Asymptotic results for a class of fourth order quasilinear difference equations

Abstract

The authors consider the fourth order quasilinear difference equation D 2 p n jD 2 x n j a21 D 2 x n þ q n jx n j b21 x n ¼ 0;where a and b are positive constants and {p n } and {q n } are positive real sequences. They classify the nonoscillatory solutions according to their asymptotic behavior for large n and then give necessary and sufficient conditions for existence of solutions of these various types. The results are illustrated with examples.

Kneser solutions of fourth-order trinomial delay differential equations

“…For a general interest in this field, we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references cited therein. For a general interest in this field, we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references cited therein.…”

“…Even though qualitative properties of solutions of binomial differential equation related to (E), namely, y (4) + q(t)y(τ (t)) = 0 and its various generalizations containing quasi-derivatives have been widely investigated in the literature (see, for example, [1,3,2]), much less is known about asymptotic behavior of (E). So far, prototypes of higher-order trinomial differential equations primarily studied in the literature are such that a difference in the derivative order between the first and the middle term is either one or two.…”

Oscillatory behavior of second order unstable type neutral difference equations

“…In most of the papers [1,2,6,7,10,11], the authors established conditions for the oscillation and nonoscillation of solutions of equation of type (1) with α = 1 and treating the deviations are constant. In [1,3,5,8,9,12], the authors consider the particular cases of equation 1in the form ∆(a n (∆y n ) α ) − g n f (y α(n) )) = 0…”

Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

“…which, in view of (9), yields (16) which obviously contradicts the fact that D 2 u(n) is decreasing.…”

“…In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the last several years, there has been a constant interest in getting sufficient conditions for oscillatory behavior of different classes of fourth-order difference equations with or without deviating arguments, (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], and the references cited therein). In particular, the authors in [5,8,13,15,18,19] established oscillation results for (1) under the following assumptions:…”