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An abstract existence theorem at resonance and its applications. (English) Zbl 0940.34056

The authors consider the operator equation in the form: \( (1) \quad Lx =Nx \) , where \( L \) is a Fredholm mapping of index zero and \( N \) is \(L\)-completely continuous. By using Brouwer degree theory and a continuation theorem based on Mawhin’s coincidence degree theory there is developed an abstract existence theorem at resonance for the equation (1). As application of this result sufficient conditions are proved for the existence of \( 2 {\pi} \)-periodic solutions to semilinear equations at resonance, where the kernel of the linear part has dimension \( \geq 2\). Finally, some ilustration examples on the theory are given.

MSC:

34K13 Periodic solutions to functional-differential equations
34K30 Functional-differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
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