Extension of some results concerning the generalized Liénard equation

Abstract

I~OLF ~]~ISSlC-(Bochum) Dedicated to Professor C~IOVAN:NI SAI'~SONE Oll his 85th birthday Summary. -A recent result of JTfawhin [7] concerning the existence of forced oscillations for a second order equation of JLidnard type with an arbitrary damping term is extended to some eases where the restoring force is not assumed to be suf/iciently weak. The results are valid, too,/or a certain class of third order equations. They are based on the 15eray-Sehauder prin. ciple, By an analogous argumentation the l~ayleigh equation with a quasiIinear restoring force, but with an arbitrary damping term, is shown to possess aperiodie solution. Again, the ~'esult admits an extension to a third order equation.

A continuation theorem with applications to periodically forced Liénard equations in the presence of a separatrix

“…In this situation, due to a theorem of Reissig [64] (see also [21], [43], [44] for previous works in this direction), we know that Equation (1.3) has at least one T -periodic solution for any bounded forcing term e. Thus, we can exclude this case from our further considerations and, for the polynomial model, we may put aside the possibility "l odd and a 0 < 0". Next, let us consider the case when lim s→±∞ g(s) = +∞.…”

On the existence of periodic solutions for a class of p-Laplacian system

“…In recent years, the existence of periodic solutions of (1.1) for p = 2 has been extensively studied (see [1][2][3][4][5][6][7]). In [5], by using Krasnoselskii's fixed point theorem, Ding proved the following result.…”

Periodically perturbed nonconservative systems of Liénard type

“…(1 n u"(t) + f(u(t))u(t) + g(u(t)) = e(t), Starting with a device due to Faure [4], a research effort of Bebernes and Martelli [1], Cesari and Kannan [2], Lazer [7], Mawhin [8], Mawhin and Ward [10], Reissig [12,13], and others [6,11], leads to the following interesting nonresonance result.…”

“…The proof of our main result ( §2) makes use of Leray-Schauder topological degree [9]. On the other hand, we avoid the standard line of proof that relies heavily on the fact that equations considered are scalar [6,10,12,13], or with restoring term weakly coupled [1,2], or involving the symmetry of the restoring term [3,5].…”

“…^ n2 , r-'/2 l/2n I I ,111-1 2nb\u\ -2nb6 n Rx < i?, + 6 n Rx\e\Y + \e\Y\u\ which implies, since b > \e\Y , that (2.8) |ïï| < K for some constant K > 0 depending only on a, b, and e. Since \u\c < |ïï| + \u\c , we deduce from (2.6), (2.8), and Sobolev inequality, that there exists a constant R2 > 0 depending only on a, b, and e such that \u\c < R2 for all possible solutions of (2.5) which are such that \u(t)\ > R for all t e [0, 2n], and the claim is proved. Now, assume that for some u e X solution of (2.2) there exists 7 6 [0, 2n] such that \u(t)\ < R. Then standard arguments [1,6,10,12] and CauchySchwarz inequality imply |ïï,.| <i? + (27r)1/2|w|.|i2, l<i<N.…”