# 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)
Periodic solution of certain nonlinear differential equations: via topological degree theory and matrix spectral theory. (English) Zbl 1258.34114
Summary: The main purpose of this article is to establish the existence and stability of a periodic solution of nonlinear differential equation connected with a problem from mathematical biology. The existence and stability conditions are given in terms of spectral radius of explicit matrices, which are better than conditions obtained by using classic norms. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory and Lyapunov functional. It should be noted that the new problem appears due to the introduction of Gilpin-Ayala effect. The standard methods used in the previous literature cannot be used to analyze the asymptotic stability of such systems. To handle this problem, two novel techniques should be employed. One is to rescale the system by ${w}_{i}=\frac{{y}_{i}}{{d}_{i}}$, not ${w}_{i}=\frac{{y}_{i}}{d}$. The other is to analyze the maximal eigenvalue of a matrix. Finally, some examples and their simulations show the feasibility of our results.
##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 34C25 Periodic solutions of ODE 47N20 Applications of operator theory to differential and integral equations 34D20 Stability of ODE 92D25 Population dynamics (general) 34D23 Global stability of ODE