Funkcialaj Ekvacioj, 37 (1994) 415-446 Hopf Bifurcation for Parametrized Equivariant Coincidence Problems and Parabolic Equations with Delays By W. KRAWCEWICZ*, T. SPANILY and J. WU* (University of Alberta, Canada, Gdansk University, Poland and York University, Canada) §1. Introduction In this paper, we consider the following parametrized equivariant coincidence problem (1.1) $L_{ lambda}( mathrm{x})=F( lambda, x)$ , $( lambda, x) in theta$ , where $ mathrm{E}$ and $ mathrm{F}$ are real Banach spaces which are also isometric representations of the group $S^{1}= {z in C;|z|=1 }$ , $ {L_{ lambda} }_{ lambda in R^{2}}$ is a continuous family of equivariant Fredholm operators of index zero from $ mathrm{E}$ to $ mathrm{F}$ , and $F$ is a completely continuous equivariant mapping from the locally trivial $ mathrm{S}^{1}$ -vector bundle $ ovalbox{ tt small REJECT}:= {( lambda, x) in R^{2} times mathrm{E}; mathrm{x} in mathrm{E}_{ lambda} }$ to $ mathrm{F}$ , where $ mathrm{E}_{ lambda}$ is the space Dom $(L_{ lambda})$ equipped with the graph norm. We further assume that there exists a 2-dimensional submanifold $M subseteq R^{2} times mathrm{E}^{S^{1}}$ where $ mathrm{E}^{S^{1}}= {x in mathrm{E};gx$ $= mathrm{x}$ for all $g in S^{1} }$ , such that (i) for every $( lambda, x) in M$ , $L_{ lambda}x=F( lambda, x)$ ; (ii) if $( lambda_{0}, x_{0}) in M$ then there exist open neighbourhoods $U_{ lambda_{0}}$ of $ lambda_{0}$ in $R^{2}$ and $U_{x_{0}}$ of $x_{0}$ in $ mathrm{E}^{S^{1}}$ and a $C^{1}-$ map $ eta:U_{ lambda_{ mathrm{O}}} rightarrow mathrm{E}^{S^{1}}$ such that $ M cap(U_{ lambda_{ mathrm{o}}} times$ $U_{x mathrm{o}})= {( lambda, eta( lambda)); lambda in U_{ lambda_{0}} }$ . Under this assumption, all points $( lambda, mathrm{x}) in M$ are solutions of (1.1) (called trivial solutions). One of the main purposes of this paper is to develop a Hopf bifurcation theory which provides very sharp information about the maximum continuation of nontrivial solutions (solutions which are not in $M$ ) of (1.1). Our approach to the bifurcation problem of (1.1) is to employ an equivariant resolvent $K$ of $L= {L_{ lambda} }_{ lambda in R^{2}}$ to reduce the problem to a corresponding problem for a certain completely continuous equivariant vector field $ Theta_{K}(F):R^{2} times mathrm{F} rightarrow mathrm{F}$ , and then to appeal to the method of Gpba and Marzantowicz [21] based on the notion of the $S^{1}-$ equivariant degree of [11] as well as the complementing function method of Ize (cf. [25], [26]). * Research supported by NSERC-Canada