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Global asymptotic stability in a class of Putnam-type equations. (English) Zbl 1125.39014
The main result in this paper is a study of a sufficient condition for global asymptotic stability of the equilibrium point $$c=1$$ for the class of a Putnam difference equation [cf. G. Ladas, J. Difference Equ. Appl. 4, No. 5, 497–499 (1998; Zbl 0925.39004)] of the form
$x_{n+1}=\frac{A_1x_n+A_2x_{n-1} + A_3x_{n-2}x_{n-3}+ A_4}{B_1x_nx_{n-1}+ B_2x_{n-2}+B_3x_{n-3}+B_4}, \qquad n=0,1,2,\dots,$
where $$A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4$$ are positive constants that satisfy the following conditions (c1) $$A_1+A_2+A_3+A_4=B_1+B_2+B_3+B_4$$; (c2) $$A_1+A_2 > B_1$$; (c3) $$A_3< B_1+ B_3< A_3+A_4$$; (c4) $$A_3, B_3$$; $$x_{-3},x_{-2},x_{-1},X_0$$ are positive numbers.

MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
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References:
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