On a fuzzy logistic difference equation
The paper deals with the difference equation \[ u_{k+1}-u_k= a(b- cu_k)u_{k+1} \] with fuzzy coefficients fulfilling the conditions: \[ \text{A}1)\quad a<1/b; \qquad \text{A}2)\quad u_k<b/c; \qquad \text{A}3)\quad u_k,\;a,\;b\text{ and \(c\) are positive}. \] The problem is reduced to the system of interval recurrences. Unfortunately the final result (Lemma 4.6) is false because of the following mistakes: 1) Wrong interpretation of the inequality A2) for fuzzy numbers. 2) The next term of the recurrence may not belong to its domain. (E.g. let \(a=[1/4,1/3]\), \(b=[2.5,3]\), \(c=[0.5,1.5]\), \(b/c=[5/3,6]\). If \(u_k=[L_k,R_k]= [0.5,1.5]\), then \(u_k<5/3\leq b/c\), \(u_{k+1}= [8/15,18]\) and \(u_{k+1}\) does not fulfil the condition A2). 3) Bad approximation of the final recurrence which gives a system of two simultaneous recurrences of the form \(L_{k+1}= f(L_k,R_k)\), \(R_{k+1}= g(L_k,R_k)\).
scientific article; zbMATH DE number 990868