Affine-Periodic Solutions For Dissipative Systems

As profound and important systems, dissipative systems [1,2,3] are wide-ly found in nature. The existence of periodic solutions is one of the most im-portant issues in the field of differential equations. To study the existence of periodic solutions of dissipative systems has important significance.Prom the verification of Kepler’s Laws of planetary motion by Newton to the famous Poincare periodicity theorem, mathematicians study periodic solutions of differential equations for a long time period. In the beginning of the last century, Hilbert’s sixteenth problem about the number of limit cycles of polynomial system was raised. Since the1940s, N. Levinson, T. Yoshizawa, J. L. Massera, T. A. Buiton, Chow. S. N. and many other scholars have done a lot of work.There are a large number of articles have been published, such as [4,5,6,7] and [11]-[21]. The most classic results are Massera criterion [8] and Yoshizawa Theorem [9]. In1950, J. L. Massera established the existence theorem of periodic solutions for plane periodic system: x’＝f (t, x),f (t+ωx)＝f (t, x).He proved that if this equation admits a forward bounded solution, then there is at least one ω-periodic solution. He also got similar results on high-dimensional linear periodic system. Many scholars do further research on these two important theorems from the following two aspects:extending these two theorems to a wider range of equations; weakening the conditions in theorems. In1973, Chow S. N. consider one dimensional linear differential equations with finite delay [7]: x’(t)=L(t,xt)+f(t), where xt(.)=x(t+.)€C([-Υ,0],R), w≥r, L:(-∞,+00)�'R and f are continuous and ω-periodic with respect to t. A similar result can be obtained by using constant variation formula:If the differential equation admits a bounded solution and r≤ω, then there exists an w-periodic solution. However, this result is limited by the condition r≤ω.In1999, Li Yong and his collaborators extended the above results [22]. They removed the restriction r≤ω and proved that the functional differential equation with advance and delay exists periodic solutions if and only if the bounded solutions exist. Further, they extended Massera criterion to the case of periodic linear evolution equations in Banach spaces. So far, Massera crite-rion for Lienard equations, non-autonomous differential equations with delay, almost automorphic bounded non-autonomous differential equations and so on are all given [23,24].This is an ideal state that the model described by ordinary differential equations is only associated with the current state and is regardless of the state of the system before. But in fact, changes in the law of things is closely related to its historical status. So, we need a new way to deal with the impact on the system caused by delay. This has led to the interest in the functional d-ifferential equations, which can characterize the system depends on the current status and the past states and play a very important role in many fields. It is of great significance to extend the conclusion of ordinary differential equations to functional differential equations.In1966, T. Yoshizawa proved that N-dimensional periodic differential equations admit periodic solutions if they are uniformly ultimately bounded [26] by using Browder fixed point theorem [25]. In the same year, T. Yoshizawa considered the existence of periodic solutions for periodic functional differential equations with finite delay [27] and extended Massera criterion to the case of functional differential equations. The following is the famous Yoshizawa Theorem:Considered the following functional differential equations with finite delay: x=f(t,xt), t≥0, where xt(9)=x(t+η),9€[-Υ,0],||θ||=sup|θ(θ)|, θ> E C, C is the space consists of all continuous functions mapping [-Υ,0] to Rn.Yoshizawa proved that if the solutions of this equation are uniformly bounded and uniformly ultimately bounded with respect to B, then there is at least one periodic solution.Yoshizawa theorem for numerous types of periodic system have been built up. In1985, T. A. Burton [28] considered some ω-periodic integral differential equations with infinite delay and proved that if the initial function is bounded and the solutions of equations are uniformly bounded and uniformly ultimate-ly bounded, then there exist mw-periodic solutions. Further, T. A. Burton obtained the existence of w-periodic solutions [29,30,31]. In1989, T. A. Burton[32] established Yoshizawa theorem for functional differential equations with infinite delay and the nature of the weak decays of memories in the space Cg. In1988, Yoshizawa theorem is generalized to neutral functional differential equations: where xt(θ)=x(t+θ), f satisfies some proper conditions. When θ∈[-Υ,0], it is neutral functional differential equations with finite delay, when θ∈[-∞,0], it is neutral functional differential equations with infinite delay. Obviously, when certain conditions are met, neutral functional differential equations are exactly the functional differential equations with delay. During1994-2000, Liu [33,34,35] established Massera criterion and Yoshiza-wa theorem for periodic evolution equations without delay in Banach space, periodic evolution equations with finite delay and infinite delay. He proved that if there exist bounded and ultimately bounded solutions of the periodic evolution equations mentioned above, then there exist periodic solutions.In this paper, we establish Massera criterion and Yoshizawa theorem for Q-affine periodic and affine dissipative system, study the existence of affine periodic solutions. If f(t, x) satisfies f(t+T, x)=Qf(t, Q-1x), then f(t, x) is said to be Q-affine periodic. Let Q=I or Q=-I, Q-affine periodic function is T-periodic or anti-periodic.Affine dissipative system x’=f(t, x) is defined as follows:Let Bo>0, for any B>0, there exist M=M(B)>0, L=L(B)>0, such that for all|x0|≤B, there hold|x(t,x0)|≤M, Vt∈[0,L],|Q-mx(t+mT,x0)|≤B0, Vt+mT∈[L,∞), m∈Z.In this paper, Horn’s fixed point theorem is crucial for our proofs. In1970, W. A. Horn proved the following theorem according to the result given by F. E. Browder in1959:Let X be a finite-dimensional vector space, S0∈S1∈S2be the bounded convex subsets of X and S0, S2are closed, S1is open relative to S2. If for some integer m, continuous mapping f:S2�'X satisfies:(1) fj(S1)∈S2,1≤j<m-1,(2) fj(Si)∈S0, m≤j≤2m-1. Then there exist fixed points of f in So.For a clearer understanding of the work of this article. We will give a complete proof of Horn’s fixed point theorem. By constructing the sets S0, S1, S2, we get the result on the existence of Q-affine periodic solutions of affine periodic and affine dissipative system:Theorem1:If system x’=f(t, x) is Q-affine periodic and affine dissi-pative, then it admits Q-affine periodic solutions.and the result on the existence of Q-affine periodic solutions of affine periodic and affine dissipative system with Lyapunov function:Theorem2:Consider the system x’=f(t,x). Suppose that there exist Lyapunov function V:R1+×Rn�'R1+such that:i)V(t,x)∈C1;ii)V’(t,x)≤-W(t,x),|x|> M>0, W(t,x) is continuous on R1+×{|x|<M}, W(t, x)≥α>0for all|x|> M;iii)For all t, there holds: Then system admits Q-affine periodic solutions. According to the above results, we give some Q-affine periodic and affine dissipative equations, for example: x’+2x=e-t and which satisfies certain conditions. The existence of Q-affine periodic solutions of these equations can be confirmed clearly and quickly.We prove the existence of Q-affine periodic solutions for affine periodic and affine dissipative functional system:Theorem3:If system x’=F(t, x) is Q-affine periodic and affine dissi-pative, then it admits Q-affine periodic solutions.We also give Massera criterion for linear Q-affine periodic system x’=A(t)x+g(t). Theorem4:If the linear Q-affine periodic system x’=A(t)x+g(t) has a Q-affine bounded solutions x0(t), then it admits Q-affine periodic solution x. e co{Q-mx0(mT)}%=0.Based on this result, we discuss the method of upper and lower solutions for the existence of Q-ane periodic solutions for nonlinear Q-amne periodic system:Theorem5:Let Q€SO(n). Suppose that system x’=f(t,x) has Q-affine periodic solution. Ifi) system x’=g(t, x) has C1upper and lower solutions fi and a: n(t+T)=QQ(t) Vt;ii) g(t,x) is a Kamke-function with respect to Cl(t);iii) the following hold: Then system x’=f(t,x) has a Q-affine periodic solution x*(t) such that a(t)<x■(t)</3(t) Vi.