Let \(M\) be a smooth compact connected and simply-connected
manifold with simply-connected boundary \(\partial M\), \(r\) be a
fixed odd natural number. We consider \(f\), a smooth self-maps of
\(M\), preserving \(\partial M\). Under the assumption that the
dimension of \(M\) is at least \(4\), we define the invariant
\(D_r(f;M,\partial M)\) that is equal to the minimal number of
\(r\)-periodic points for all maps preserving \(\partial M\) and
smoothly homotopic to \(f\). We estimate the value of
\(D_r(f;M,\partial M)\) for some values of \(r\).
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