Anti-Periodic Solutions Of Differential Equations

The anti-periodic problems of differential equations widely used in many fields such as physics, biology, ecology and economy and so on. The methods of periodic problems of differential equations have been extended to anti-periodic problems, and have gained a lot of results.In this paper, we are mainly concerned with the existence of anti-periodic solutions. The article contains three sections. In section 1, we prove an anti-periodic LaSalle oscillation theorem. In section 2, we consider the existence problem for anti-periodic second order differential equation. Finally we study anti-periodic solutions for evolution equations, and prove the Massera theorem.§1 An Anti-periodic LaSalle Oscillation TheoremConsider the equation x'=f(t,x), (1)where f : R×Rⁿ→Rⁿ is continuous and -f(t + T,x) = f(t, -x); t∈R. Theorem 1. If Eq.(1)has the solution x₀(t) on [0,∞), and there exists a nonnegative functionα(t) on [0,∞) with (?), such that any two solutions x(t), y(t) of Eq.(1) satisfyThen Eq.(1) admits a T-anti-periodic solution.Theorem 2. Let f(t,x) satisfy the Kamke monotonicity, that is, for any x, y∈Rⁿ, if x_j≤ y_j, j = 1,2,…, n; x_i = y_i, there is f_i(t, x)≤f_i(t, y). Further assume that there exist C¹ functionsα,β: [0, T]→Rⁿ, such that(H1)β≥0,α=-β,β(T)≤β(0);(H2)α'≤f(t,α),β'≥f(t,β).Then Eq.(1) has T-anti-periodic solution x_*(t), withα≤x_*≤β.Definition 1. We say that a pair of functionsβ,α∈AC([0, T]) are related lower and upper solutions for the anti-periodic problem (1) ifandTheorem 3. Suppose that there existβ,α∈AC([0,T]) related lower and upper solutions for the anti-periodic problem (1). Further assume that f (t, x) satisfy the Kamke monotonicity, that is, for any x, y∈Rⁿ, if x_j≤y_j, j = 1,2,…,n; x_i = y_i, there is f_i(t,x)≤f_i(t,y). Then Eq.(1) has T-anti-periodic solution x_*(t), withα≤x_*≤β. §2 The existence of anti-periodic second order differential equationIn the natural world, the essence of many phenomenons is the study of solutions of nonlinear differential equations.Consider the following equation:(?) (2) where f : R¹Ã—R¹Ã—R¹â†’R¹ is continuous and f(t+T,x,p) = -f(t,-x,-p), for any (t,x,p).Theorem 4 Assume that there exist C² functionsα,β: [0, T]→R¹, such that(H1)α(0) =α(T),β(0) =β(T),α'(0)≥α'(T),β'(0)≤β(T);β(t)≥0,α(t) =-β(t), t∈[0,T];(H2)α"≥f(t,α,α'),β"≤f (t,β,β'), t∈[0,T]; (H3) |f(t,x,p)|≤c(1 + |p|²),α≤x≤β.Then the Eq.(2) has a T-anti-periodic solution x_*(t) withα≤x_*≤β, t∈[0,T].Consider the second order differential equations(p(t)x')' + f(t,x) = 0. (3)We have the following assumptions:(A1) f∈C¹(R×R) is T-anti-periodic with respect to the first variable, that is -f(t + T,x) = f(t, -x):p∈C ¹(R) is T-periodic, that is p(t + T) = p(t), and p(t) >0,t∈R.(A2) There exist two constants a and b such that(?) on R×R, and there exists a nonnegative integer m satisfying the conditionTheorem 5 Let the assumptions (A1)and (A2) hold. Them Eq.(3) has a unique T-anti-periodic solution.As a direct generalization of Theorem 5, we can consider a general even order differential equation as follows:Theorem 6 Let f∈C¹(R×R) with -f(t + T, x) = f(t, -x), and p∈Cⁿ(R) with p(t + T) = p(t), and p(t + T)= p(t), and p(t) > 0,t∈R Assume that there exist constants a and b, and a nonnegative integer m such that the inequalityholds on R×R for constantsα_j j = 1, 2,…, n - 1. Then Eq.(4) has a unique T-anti-periodic solution.Consider the following equation (?), (5)where x∈Rⁿ, f∈C¹(R×Rⁿ), -f(t + T, x) = f(t, -x). Theorem 7 Assume that there exist two constants 7 > 0 and v > 0 such thatγI < p(t) <νI, where I is a n×n unit matrix. There exist two constant symmetric n×n matrices A and B such thatwhere f_x = (f_ixj) is a Jacobian matrix, and f_x is a symmetric n×n matrix. If k₁≤k₂≤…≤k_n andμ₁≤μ₂≤…≤μ_n are the eigenvalues of A and B respectivly. There exist integers N_i,i = 1,2,…,n, satisfying the conditionThen Eq.(5) has only one T-anti-periodic solution.§3 Anti-periodic solutions for evolution equationsLet X be a Banach space with the norm ||·||. We denote by C≡C([-r, 0],X) the Banach space of all continuous map from [-r, 0]→X with the usual supremum norm ||·||. Here r≥0.Consider the linear evolution equation with delayx' = A(t)x_t + B(t), (6)where A(t) : D(A(t))→X,t∈R, is linear: B : R→X, and A(t) isωperiodic, that is A(t) = A(t +ω); B(t) isωanti-periodic, that is, -B(t) =B(t+ω):ω>0,t∈R; x_t : [-r,0]→X, and x_t(θ) = x(t +θ),θ∈[-r,0]; D(A(t)) is the domain of A(t), is a linear subspace of C. For Eq.(6), we have the following assumptions:(H1)There exists a unique continous linear operator function U : R⁺×R⁺→L(X, X) (L(X, X) is the Banach space of all bounded linear operator from X into X with the supremum norm ||·||) such that(a)(the indentical operator), t = s;(b)∪(t, t₁)∪(t₁,s) = U(t, s) 0≤s≤t₁ 0, andφ∈D(A(σ)),there exists a unique continuous solution u(t,σ,φ) of the problem,(?), (7)that is, when t≥σ, u(t,σ,φ)satisfies the equation u' = A(t)_{ut, uσ}=φ, and the conition u_σ=φ.Definition 2 A mild (strict) solution x(t) is said to be a mild (strict)ωanti-periodic solution of Eq.(6), if x(t +ω) =-x(t), t∈R.Theorem 8 Letω≥r, and let the condition (H1) and (H2)hold. Moreover, assume that there exists a mild (strict) solution of Eq.(6) defined on R, with the initial value x₀ =φ∈D(A(0)), such tant the set {(x_k)t}is precompact in C, where x_k(t) = x(t + kω) = (-1)^kx(t) t∈R⁺. Then Eq.(6) admits aωanti-periodic solution.Consider the following equationx' + A(t)x = f(t,x), (8)where x∈X,t∈R, {A(t) : A(t +ω) = A(t), t∈R} is a family of closed densely defined linear operators in X such that the domain D(A(t)) of A(t) is independent of t; f : R×C→X, and -f(t +ω, x) = f(t, -x); X is a Banach space with the norm ||·||.We have the following assumption:(H3) The bounded operator A(t)A(s)^-1 is Holder continuous in t for each fixed s.(H4) For each t∈[0,ω], R(λ,A(t)) exists for allλwith Reλ> 0,and there exists M > 0 such that (?). R(λ,A(t)) =(λI - A(t))^-1 is the resolvent of A(t).(H5) For each t∈[0,ω] and someλ∈ρ(A(t)), R(λ, A(t)) is a compact operator.(H6) For someω> 0, A(t +ω) = A(t), f(t +ω,Φ) = -f(t, -Φ); there exists a constantν∈(0,1); and for some M₀ > 0 and everyρ> 0, there exists a constant C(ρ) > 0 such thatfor t∈R,Φ,φ∈C_αsatisfyingwhere C_α= C([-r,0),X_α), with the norm ||·||_∞,α definded ||Φ|| _∞,α= (?).Theorem 9 Assume that (H3)-(H6) hold. If there existσ> 0 andβ∈(0,1] such that for every x₀∈X_β, every continuousω-anti-periodic functionΦ: R→X, with ||Φ||_∞≤σ, and the mild solution x(t,Φ) of the Cauchy problemx' + A(t)x = f(t,Φ), the following holds:(?). (9)Then Eq. (8) admits aω-anti-periodic solution x_*(t) with the norm ||x_*(t) ||≤σ,t∈R.