# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems. (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) ${d\over dt} (x(t)- b(x\sb t,\alpha))= F(x\sb t,\alpha)$, $t\in \bbfR$, where $\alpha$ is a parameter. For this sake, the authors apply the notion of the $S\sp 1$-composite coincidence degree to the $S\sp 1$-equivariant nonlinear problem (2) $L(\pi(x)- B(x))= N(x)$, 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 $${d\over dt} (x(t)- qx(t- r))= -ax(t)- bqx(t- r)- g(x(t))+ qg(x(t- r)),\tag3$$ where $a,b,r>0$, $q\in (0,1)$, $g(x)\in C\sp 1(\bbfR)$, is nondecreasing, $g(o)= g'(o)= 0$, $xg(x)>0$ for $x\ne 0$, ${g(x)\over x}\to\infty$ 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)