Báo cáo hóa học: " PERIODIC SOLUTIONS OF SECOND-ORDER LIÉNARD EQUATIONS WITH p-LAPLACIAN-LIKE OPERATORS" pot

PERIODIC SOLUTIONS OF SECOND-ORDER LIÉNARD EQUATIONS WITH p-LAPLACIAN-LIKE OPERATORS YOUYU WANG AND WEIGAO GE Received 12 April 2005; Accepted 10 August 2005 The existence of periodic solutions for second-order Li ´ enard equations with p-Laplacian- like operator is studied by applying new generalization of polar coordinates. Copyright © 2006 Y. Wang and W. Ge. This is an op en access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In recent years, the existence of periodic solutions for second-order Li ´ enard equations u  + f (u,u  )u  + g(u) =e(t,u,u  ) (1.1) and its special case have been studied by many researchers, we refer the readers to [1, 3, 4, 6, 7, 9–12] and the references therein. Let us consider the so-called one-dimensional p-Laplacian operator (φ p (u  ))  ,where p>1andφ p : R →R is given by φ p (s) =|s| p−2 s for s =0andφ p (0) = 0. Periodic bound- ary conditions containing this operator have been considered in [2, 5]. In [8], Man ´ asevich and Mawhin investigated the existence of periodic solutions to some system cases involving the fairly general vector-valued operator φ. They considerd the boundary value problem  φ(u  )   = f (t,u,u  ), u(0) = u(T), u  (0) = u  (T), (1.2) where the function φ : R N → R N satisfies some monotonicity conditions which ensure that φ is a homeomor phism onto R N . Recently, in [16] we studied the existence of periodic solutions for the nonlinear dif- ferential equation with a p-Laplacian-like operator  φ(u  )   + f (t, u,u  ) = 0. (1.3) Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 98685, Pages 1–17 DOI 10.1155/JIA/2006/98685 2 Periodic solutions for Li ´ enard equations Motivated by the work of [13], in this paper we use new polar coordinates [13]to investigate the existence of periodic solutions for the second-order generalized Li ´ enard equations with p-Laplacian-like operator  φ(u  )   + f (u,u  )u  + g(u) =e(t,u,u  ), t ∈ [0, T]. (1.4) Throughout this paper, we always assume that φ, g ∈ C(R,R), f ∈ C(R 2 ,R), e ∈ C ([0,T] ×R 2 ,R). And the following conditions also hold. (H1) φ is continuous and strictly increasing, yφ(y) > 0fory = 0, and there exist p>2, m 2 ≥ m 1 > 0, such that m 1 |y| p−1 ≤   φ(y)   ≤ m 2 |y| p−1 . (1.5) (H2) e ∈ C([0,T] ×R 2 ,R), periodic in t with period T, there exist α 1 ,β 1 ,γ 1 > 0, and p>k>2suchthat   e(t, x, y)   ≤ α 1 |x| p−1 + β 1 |y| k−1 + γ 1 for (t,x, y) ∈[0,T] ×R 2 . (1.6) (H3) f ∈ C(R 2 ,R), there exist α 2 ,β 2 ,γ 2 > 0suchthat   f (x, y)   ≤ α 2 |x| p−2 + β 2 |y| k−2 + γ 2 for (x, y) ∈R 2 . (1.7) (H4) There exist λ,μ,andn ≥0suchthat m 2 m 1  p  p  −1  p−1  2nπ p T  p + α 1 m 1 + p −1 p  α 2 m 1  p/(p−1)  m 2 m 1  1/(p−1) 2 <λ ≤ g(x) φ(x) ≤ μ< m 1 m 2  p  p  +1  p−1  2(n +1)π p T  p − α 1 m 2 − p −1 p  α 2 m 2  p/(p−1)  m 2 m 1  1/(p−1) , (1.8) where p  = p(p −1), π p = 2π(p −1) 1/p psin(π/p) . (1.9) (H5) Solutions of (1.4) are unique with respect to initial value. In this paper, we use a new coordinate to estimate the time when a point moves along a trajectory around the origin and then give some sufficient conditions for the existence of periodic solutions of (1.4). 2. Periodic solutions with a Laplacian-like operator Let v = φ(u  ). Then (1.4) is equivalent to the system u  = φ −1 (v), v  =−g(u) − f  u,φ −1 (v)  φ −1 (v)+e  t,u,φ −1 (v)  . (2.1) Y. Wan g and W. Ge 3 Let u(t, ξ,η) denote the solution of (1.4) which satisfies the initial value condition u(0,ξ,η) = ξ, v(0,ξ,η) =η, (2.2) then we have the following conclusion. Lemma 2.1. Suppose (H1)–(H5) hold, then for all c>0, there exists constant A>0 such that if 1 p |ξ| p + p −1 p |η| p/(p−1) = A 2 , (2.3) then 1 p   u(t,ξ,η)   p + p −1 p   v(t,ξ,η)   p/(p−1) ≥ c 2 for t ∈ [0,T]. (2.4) Proof. Let (u(t),v(t)), t ∈ [0, T], be a solution of (2.1) satisfying u(0,ξ,η) =ξ, v(0,ξ,η) = η. Let r 2 (t) = 1 p   u(t)   p + p −1 p   v(t)   p/(p−1) . (2.5) It is clear that (H1) implies  | v| m 2  1/(p−1) ≤   φ −1 (v)   ≤  | v| m 1  1/(p−1) . (2.6) So we have     dr 2 (t) dt     =      u(t)   p−2 u(t)u  (t)+   v(t)   (2−p)/(p−1) v(t)v  (t)    ≤| u| p−1   φ −1 (v)   + |v| 1/(p−1)    − g(u) − f  u,φ −1 (v)  φ −1 (v)+e  t,u,φ −1 (v)     ≤| u| p−1   φ −1 (v)   + μ|v| 1/(p−1)   φ(u)   + |v| 1/(p−1)  α 2 |u| p−2 + β 2   φ −1 (v)   k−2 + γ 2    φ −1 (v)   + |v| 1/(p−1)  α 1 |u| p−1 + β 1   φ −1 (v)   k−1 + γ 1  ≤| u| p−1  | v| m 1  1/(p−1) + μm 2 |v| 1/(p−1) |u| p−1 + α 2 m −1/(p−1) 1 |v| 2/(p−1) |u| p−2 + β 2 m (1−k)/(p−1) 1 |v| k/(p−1) + γ 2 m −1/(p−1) 1 |v| 2/(p−1) + α 1 |v| 1/(p−1) |u| p−1 + β 1 m (1−k)/(p−1) 1 |v| k/(p−1) + γ 1 |v| 1/(p−1) = l 1 |u| p−1 |v| 1/(p−1) + l 2 |v| k/(p−1) + l 3 |v| 2/(p−1) |u| p−2 + l 4 |v| 2/(p−1) + γ 1 |v| 1/(p−1) , (2.7) 4 Periodic solutions for Li ´ enard equations where l 1 = m −1/(p−1) 1 + μm 2 + α 1 , l 2 = β 1 m (1−k)/(p−1) 1 + β 2 m (1−k)/(p−1) 1 , l 3 = α 2 m −1/(p−1) 1 , l 4 = γ 2 m −1/(p−1) 1 , (2.8) while l 1 |u| p−1 |v| 1/(p−1) ≤ l 1  1 p |v| p/(p−1) + p −1 p |u| p  ≤ l 1 max  p −1, 1 p −1   1 p |u| p + p −1 p |v| p/(p−1)  = l 1 max  p −1, 1 p −1  r 2 , l 2 |v| k/(p−1) ≤ k p |v| p/(p−1) + p −k p l p/(p−k) 2 ≤ k p −1 r 2 + p −k p l p/(p−k) 2 l 3 |v| 2/(p−1) |u| p−2 ≤ l 3  2 p |v| p/(p−1) + p −2 p |u| p  ≤ l 3  2 p −1 + p −2  r 2 , l 4 |v| 2/(p−1) ≤ 2 p |v| p/(p−1) + p −2 p l p/(p−2) 4 ≤ 2 p −1 r 2 + p −2 p l p/(p−2) 4 , γ 1 |v| 1/(p−1) ≤ 1 p |v| p/(p−1) + p −1 p γ p/(p−1) 1 ≤ 1 p −1 r 2 + p −1 p γ p/(p−1) 1 . (2.9) So,     dr 2 (t) dt     ≤ br 2 (t)+a, (2.10) where a = p −k p l p/(p−k) 2 + p −2 p l p/(p−2) 4 + p −1 p γ p/(p−1) 1 , b = l 1 max  p −1, 1 p −1  + l 3  2 p −1 + p −2  + k +3 p −1 . (2.11) It follows that  r 2 (0) + a b  e −bT ≤  r 2 (0) + a b  e −bt ≤  r 2 (t)+ a b  ≤  r 2 (0) + a b  e bt ≤  r 2 (0) + a b  e bT ,0≤ t ≤ T. (2.12) Let A = [(c 2 + a/b)e bT −a/b] 1/2 ,thenr(0) = A implies r(t) ≥c.  Y. Wan g and W. Ge 5 Lemma 2.2. Let (u(t),v(t)) be a solution of (2.1). Suppose the conditions of (H1)–(H5) are satisfied. Then there is R such that under the generalized polar coordinates, r(0) ≥ R implies that dθ(t) dt ≤ 0, t ∈ [0,T]. (2.13) Proof. Applying generalized polar coordinates, u = p 1/p r 2/p |cos θ| (2−p)/p cos θ, v =  p p −1  (p−1)/p r 2(p−1)/p |sinθ| (p−2)/p sinθ, (2.14) or r cosθ = 1 √ p |u| (p−2)/2 u, r sinθ =  p −1 p |v| (2−p)/2(p−1) v. (2.15) Then θ = tan −1 [  p −1(|v| ((2−p)/2(p−1)) v/|u| ((p−2)/2) u)]. So we have θ  = | u| ((p−2)/2) |v| ((2−p)/2(p−1)) 2  p −1r 2  uv  −(p −1)u  v  =− | u| ((p−2)/2) |v| ((2−p)/2(p−1)) 2  p −1r 2  ug(u)+uf  u,φ −1 (v)  φ −1 (v) +(p −1)vφ −1 (v) −ue  t,u,φ −1 (v)   (2.16) as ug(u)+uf  u,φ −1 (v)  φ −1 (v)+(p −1)vφ −1 (v) −ue  t,u,φ −1 (v)  ≥ λuφ(u)+(p −1)vφ −1 (v) −|u|  α 2 |u| p−2 + β 2   φ −1 (v)   k−2 + γ 2    φ −1 (v)   −| u|  α 1 |u| p−1 + β 1   φ −1 (v)   k−1 + γ 1  ≥ λm 1 |u| p +(p −1)m −1/(p−1) 2 |v| p/(p−1) −α 2 m −1/(p−1) 1 |u| p−1 |v| 1/(p−1) −γ 2 m −1/(p−1) 1 |u||v| 1/(p−1) −α 1 |u| p −  β 1 + β 2  m (1−k)/(p−1) 1 |u||v| (k−1)/(p−1) −γ 1 |u| =  λm 1 −α 1  | u| p +(p −1)m −1/(p−1) 2 |v| p/(p−1) −α 2 m −1/(p−1) 1 |u| p−1 |v| 1/(p−1) −γ 2 m −1/(p−1) 1 |u||v| 1/(p−1) −  β 1 + β 2  m (1−k)/(p−1) 1 |u||v| (k−1)/(p−1) −γ 1 |u|. (2.17) 6 Periodic solutions for Li ´ enard equations Let τ = p(p −1) 4(k −1) m −1/(p−1) 2 , β  = 4  β 1 + β 2  (k −1) p(p −1) m (1−k)/(p−1) 1 m 1/(p−1) 2 , (2.18) so we have  β 1 + β 2  m (1−k)/(p−1) 1 |u||v| (k−1)/(p−1) = τ|u|  |v| (k−1)/(p−1) β   ≤ τ|u|  k −1 p −1 |v|+ p −k p −1 β (p−1)/(p−k)  = 1 4 pm −1/(p−1) 2 |u||v|+ p(p −k) 4(k −1) m −1/(p−1) 2 β (p−1)/(p−k) |u| ≤ 1 4 pm −1/(p−1) 2  1 p |u| p + p −1 p |v| p/(p−1)  + p(p −k) 4(k −1) m −1/(p−1) 2 β (p−1)/(p−k) |u|. (2.19) Let τ 1 = 1 4 p(p −1)m −1/(p−1) 2 , β  1 = 4γ 2 p(p −1)  m 2 m 1  1/(p−1) , (2.20) then γ 2 m −1/(p−1) 1 |u||v| 1/(p−1) = τ 1 |u|  |v| 1/(p−1) β  1  ≤ τ 1 |u|  1 p −1 |v|+ p −2 p −1 β (p−1)/(p−2) 1  = 1 4 pm −1/(p−1) 2 |u||v|+ p(p −2) 4 m −1/(p−1) 2 β (p−1)/(p−2) 1 |u| ≤ 1 4 pm −1/(p−1) 2  1 p |u| p + p −1 p |v| p/(p−1)  + p(p −2) 4 m −1/(p−1) 2 β (p−1)/(p−2) 1 |u|. (2.21) Let τ 2 = 1 4 p(p −1)m −1/(p−1) 2 , β  2 = 4α 2 p(p −1)  m 2 m 1  1/(p−1) (2.22) Y. Wan g and W. Ge 7 then α 2 m −1/(p−1) 1 |u| p−1 |v| 1/(p−1) = τ 2  | v| 1/(p−1) β  2 |u| p−1  ≤ τ 2  1 p |v| p/(p−1) + p −1 p  β  2 |u| p−1  p/(p−1)  ≤ 1 4 pm −1/(p−1) 2  1 p |u| p + p −1 p |v| p/(p−1)  + p −1 p τ 2 β  2 p/(p −1) |u| p . (2.23) We selec t λ large enough such that δ = λm 1 −α 1 − p −1 p τ 2 β  2 p/(p −1) −m −1/(p−1) 2 > 0, (2.24) Let d = γ 1 +(p(p − k)/4(k − 1))m −1/(p−1) 2 β (p−1)/(p−k) +(p(p − 2)/4)m −1/(p−1) 2 β (p−1)/(p−2) 1 ,wealsohave d |u|=δp|u|  d δp  ≤ δ|u| p +(p −1)δ  d pδ  p/(p−1) , (2.25) therefore ug(u)+uf  u,φ −1 (v)  φ −1 (v)+(p −1)vφ −1 (v) −ue  t,u,φ −1 (v)  ≥ 1 4 pm −1/(p−1) 2  1 p |u| P p + p −1 p |v| p/(p−1)  − (p −1)δ  d pδ  p/(p−1) = 1 4 pm −1/(p−1) 2 r 2 (t) −(p −1)δ  d pδ  p/(p−1) . (2.26) Lemma 2.1 implies that there is R > 0, such that 1 4 pm −1/(p−1) 2 r 2 (t) > (p −1)δ  d pδ  p/(p−1) (2.27) when r(0) > R, then our assertion is verified.  Lemma 2.3. Assume that (H1)–(H5) hold, and 1 p |ξ| p + p −1 p |η| p/(p−1) = A 2 (A  1) (2.28) 8 Periodic solutions for Li ´ enard equations then  u(T,ξ,η),v(T,ξ,η)  = (λ 2/p ξ,λ 2(p−1)/p η), (2.29) where λ is an arbitrary posit ive number. Proof. It follows from Lemma 2.1 that if 1 p |ξ| p + p −1 p |η| p/(p−1) = A 2 , (2.30) then 1 p   u(t,ξ,η)   p + p −1 p   v(t,ξ,η)   p/(p−1) ≥ c 2 for t ∈ [0, T]. (2.31) According to the generalized polar coordinates (2.14), we have r(t) ≥ c for t ∈ [0, T]ifr(0) = A. (2.32) On the other hand, when r(0) →∞, it holds uniformly from (H1)–(H3) that −θ  = | u| (p−2)/2 |v| (2−p)/2(p−1) 2  p −1r 2  ug(u)+uf  u,φ −1 (v)  φ −1 (v) +(p −1)vφ −1 (v) −ue  t,u,φ −1 (v)   ≥ | u| (p−2)/2 |v| (2−p)/2(p−1) 2  p −1r 2   λm 1 −α 1  | u| p +(p −1)m −1/(p−1) 2 |v| p/(p−1) −α 2 m −1/(p−1) 1 |u| p−1 |v| 1/(p−1) −γ 2 m −1/(p−1) 1 |u||v| 1/(p−1) −  β 1 + β 2  m (1−k)/(p−1) 1 |u||v| (k−1)/(p−1) −γ 1 |u|  (2.33) as α 2 m −1/(p−1) 1 |u| p−1 |v| 1/(p−1) = m −1/(p−1) 2  | v| 1/(p−1)   α 2  m 2 m 1  1/(p−1) |u| p−1  ≤ m −1/(p−1) 2  1 p |v| p/(p−1) + p −1 p α p/(p−1) 2  m 2 m 1  p/(p−1) 2 |u| p  = 1 p m −1/(p−1) 2 |v| p/(p−1) + p −1 p α p/(p−1) 2 m 1 −p/(p−1) 2 m 2 1/(p −1) 2 |u| p . (2.34) Y. Wan g and W. Ge 9 So −θ  ≥ | u| (p−2)/2 |v| (2−p)/2(p−1) 2  p −1r 2   λm 1 −α 1 − α  | u| p + p  −1 p  (p −1)m −1/(p−1) 2 |v| p/(p−1) −γ 2 m −1/(p−1) 1 |u||v| 1/(p−1) −  β 1 + β 2  m (1−k)/(p−1) 1 |u||v| (k−1)/(p−1) −γ 1 |u|  = p|sinθ| (2−p)/p |cos θ| (p−2)/p 2(p −1) 1/p   λm 1 −α 1 − α  cos 2 θ + p  −1 p  m −1/(p−1) 2 sin 2 θ  − γ 2 m −1/(p−1) 1 p 2/p 2(p −1) 2/p r 2(p−2)/p |cos θ||sinθ| (4−p)/p −  β 1 + β 2  m (1−k)/(p−1) 1 p k/p 2(p −1) k/p r 2(p−k)/p |cos θ||sinθ| (2k−p)/p − γ 1 p 1/p 2(p −1) 1/p r 2(p−1)/p |cos θ||sinθ| (2−p)/p = a 1  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p − γ 2 m −1/(p−1) 1 p 2/p 2(p −1) 2/p r 2(p−2)/p |cos θ||sinθ| (4−p)/p −  β 1 + β 2  m (1−k)/(p−1) 1 p k/p 2(p −1) k/p r 2(p−k)/p |cos θ||sinθ| (2k−p)/p − γ 1 p 1/p 2(p −1) 1/p r 2(p−1)/p |cos θ||sinθ| (2−p)/p , (2.35) where α = p −1 p α p/(p−1) 2 m −p/(p−1) 2 1 m 1/(p−1) 2 2 , p  = p(p −1), a 1 = p(p  −1) 2p  (p −1) 1/p m 1/(p−1) 2 , b 1 = p  p  −1  λm 1 −α 1 − α  m 1/(p−1) 2 . (2.36) Denote  b =min{b 1 ,1},thenwehave −θ  ≥ a 1  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p − γ 2 m −1/(p−1) 1 p 2/p 2  b(p −1) 2/p r 2(p−2)/p  b 1 cos 2 θ + sin 2 θ  | cos θ||sinθ| (4−p)/p 10 Periodic solutions for Li ´ enard equations −  β 1 + β 2  m (1−k)/(p−1) 1 p k/p 2  b(p −1) k/p r 2(p−k)/p  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p − γ 1 p 1/p 2  b(p −1) 1/p r 2(p−1)/p  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p =  a 1  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p , (2.37) where a 1 = a 1 − γ 2 m −1/(p−1) 1 p 2/p 2  b(p −1) 2/p r 2(p−2)/p −  β 1 + β 2  m (1−k)/(p−1) 2 p k/p 2  b(p −1) k/p r 2(p−k)/p − γ 1 p 1/p 2  b(p −1) 1/p r 2(p−1)/p . (2.38) Assume that it takes time Δt for the motion (r(t), θ(t))(r(0) = A, θ(0) = θ 0 )tocom- plete one cycle around the origin. It follows from the above inequality that Δt<  θ 0 +2π θ 0 dθ a 1  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p = 4 a 1  π/2 0 dθ  b 1 cos 2 θ + sin 2 θ  | sinθ| (2−p)/p |cos θ| (p−2)/p . (2.39) Let η = tan −1 1  b 1 tanθ, (2.40) then Δt< 4 a 1 b 1/p 1  π/2 0 dη |tanη| (2−p)/p = 2 a 1 b 1/p 1 B  1 p , p −1 p  = 2π a 1 b 1/p 1 sin(π/p) , (2.41) from (H4), we have a 1 b 1/p 1 sin π p = π π p  p  −1 p   (p−1)/p  λm 1 −α 1 − α m 2  1/p > 2nπ T . (2.42) So there exists σ>0suchthat(a 1 −σ)b 1/p 1 sin(π/p) > 2nπ/T.Fortheσ>0, there exists R  > 0suchthat 0 < γ 2 m −1/(p−1) 1 p 2/p 2  b(p −1) 2/p r 2(p−2)/p +  β 1 + β 2  m (1−k)/(p−1) 2 p k/p 2  b(p −1) k/p r 2(p−k)/p + γp 1/p 2  b(p −1) 1/p r 2(p−1)/p <σ (2.43) [...]... 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A E Murua, Existence and multiplicity of solutions with a prescribed period for a second order quasilinear ODE, Nonlinear Analysis Theory, Methods & Applications 18 (1992), no 1, 79–92 ´ [3] M A del Pino, R F Man´ sevich, and A Murua, On the number of 2π periodic solutions for a e u + g(u) = s(1 + h(t)) using the Poincar´-Birkhoff theorem, Journal of Differential Equations 95 (1992), no 2, 240–258 [4]... different solutions to (3.1) satisfying x1 (t0 ) = x2 (t0 ) = x0 , x1 (t0 ) = x2 (t0 ) = x0 (3.5) 16 Periodic solutions for Li´ nard equations e Let y = φ(x ), then (xi (t), yi (t)) = (xi (t),φ(xi (t))) (i = 1,2) are two different solutions to the system x = φ−1 (y), (3.6) y = −g(x) − f x,φ−1 (y) φ−1 (y) + e t,x,φ−1 (y) , satisfying (xi (t0 ), yi (t0 )) = (x0 ,φ(x (t0 ))) (i = 1,2) Without loss of generality,... one 1 -periodic solution Acknowledgments The authors of this paper wish to thank the referee for his (or her) valuable suggestions regarding the original manuscript The project is supported by the National Natural Science Foundation of China (10371006) Y Wang and W Ge 17 References [1] T A Burton and C G Townsend, On the generalized Li´nard equation with forcing function, e Journal of Differential Equations. .. easy to see that u(t) = u(t,ξ ∗ ,η∗ ) is a T -periodic solution of (1.4) Y Wang and W Ge 15 If we let φ(u) = ϕ p (u) = |u| p−2 u, p > 2, then we have the following special cases of (1.4): ϕ p (u ) + f (u,u )u + g(u) = p(t,u,u ) t ∈ [0,T], (2.70) so we can easy get the following results Theorem 2.5 Assume (H2) and (H3) hold and solutions of (2.70) are unique with respect to initial value, moreover suppose . PERIODIC SOLUTIONS OF SECOND-ORDER LIÉNARD EQUATIONS WITH p-LAPLACIAN-LIKE OPERATORS YOUYU WANG AND WEIGAO GE Received 12 April 2005; Accepted 10 August 2005 The existence of periodic solutions. for periodic solutions of semilinear Duffing equations, JournalofDiffer ential Equations 105 (1993), no. 2, 364–409. [5] C. Fabry and D. Fayyad, Periodic solutions of second order differential equations. Mawhin, Periodic solutions for nonlinear sy stems with p-Laplacian-like operators,JournalofDifferential Equations 145 (1998), no. 2, 367–393. [9] J.MawhinandJ.R.WardJr. ,Periodic solutions of some