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On resonant differential equations with unbounded non-linearities. (English) Zbl 1025.34032
The authors study asymptotically linear degenerate problems with sublinear unbounded nonlinearities. More precisely, let \(x\) be in some Banach space and consider the equation \(x=T(x)\) where \(T(x)=Ax+F(x)\), \(A\) is linear and \(F\) is sublinear, i.e., \(\|F(x)\|\|x\|^{-1}\to 0\) at infinity. If the main linear part \(x=Ax\) has no nontrivial solutions then various problems related to this equation are rather simple. The degenerate case where 1 is an eigenvalue of \(A\) is more complicated. Also, the unboundedness of \(F\) complicates essentially then analysis of many problems.
The authors prove several deep results on the uniform convergence to zero of projections of nonlinearity increments onto finite-dimensional spaces in the case of degenerate \(A\) and unbounded \(F\). These results are used to prove existence theorems, notably for forced periodic oscillations in differential equations, nontrivial cycles for higher-order autonomous ODEs and Hopf bifurcations at infinity.
Reviewer: W.Govaerts (Gent)

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
45M15 Periodic solutions of integral equations
47H11 Degree theory for nonlinear operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Full Text: DOI
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