Periodic Solutions Of Differential Equations

This paper deals with the problems of finding periodic solutions for vector ordinary differential equations of the form(?) = f(t,x) = f(t + T,x),where T is a fixed positive number and f satisfies some additional conditions which will be given later.As is well known, the periodicity problem plays a central role in the qualitative theory of ordinary differential equations for its significance in the physical science. Hence finding periodic solutions of ordinary differential equations is naturally an attracting topic.This paper consist of four parts.In the first part reference language.In the second part, we consider the existence periodic solution to a class of neutral functional differential equation by using the fixed point theorem, the relevant results are extended.Theorem 1. For equation(2.1), if(i) Letα₁(t)is a continuous T-periodic function, for all (t,φ)∈R×C satisfyingλ_M(t,φ)≤a₁(t),and (ii) Letβ>0, such thatsatisfyingandthen equation (2.1) has a T-periodic solution.Theorem 2. For equation(2.1), if(i) Let a₂(t)is a continuous T-periodic function, for all (t,φ)∈R×C, satisfyingλ_M(t,Φ)≥α₂(t),and(ii) Letβ> 0, such thatR₁,q just like the definition of the theorem 5.1, andthen equation (2.1) has a T-periodic solution.In the third part we consider the periodic solutions of the order Hamiltoniansystem-(?)-λx = h(t)V′(x)with V being positive and superquadratic at infinity, h being continuous, 2Ï€-periodic , sign changing and satisfying {t|h(t) > 0}∩{t|h(t) < 0} =φ.Some existence and multiplicity results of periodic solutions are given. Theorem 3. Let V₁ and V₂ be C² functions satisfying the superquadratic condition (3.2), let h be a continuous 2Ï€-periodic function satisfying the thick zero condition (3.3). Set h_- = min{0, h},h₊ = max{0,h}, supposeλis not inδ(S₀), thenhas a nonzero 2Ï€-periodic solution, if eitherλis not inδ(S₀), V₁ and V₂ satisfyor there is a symmetric neighborhood U of 0 in R^m, such thatIf V₁ and V₂ are even in x, then (3.5) has an unbounded sequence of 2Ï€-periodic solutions.Lemma 1. There is a direct sum decompositionwith X₁= H₀¹(S₀¹âˆªS₊∪S₀²) andProposition 1. Let V₁, V₂∈C¹ satisfy (3.2), let h be a continuous,2Ï€-periodic function satisfying (3.3) andλis not inσ(S₀).Let C be a constant and s_n∈[0,1],x_n∈H¹(S¹), such thatThen {x_n} is bounded and contains a convergent subsequence.Proposition 2. There are constants A andδ> 0, such that for S∈[0,1], Proposition 3. Let h∈C⁰ satisfy (3.3), V₁ and V₂ be C² functions, satisfying the superquadratic condition (3.3) andλis not inσ(S₀). ThenC_*(I,∞) =0,* =0,1,2,…Theorem 4. Let V₁ be a C² function, satisfy (3.16) and (3.6),let V₂ be C² function satisfying (3.2) and (3.6). Then ifλis not inσ(S₀)∪σ(S₁),has a nonzero 2Ï€-periodic solution. Moreover, if V₁ and V₂ are even in x, then (3.17) has an unbounded sequence of 2Ï€-periodic solutions.In forth part the existence of the periodic solutions for a class of functional differentional equations with infinite delay is studied, we obtain some new results which extend and improve the related known works in the literature.Theorem 5. Suppose(H₁) Letα(t) is anω-periodic function, and not always equals to zero, such thatfor all(H₂) Let M > 0, for all t^*∈[0,ω], satisfyingandthen equation (4.1) has aω-periodic solution.Theorem 6. Suppose (H₃) Letα(t) is a continous,ω-periodic function, and not always equals to zero, such thatfor all(H₄) Let M > 0,for all t^*∈[0,ω], satisfyingandthen equation (4.1) has aω-periodic solution.