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\(T\)-periodic solutions for a second order system with singular nonlinearity. (English) Zbl 0824.34040

Lazer and Solimini proved the existence of \(T\)-periodic solutions to the system \(x''= {1\over x^ \nu}- h(t)\), \(\nu\geq 1\), \(h\) locally integrable and \(T\)-periodic, if and only if \(\int^ T_ 0 h> 0\). Mawhin generalized these results.
Here the authors study a system of the form \(d(u''+ au')= H_ v(u, v)- h(t)\), \(d(v''+ bv')= H_ u(u, v)- k(t)\), where \(d\geq 1\) and \(h\), \(k\) are locally integrable and \(T\)-periodic, \(H\) is a \(C^ 1\) function on \((0, \infty)\times (0, \infty)\). To prove either uniqueness or multiplicity of solutions the potential \(H\) is restricted to the form: \(H(u, v)= F(v)+ G(u)\). With some assumptions made on \(F\) and \(G\) the authors show either uniqueness or existence of at least \((2n+ 1)\) \(T\)-periodic solutions. Finding \(T\)-periodic solutions is equivalent to finding fixed points of a certain compact operator \(T_ 1\). This is accomplished by deforming homotopically the operator family \(T_ \tau\) as \(\tau\to 0\). The invariance of the Schauder-Leray degree under compact homotopies is used to prove the existence of fixed points. This technique was first introduced by J. Mawhin.
Reviewer: V.Komkov (Roswell)

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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