Topological identification of multiple solutions to parametrized nonlinear equations

Robert F. Brown

Vol. 131 (1988), No. 1, 51–69

DOI: 10.2140/pjm.1988.131.51

Abstract

Let L : E → F be an isomorphism of Banach spaces, let H : E × Rⁿ → F be a completely continuous mapping, and let B : E → Rⁿ be a bounded linear mapping onto a euclidean space. The solutions (y,λ) to the problem

can be represented as the fixed points of a mapping T : E × Rⁿ → E × Rⁿ. Neilsen fixed point theory may be extended to produce lower bounds for the number of fixed points of such maps. Problems of the type (*) include boundary value problems for ordinary differential systems of the form:

where y = y(x) : [0,1] → Rⁿ and λ ∈ Rⁿ, satisfying an additional condition such as y(1∕2) = 0 or ∫ ₀¹y(t)dt = A for a given A ∈ Rⁿ.

Mathematical Subject Classification 2000

Primary: 47H15

Secondary: 34B15, 34G20, 47H10, 55M20

Milestones

Received: 20 February 1986

Published: 1 January 1988