Vol. 44, No. 2, 1973

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A bifurcation theorem for k-set contractions

James William Thomas

Vol. 44 (1973), No. 2, 749–756

One of the the most often used results of bifurcation theory is the following theorem. Let C be a compact mapping from the Banach space X into itself. Suppose that C is such that C(𝜃) = 𝜃 and the derivative of C exists at x = 𝜃. Then each characteristic value, μ0, of odd multiplicity of C(𝜃) is a bifurcation point of C, and to this bifurcation point there corresponds a continuous branch of eigenvectors of C. The main result in this paper will show that the above theorem can be extended to a class of non-compact mapping of the form I f where f is a k-set contraction.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 29 October 1971
Published: 1 February 1973
James William Thomas