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Response solutions are quasi-periodic ones with the same frequency as theforcing term. The present work is devoted to the construction of responsesolutions for $d$-dimensional beam equations with nonlocal nonlinear damping,which model frictional mechanisms affecting the bodies based on the average. Byconsidering $\epsilon$ in a domain that does not include the origin andimposing a small quasi-periodic forcing with Liouvillean frequency vector,which is weaker than the Diophantine or Brjuno one, we can show the existenceof the response solution for such a model. We present an alternative approachto the contraction mapping principle (cf. [5,33]) through a combination ofreduction and the Nash--Moser iteration technique. The reason behind thisapproach lies in the derivative losses caused by the nonlocal nonlinearity.

Response solutions for beam equations with nonlocal nonlinear damping and Liouvillean frequencies