Abstract
Lx = (F + G)(x), (1.1) where L is a linear Fredholm map of index zero, defined on a subspace domL of a real normed space X with values in Z, and where F and G are two maps from X to Z such that their sum F + G is L-completely continuous (for the terminology see [l]). The proof will be based on the extended version of the Leray-Schauder continuation principle established by J. Mawhin (see [l]), that is, roughly speaking, on the comparison of equation (1.1) with an uniquely solvable linear equation of the form Lx = Ax, via a linear homotopy. As application of our abstract theorem, we shall study the problem of existence of p-periodic solutions of a vector ordinary differential equation of Rayleigh type:
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