On the global attractivity for a logistic equation with piecewise constant arguments

Abstract

In this paper, we consider the following logistic equation with piecewise constant arguments:where r > 0, a 0 , a 1 , . . . , a m 0, m j =0 a j > 0, and [x] means the maximal integer not greater than x. The sequence {N n } ∞ n=0 , where N n = N(n), n = 0, 1, 2, . . . , satisfies the difference equationUnder the condition that the first term a 0 dominates the other m coefficients a i , 1 i m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N * = 1/( m j =0 a j ).

New global stability conditions for a class of difference equations

“…[13] (and Refs. [2][3][4][5][6][7][10][11][12][13][14][15][16][17] and references therein), one can see that our results provide novel stability conditions to (4.17).…”

“…[11] for 0 < q 1 which is an extension of Lemma 3.1 in Ref. [15] for q = 1 and cf. Lemma 3.3 and Theorem 3.1 in Ref.…”

“…[15], we first present some basic lemmas in order to prove Theorem 4.3. Our proofs of Lemmas 4.1-4.5, which are necessary generalizations of Lemmas 2.1-2.5, 3.1-3.4 and Theorem A in Ref.…”

“…Our proofs of Lemmas 4.1-4.5, which are necessary generalizations of Lemmas 2.1-2.5, 3.1-3.4 and Theorem A in Ref. [15] for q = 1 to 0 < q < 1 are much technical but for the convenience of the reader, we give their proofs. Moreover, Lemmas 5.6-5.8 analyze the conditions appearing in Lemmas 5.2 and 5.3.…”

Global attractivity for discrete models of nonautonomous logistic equations

“…But these results still cannot extend the condition obtained by [5] for the case m = 0 to m ≥ 1 in the autonomous case of (1.1). Using some kind of the monotone iterative method, [6] succeeded in this problem and established sufficient condition (1.7) of global asymptotic stability for the positive equilibrium of autonomous logistic equation with piecewise constant delays.…”