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Global continua of periodic solutions to some difference-differential equations of neutral type. (English) Zbl 0778.34054
The author shows some theorems which are used to study the existence of periodic solutions of neutral equations. Among other things, he presents extensions to neutral equations of the following results: a) local Hopf bifurcation theorem of Krasnosel’skij type [M. A. Krasnosel’skij, Topological methods in the Theory of Nonlinear Integral Equations, Oxford (1964; Zbl 0111.303)]; b) global bifurcation theorems of J. C. Alexander and J. A. Yorke [Am. J. Math. 100, 263-292 (1978; Zbl 0386.34040)] or P. H. Rabinowitz [J. Funct. Anal. 7, 487-513 (1971; Zbl 0212.165)]; c) a well known result for retarded equations by S.-N. Chow and J. Mallet-Paret [J. Differ. Equations 29, 66-88 (1978; Zbl 0369.34020)]. A specific example arising from the lossless transmission lines is studied. Moreover, a good information concerning the applications of neutral equations is given. The paper finishes stating some unanswered interesting questions.
MSC:
34K99Functional-differential equations
34K40Neutral functional-differential equations
34C25Periodic solutions of ODE
34C23Bifurcation (ODE)