-valued \(k\)-set contraction with \(k\) in \([0,1)\) in a \((B)\)-space with a cone. And improvements and developments of the results on the positive fixed point theorems for the Fréchet differentiable or the semidifferentiable (single-valued) completely continuous mappings or strict set contraction for Petryshyn, Amann, Edmunds et al. and the author are obtained in this paper”.

MSC:

47H04 Set-valued operators
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
46G05 Derivatives of functions in infinite-dimensional spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J50 Fréchet and Gateaux differentiability in optimization
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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References:

[1] Petryshyn, W. V., J. math. Analysis Applic., 133, 297-305 (1988)
[2] Amann, H., SIAM Rev., 18, 620-709 (1976)
[3] Amann, H., J. fund. Analysis, 14, 162-171 (1973)
[4] Edmunds, D. E.; Potter, A. J.; Stuart, C. A., (Proc. R. Soc. London Sr.A., 328 (1972)), 67-81
[5] Yang, Sh. L., Chin. Ann. Math., 7A, 705-708 (1986)
[6] Yang, Sh. L., J. sys. Sci. math. Sci., 7, 329-334 (1987)
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