ON EXISTENCE OF PERIODIC SOLUTIONS OF THE RAYLEIGH EQUATION OF RETARDED TYPE

In this paper, we give two sufficient conditions on the existence of periodic solutions of the non-autonomous Rayleigh equation of retarded type by using the coincidence degree theory.


Introduction.
In [1,2], the authors studied the existence of periodic solutions of the differential equation (1.1) In this paper, we discuss the existence of periodic solutions of the non-autonomous Rayleigh equation of related type where τ, σ ≥ 0 are constants, f and g ∈ C(R 2 ,R), f (t,x) and g(t, x) are functions with period 2π for t, f (t,0) = 0 for t ∈ R, p ∈ C(R, R), p(t) = p(t +2π) for t ∈ R and 2π 0 p(t) = 0. Using coincidence degree theory developed by Mawhin [2], we find two sufficient conditions for the existence of periodic solutions of (1.2).

Main results
Theorem 2.1. Suppose there are positive constants K, D, and M such that D] |g(t, x)| < +∞. Then (1.2) has at least a periodic solution with period 2π .
Proof. Consider the equation where λ ∈ (0, 1). Suppose that x(t) is a periodic solution with period 2π of (2.1). Since such that x (t 0 ) = 0. In view of (2.1), we see that for any t ∈ [0, 2π], We assert that for some positive number D 1 . Indeed, integrating (2.1) from 0 to 2π and noting con- Thus letting By applying (ii), (iii), and (iv), we have for some positive number D 2 . Next, note that the last equality in (2.4) implies for some t 1 in [0, 2π]. Thus in view of condition (i), we have and in view of (ii), we have Since x(t) is a periodic solution with period 2π of (2.1), we infer that |x(t 2 )| < D for some t 2 in [0, 2π]. Therefore, Let X be the Banach space of all continuous differentiable functions of the form x = x(t), defined on R such that x(t + 2π) = x(t) for all t, and endowed with the norm x 1 = max 0≤t≤2π {|x(t)|, |x (t)|}. Let Y be the Banach space of all continuous functions of the form y = y(t), defined on R such that y(t + 2π) = y(t) for all t, and endowed with the norm y 0 = max 0≤t≤2π |y(t)|, and let Ω be the subspace of X containing functions of the form x = x(t), such that |x(t)| <D and |x (t)| <D, whereD is a fixed number greater than D + 2πD 2 . Now, let L : X ∩ C (2) (R, R) → Y be the differential operator defined by (Lx)(t) = x (t) for t ∈ R, and let N : (2.14) We know that ker L = R. Furthermore if we define the projections P : X → ker L and respectively, then ker L = Im P and ker Q = Im L. Furthermore, the operator L is a Fredholm operator with index zero, and the operator N is L-compact on the closurē Ω of Ω (see, e.g., [2, p. 176]). In terms of valuation of bound of periodic solutions as above, we know that for any λ ∈ (0, 1) and any x = x(t) in the domain of L, which also belongs to ∂Ω, Lx ≠ λNx. Since for any x ∈ ∂Ω ∩ ker L, x =D or x = −D, then in view of (ii), (iii), and 2π 0 p(t)dt = 0, we have (2.16) In particular, we see that (ii) xg(t, x) > 0 and |g(t, x)| > K for t ∈ R, |x| ≥ D; (iii) g(t, x) ≤ M for t ∈ R, x ≥ D; (iv) sup (t,x)∈R×[−D,D] |g(t, x)| < +∞.
Then (1.2) has at least a periodic solution with period 2π .
The proof of Theorem 2.2 is similitude of Theorem 2.1, and so, we omit the details here.