On a bound for the spectral radius of a linear operator in the space of continuous functions (Q1390430)

For a linear bounded isotonic operator \(A\) on \(C[ab]\) the spectral radius \(\rho(A)\) is less than 1, iff for some \(v\in C[ab]\) and all \(t\in [a,b]\), \(v(t)>0\) and \(v(t)- Av(t)> 0\). This lemma is proved and generalized to make it easier to apply, for example, to the de la Vallée Poussin problem \[ \ddot x (t)+ q(t)\dot x(t)+ p(t) x(t)- Tx(t)= f(t),\quad x(a)= x(b)= 0, \] where \(T\) is a linear bounded isotonic operator with values in \(L^1[a,b]\).