*(English)*Zbl 0801.34069

The main goal of the paper is to establish an analog of the Alexander- York’s global bifurcation theory for the neutral functional differential equation (1) $\frac{d}{dt}(x\left(t\right)-b({x}_{t},\alpha ))=F({x}_{t},\alpha )$, $t\in \mathbb{R}$, where $\alpha $ is a parameter. For this sake, the authors apply the notion of the ${S}^{1}$-composite coincidence degree to the ${S}^{1}$-equivariant nonlinear problem (2) $L\left(\pi \right(x)-B(x\left)\right)=N\left(x\right)$, where $\pi $ is the natural projection, $L$ is an equivariant Fredholm operator, $B$, $N$ are equivariant mappings, in order to obtain local and global bifurcation theorems for the problem (2) which are characterized as belonging to Krasnosel’skij or Rabinovitz type correspondingly.

With the help of these two theorems, local and global bifurcation theorems for the problem (1) are proved. As an example of the application of these results, the following difference differential equation of neutral type

where $a,b,r>0$, $q\in (0,1)$, $g\left(x\right)\in {C}^{1}\left(\mathbb{R}\right)$, is nondecreasing, $g\left(o\right)={g}^{\text{'}}\left(o\right)=0$, $xg\left(x\right)>0$ for $x\ne 0$, $\frac{g\left(x\right)}{x}\to \infty $ as $x\to \pm \infty $, is considered. Under some conditions, the authors establish the number of periodic solutions and evaluate there periods.

##### MSC:

34K99 | Functional-differential equations |

34C23 | Bifurcation (ODE) |

34K40 | Neutral functional-differential equations |

47J05 | Equations involving nonlinear operators (general) |