The existence of optimal parameters of the generalized logistic function (Q1918318)

The authors consider the generalized logistic function \[ f(t)= {A\over (1+ b\exp\{- c\gamma t\})^{1/\gamma}},\quad A, b, c, \gamma> 0. \] Let us have available the data \((p_i, t_i, f_i)\), \(i= 1,\dots, m\), where \(p_i\) are some positive weights, \(t_1<\cdots< t_m\) and \(f_i> 0\). The authors try to respond to the following question: Does there exist a pair of numbers \((b^*, c^*)\in {\mathcal B}\) such that \[ F(b^*, c^*)= \inf_{(b, c)\in {\mathcal B}} F(b, c), \] where \({\mathcal B}= \{(b, c)\in \mathbb{R}^2\mid b> 0, c> 0\}\) and \[ F(b, c)= {1\over 2} \sum^m_{i= 1} p_i\Biggl[f(i)- {A\over (1+ b\exp \{- c\gamma t\})^{1/\gamma}}{\Biggr]^2}. \] The above problem does not always have a solution. The authors show the existence of the minimum of the functional \(F\) under conditions which are considerably weaker than those given earlier for the same problem.