Periodic solutions for generalized high-order neutral differential equation in the critical case

On high-order generalized neutral singular differential equations with time-dependent deviating argument

“…The existence of periodic solutions for equation (1.1) was obtained in the general case (i.e., |c| = 1 in [12]) and in the critical case (i.e. |c| = 1 in [13]), respectively. In 2015, Cheng and Ren [4] investigated a kind of fourth-order generalized neutral Liénard equation (φ p (x(t) − c(t)x(t − τ(t))) ) + f (x(t))x (t) + g(t, x(t), x(t − σ (t)), x (t)) = p(t), (1.2) where c ∈ C(R, R) is a T -periodic function for some T > 0 and c(t) = 1.…”

“…The periodic solution problem for neutral differential equations attracted much attention during the past years, see [1]- [13]. In 2009, Ren and Cheng [13] discussed the following high-order neutral differential equation (φ p (x(t) − cx(t − τ)) (l) ) (n−l) = F(t, x(t), x (t), • • • , x (l−1) (t)), (1.1) where p ≥ 2, φ p (x) = |x| p−2 x for x = 0 and φ p (0) = 0. The existence of periodic solutions for equation (1.1) was obtained in the general case (i.e., |c| = 1 in [12]) and in the critical case (i.e.…”

Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities

“…In recent years, the study of periodic solutions for neutral differential equations has attracted the attention of many researchers; see [ 2 – 9 , 14 , 16 – 18 , 20 , 21 ] and the references cited therein. For related books, we refer the reader to [ 1 , 12 ].…”

“…For related books, we refer the reader to [ 1 , 12 ]. Most work concentrated on the neutral operator (see [ 6 , 7 , 14 , 16 , 21 ]) or the neutral operator with variable parameter (see [ 3 , 8 ]) or the neutral operator with variable delay (see [ 4 , 5 ]). However, the study of a neutral operator with linear autonomous difference operator is relatively rare.…”

Existence and Lyapunov Stability of Periodic Solutions for Generalized Higher-Order Neutral Differential Equations

“…In recent years, there is a good amount of work on periodic solutions for neutral differential equations see [1][2][3][4][5][6][7][8][9][10][11] and by using topological degree theory and some analysis skills, existence results of periodic solutions for 1.2 have been presented. In general, most of the existing results are concentrated on lower-order neutral functional differential equations, while studies on higher-order neutral functional differential equations are rather infrequent, especially on higher-order p-Laplacian neutral functional differential equations.…”