Bounded and almost automorphic solutions of a Liénard equation with a singular nonlinearity

Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

“…1. Corollary 7.3 (item (iii)) improves and generalizes some of the results from [4,12,14,19] when the function f is strictly increasing with respect to the second variable.…”

“…If we suppose the opposite, then there are ε 0 > 0, δ n → 0 (δ n > 0) x n ∈ X and t n → +∞ such that ρ(x n , M) < δ n and ρ π(t n , x n ), M ε 0 . (14) Let m n ∈ M be a point such that ρ(x n , m n ) = ρ(x n , M), and denote by y n := h(x n ). Since the set M is compact, taking into account (14), we can assume that the sequences {x n }, {y n } and {m n } are convergent.…”

“…(4) with almost automorphic forcing term p was studied by Cieutat et al in [14], where they proved the asymptotically almost automorphy [9,Ch. I] of every solution which is bounded on R + , and the existence of a unique almost automorphic solution of Eq.…”

Almost periodic and asymptotically almost periodic solutions of Liénard equations

“…for almost periodic F ) these results were generalized by P. Cieutat in [8]. The almost automorphic and asymptotically almost automorphic solutions of equation (1) were studied by P. Cieutat et al [9], while the existence of pseudo almost periodic solutions of equation (1) was analyzed by El Hadi Ait Dads et al [9].…”

“…(iii) the equation ( 9) admits a solution u 0 which is bounded on R + . Then, (i) the equation ( 9) is convergent, i.e., the non-autonomous dynamical system (cocycle) generated by ( 9) is convergent; (ii) if the point y 0 ∈ Y is a τ -periodic (quasi periodic, almost periodic in the sense of Bohr, almost automorphic, recurrent, pseudo recurrent) point, then equation (9) has a unique τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent) solution u such that M y0 ⊆ M u ; (iii) every solution of equation (9), which is bounded on R + , is asymptotically τ -periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent).…”