Periodic solutions for coupled first order nonlinear differential systems of Hamiltonian type. (English) Zbl 0562.34030

The author shows there are periodic solutions of periodic systems \(- x'+f(t,x,y)=p(t)\), \(y'+g(t,x,y)=q(t)\) under various sign and growth conditions. These results follow from analysis of nonlinear equations in Banach spaces. Abstract theorems use techniques of alternative methods and Leray-Schauder-Mawhin continuation: the latter involves a priori estimates derived from hypothesized operator estimates. It seems possible to alter the results to allow the removal of the assumption that p, q have mean value zero. In the proofs found in the first section, the author implicitly assumes that (\(\bullet\),\(\bullet)\) is positive definite on \(X_ 2\); this causes no difficulty in the second section. It seems to the reviewer that for Corollary 2.4 to imply Corollary 2.5 one must assume \(\int^{T}_{0}\Gamma (t)dt<-T+1/4\) in the latter. The paper is written well and almost completely self-contained.
Reviewer: L.Turyn


34C25 Periodic solutions to ordinary differential equations
47J05 Equations involving nonlinear operators (general)
55M20 Fixed points and coincidences in algebraic topology
34B15 Nonlinear boundary value problems for ordinary differential equations
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