| This paper is an comprehensive survey of the recent results obtained from the study of periodic solutions of Lienard equation. We briefly review of the studies and the results about periodic solutions of Lienard equations in the recent decade. We mainly discuss the periodic solutions problem of general Lienard equations,generalized Lienard and Lienard equations with deviating variables.The following is our main results.? .The periodic solutions of general Lienard equations and generalized lienard problem.For the general Lienard equation: x" + f(x)x' + g(x) = p(t) (1)and x" + Cx'+ g(x,t) = s (2)The establishment of the following condlusions:Theorem 1 Assume g(x)∈C1 (R) satisfied g(0) = 0, g'(x) < 0 andf∈C(R),p(t)∈C2n,then equation (1) have only 2π- periondic solutions of the necessary and sufficient conditions are (?)∈g(R).Theorem 2 Assume g(x,t) satisfy the following conditions:(1)g(x,t)∈C(R×R),and on 2πcycle in t,strictly convex in x; (2)(?) g(x,t) = +∞and uniform in t;Then (?)s0,such at( i )if s < s0 ,equation (2) has no periodic solutions; (ii )if s = s0,equation (2) has only 2π- periodic solutions; (iii)if s > s0 ,equation (2) exactly two 2n - periodic solutions.For the generalized Lienard equation:a(t)x" + f(x,x')x' + g(t,x) = e(t) x(0) - x(2π) = x'(0) - x'(2π) = 0Assume:C[0,2π] express one continuous function space on [0,2π],C1 [0,2π] express one continuously differentiable function space on [0,2π].C2[0,2π] = {ω| [0,2π]→C2 andωis continuous}, and normal number is ||ω||= sup{||ω(t) |||t∈[0,2π]} 'C21[0,2π] = {ω| [0,2π]→C2 andω' is continuous},and normal number is ||ω||= sup{||ω(t) || + ||ω'(t) |||t∈[0,2π]}(?)[0,2π] = {ω∈C21[0,2π]|ω(0) =ω(2π)} ,apparently (?)[0,2π] is one closedsubspace in C21 [0,2π].Assume:(H1) a:R→R is continuously differentiable in t and cycle to2π; f : R x R→R is continuous,g: R x R→R is continuous in t and cycle to 2π,and is continuously differentiable in x; e: R→R and is continuous in t and cycle to 2π. (H2) exist constants b and c1 < 0 such at (?)x∈C1[0,2π] ,andor exist constants b and c2 > 0 such at (?)x∈C1 [0,2π] ,andwhere J(t) = -g'x (t, x(t)) + (b- a'(t))f(x(t), x'(t)) - b(b - a'(t)) +1 The establishment of the following condlusions: Theorem 3 If we assume (H1),(H2) establish,then generalized Lienard equation with the boundary value problem (3) has only periodic solutions.For the generalized Lienard equation:x" + f(x, x')x' + g(t, x, x') = e(t) (4)x(0)-x(2π) = x'(0)-x'(2π) = 0 (5)Assume:C[0,2π] express one continuous function space on [0,2π];C2[0,2π] = {w [0,2π]→C2 and w is continuous}, and normal number is || w||=sup{|| w(t) |||t∈[0,2π]};(?)[0,2π] = {w∈C2 [0,2π]|w(0) = w(2π)}, apparently (?)[0,2π] is one closed subspace on [0,2π].Assume:(H1) f : R×R→R and continuous,g:R×R×R→R is continuous and cycleto 2π,and is continuously differentiable in x, x'; e: R→R is continuous,on 2πcycle in t.(H2) There exist constant b >ε>0 such atOr exist constant b≤ε≤0 such atWe can use the folling weaker conditions instead of (H2) and (H3). (H4)Exist constant b and c1 < 0 such at Or exist constant b and c2 > 0 such atwhere J = -g'u' + ig'u +b)(f-b) + 1The establishment of the following condlusions:Theorem 4 If (H1)-(H3) hold ,then problem (4),(5) has only periodic solutions.Theorem 5 If (H1) and (H4) hold, then problem (4),(5) has only periodic solutions.?. For the Perodic solutions of the Lienard equation with deviating variables.x" + f[x(t -σ)]x'(t -σ) + P(t)g[x(t -τ(t))] = Pit) = p(t + T) (6)Where respectively, f,g,p,β,τregard their respective variable continuously in R,and p(t),β(t),τ(t) are cycle to T, minβ(t)>0,σis constant. Assume that X = {x(t)∈C1 (R,R): x(t + T) = x(t)} ,We prescribe normal number is ||x|| = max{|x|∞,|x'|∞} in X ,where |x|∞= max|x(t)|,then X is Banach spacesince the normal number. We consider operator equation in X.Lx =λN(x,λ),λ∈(0,1) ,where L: DomL∩X→X is linear operator,and defineprojection operator P, Q:P: X∩DomL →KerL,x →Px = (?) (?)x(t)dt,Q: X →X/ImL,x→Qx = (?) (?)(t)dt The establishment of the following condlusions:Theorem 6 Assume that f1 = sup|f(x)| < (?) ,and conditionshold,where r<(?) ,β1=maxβ(t) ,then equation (6) at least has oneT -periodic solutions. Theorem 7 Assumeσ= 0 ,and conditon(8) and the following conditionHold , where r<(?),β1 = (?)β(t) ,then equation (6) at least has oneT -periodic solutions.Theorem 8 Assumeσ= 0, (?) p{t)dt = 0 ,and exist constant A , B > 0 ,such at(l)if x≤-A,then g(x)≥-B;(2)if |x|≥A,then xg(x)>0, Then equation(6) at least has one T -periodic solutions.Theorem 9 Assumeσ= 0, (?) p(t)dt = 0, and exist constant A , B > 0 ,such at(l)if x≥A ,then g(x)≤B, (2)if |x|≥A,then xg(x)>0,Then equation(6) at least has one T -periodic solutions.For perodic solutions to a class of higher order Lienard equation with deviating variables:xm+(?)fi(x(t-σi))xi(t-σi) + g(t,x(t-τ(t)) = p(t) (9)Where fi∈C(R,R), and (?)u∈R , |fi(u)|≤ai, ai ,σi is real number andσi is notnegative number,i = l,2,(?),m-l , g∈C(R×R,R) ,and g(t + T,·) = g(t,·) , p(t),τ(t) are continuousand cycle to T, (?)p(t)dt = 0.Assume X = {x∈C(R,R)|x(t + T) = x(t)} ,where normal number is |x| = max| x(t)| ,then X is Banach space since the normal number.Define operator:Lx = xm and define projection operator:P: X →KerL ,x1-→Px = (?) (?)x(t)dtQ:X→X/ImL,x1-→Q(x) = (?) (?) x(t)dt then KerL = ImP = R,ImL = KerL, = Im(I - P) ,thus havedim(Ker)L = co dim ImL = 1, that is,f is Fredholm operator which indicator to zero ,and N is L-compact in(?) ,whereΩis bounded open set in X.The establishment of the following condlusions:Theorem 10 Assume exist constants A ,B > 0 ,such at(l)if x≤-A,t∈R,then g(t,x)≥-B, (2)if |x|≥A,t∈R,then xg(t,x)>0,Then equation(9) at least has one T -periodic solutions.Theorem 11 Assume exist constants A ,B > 0 ,such at(l)if x≥A,t∈R,then g(t,x)≤B, (2)if |x|≥A ,t∈R ,then xg(t,x)>0,Then equation(9) at least has one T -periodic solutions.In the paper ,we also metioned some other related results,here no longer describe in detail one by one.Nx = -(?)fi(x(t-σi))xi(t-σi)-g(t,x(t-τ(t))+p(t)... |
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