An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type (Q1948421)

In this paper, the bifurcation equation \( P(x,\varepsilon) + \varepsilon Q(x, \varepsilon)= 0 \) is studied under the assumption that \( P \) is twice differentiable with respect to \( x \) continuously in \( (x,\varepsilon) \) and \( Q \) is differentiable with respect to \( x \) continuously in \( (x,\varepsilon) \). For this equation, the authors formulate and prove general bifurcation results by means of the classical Implicit Function Theorem. These bifurcation results are applied to functional differential equations of neutral type in order to prove the existence of periodic solutions. This is obtained by converting the problem of finding periodic solutions into the problem of finding fixed points of a suitably introduced map. The results are based on methods from the theory of condensing operators.