The asymptotic behavior of the positive solutions for a class of difference equations (Q2748800)

The stability property of the positive solutions of the initial value problem NEWLINE\[NEWLINEx_{n+1}=x^\alpha_n \exp\left \{\gamma_n{1-x_{n-k_n} \over 1-\lambda x_{n-k_n}} \right\},\;n=0,1,2, \dots,\tag{1}NEWLINE\]NEWLINE is investigated. It is assumed \(\alpha\in (0,1]\), \(\lambda\in(0,1)\), \(\{\gamma_n\}\) is a nonnegative sequence and \(\{k_n\}\) is a positive integer sequence such that \(\lim_{n\to \infty} (n-k_n)= +\infty\).NEWLINENEWLINENEWLINEIt is shown that if \(\sum^\infty_{n=0} \gamma_ n=+ \infty\), if \(\{k_n\}\) is bounded above by an integer \(k\) and if NEWLINE\[NEWLINE\limsup_{n \to \infty} \sum^n_{i=n-k_n} \alpha^{n-k} \gamma_i <d_k,NEWLINE\]NEWLINE where \(d_k\) is the unique positive root of the equation NEWLINE\[NEWLINE\exp\left\{{ (k+1)x^2\over 2(k+2)(1-\lambda)^2} \right\}= {1+x\over \lambda+x},NEWLINE\]NEWLINE then all positive solutions of (1) converge to 1.NEWLINENEWLINENEWLINEIt is also shown that if \(\sum^\infty_{n=0} \gamma_n=+ \infty\), if \(\{k_n\}\) is unbounded and if NEWLINE\[NEWLINE\limsup_{n\to \infty} \sum^n_{i=n-k_n} \alpha^{n -i} \gamma_i<D,NEWLINE\]NEWLINE where \(D\) is the unique positive root of the equation NEWLINE\[NEWLINE\exp \left \{{x^2\over 2(1-\lambda)^2} \right\}= {1+x \over\lambda+ x},NEWLINE\]NEWLINE then every positive solution of (1) tends to 1.