Affine-periodic Solutions For Differential Equations With Exponential Dichotomy And Exponential Trichotomy

Exponential dichotomy of differential equations was first s-tudied by Lyapunov and Poincare in the late nineteenth century. It is an important tool to study non-autonomous dynamical sys-tems. Perron ([24]) developed the exponential dichotomy of linear differential equations and studied the problem of conditional sta-bility of linear systems. Since then, exponential dichotomy has been widely studied and applied in the field of differential equa-tions; for some literature, see [12,13,25] and references therein. The exponential trichotomy is a generalization of the concept of exponential dichotomy. Sacker and Sell ([28]) established a funda-mental theory of trichotomy. Elaydi and Hajek ([17]) introduced and studied the exponential trichotomy of differential systems. The exponential dichotomy and exponential trichotomy are very important for the qualitative theory. So it is very necessary to in-vestigate exponential dichotomy and exponential trichotomy. In this thesis, we give in-depth study of exponential dichotomy, ex-ponential trichotomy and Affine-periodic solutions for differential equations.In order to make our paper more independent, this disserta-tion is divided by four chapters. In the first chapter, we first in-troduce some important works, and some useful definitions, such as affine-periodic, and some useful properties. Then, we recall the notions of exponential dichotomy and exponential trichoto-my. Finally, pointed out in this article, we take the affine-periodic solutions as the research object, discuss the differential equation-s with exponential dichotomy and exponential trichotomy, and investigate the existence of affine-periodic solutions and pseudo affine-periodic solutions.In the second chapter, we discuss the first-order affine-periodic system with exponential dichotomy. First, we consider first-order nonhomogeneous linear differential equation x’=A(t)x+f(t), where A(t):R1â†'Rn×Rn is a bounded and continuous function, f(t):R1â†'Rn is a (Q,T)-affine-periodic function. If the cor-responding homogeneous linear differential equation x’=A(t)x satisfies exponential dichotomy,we give the following theoremTheorem 1 If the homogeneous linear differential equation x’=A(t)x has an exponential dichotomy with projection P,A(t), f(t) are(Q,T)-affine-periodic,then the nonhomogeneous linear differential equation x’=A(t)x+f(t)admits a(Q,T)-affine-periodic solutionSecond,we consider first-order nonhomogeneous semi-linear differential equation x’=A(t)x+g(t,x(t)),where g:R1×Rnâ†' Rn is coninuous,A(t)and g(t,x)are(Q,T)-affine-periodic.If the homogeneous linear differential equation x’=A(t)x has an exponential dichotomy,we haveTheorem 2 Under the assumptions of Proposition 2.3,lin-ear differential equation x’=A(t)x has an exponential dichotomy with projection P and constants K,L,α,β>0.Moreover,as-sume that A(t),g(t,x) are(Q,T)-affine-periodic periodic,g(t,x) is bounded and satisfies that |g(t,x)-g(t,y)|≤N|x-y|,(?)t,x,y, where Q∈GL(n),N>0 satisfies a given condition. Then equation x’=A(t)x+g(t,x(t))admits a unique(Q,T)-affine-periodic solution.The conditions of g(t,x(t))in Theorem 2 can be replaced by condition of linear growth condition. We make the following restatement of Theorem 2.Theorem 3 Assume the linear differential equation x’= A(t)x has an exponential dichotomy with projection P and con-stants K, L, α,β> 0. Moreover, assume that A(t), g(t,x) are (Q,T)-affine-periodic, where Q∈O(n). If g(t,x) satisfies condition (C1), then equation x’= A(t)x+g(t,x(t)) admits a (Q,T)-affinc-periodic solution.In the third chapter, we give some applications in higher order differential equations. We consider the n-dimensional sec-ond order differential equation x"+p(t)x’+q(t)x= e(t), where p(t), q(t):R1â†'Rn×n, e(t):R1â†'Rn are continuous and (Q,T)-affine-periodic, we have the following resultTheorem 4 Assume that p(t),q(t) and e(t) are continuous (Q, T)-affine-periodic functions, F(t) and G(t) are bounded for all t∈R1, and F(t) satisfies some given conditions. Assume p(t) and q(t) satisfy one of the following:1). p(t) and q(t) are positive definite or negative definite for all t∈R1;2). q(t) is negative definite for all t∈R1Then equation x"+p(t)x’+q(t)x=e(t) admits a (Q, T)-affine- periodic solution.Next, we consider the following n-dimensional higher order differential equation x(m)= a(t)x+e(t), where a(t):R1â†'Rn×n, e(t):R1â†'Rn are continuous and (Q, T)-affine-periodic. Specif-ically, we have the following theorem.Theorem 5 Assume that a(t) and e(t) are continuous and (Q,T)-affine-periodic functions, and for all t∈R1, A(t), G(t) are bounded. Assume1). when m= 4k, k∈Z, a(t) is negative definite for all t∈R1;2). when m= 4k+2, k∈Z, a(t) is positive definite for all t∈R1;3). when m= 4k+1 or 4k+3, k∈Z, a(t) is positive definite or negative definite for all t∈R1.Then equation x(m)= a(t)x+e(t) admits a (Q, T)-affine-periodic solution.For the exponential trichotomy, we consider the existence of (Q, T)-affine-periodic solutions for differential equations. We have the following result about the semilinear differential equation x’= A(t)x+g(t,x(t)), where g:R1×â†'Rn is a continuous function, A(t) and g(t,x) are (Q,T)-affine-periodic functions, and the corresponding homogeneous linear differential equation is x’= A(t)x.Theorem 6 Assume that equation x’= A(t)x has an expo-nential trichotomy with projections P1,P2 and constants K,α. Moreover, assume that A(t), g(t,x) are (Q,T)-affine-periodic, g(t,x) is bounded and satisfies that where Q E GL(n), N>0 is a constant such that Then the equation x’= A(t)x+g(t,x(t)) admits a unique (Q, T)-affine-periodic solution.Next, let us consider the existence of pseudo affine period-ic solutions. We first introduce the definition of pseudo affine-periodic solutions. Furthermore, we have the following theorem on the existence of pseudo (Q, T)-affine-periodic solutions.Theorem 7 For the equation x’= A(t)x+g(t, x(t)), assume that A(t) is (Q, T)-affine-periodic and g(t, x) is pseudo (Q, T)-affine-periodic with decomposition g(t,x)= g1(t,x)+g2(t,x), where Q∈GL(n), T>0 is a constant, g1(t,x)∈CT and 92(t,x)∈C0.Moreover, g(t,x) and g1(t,x) are uniformly con-tinuous in any bounded subset of Rn uniformly for t∈R1, g(t, x) is bounded and satisfies that |g(t,x)-g(t,y)|≤N|x-y|,(?)t, x, y, where N>0 is a constant. If the corresponding homogeneous lin-ear differential equation x’= A(t)x has an exponential trichotomy with projections P1,P2 and constants K, a satisfying some given conditions, then equation x’= A(t)x+g(t, x(t)) admits a unique pseudo (Q, T)-affine-periodic solution.We also have the following results.Corollary 1 Assume that equation x’= A(t)x has an ex-ponential trichotomy with projections P1, P2 and constants K, α. Moreover, assume that A(t), g(t,x) are (Q,T)-affine-periodic, g(t,x) is uniformly continuous with respect to x uniformly for t∈R1 and satisfies that |g(t, x)|≤a|x|+b,(?)t, x, where Q∈O(n):a,b are positive constants such that 2ka/α<1. Then equation x’= A(t)x+g(t,x(t)) admits a unique pseudo (Q, T)-affine-periodic solution.Corollary 2 For the equation x’=A(t)x+g(t, x(t)), assume that A(t) is (Q,T)-affine-periodic and g(t,x) is pseudo (Q,T)-affine-periodic with decomposition g(t,x)= g1(t,x)+g2(t,x), where Q ∈ GL(n), T>0 is a constant, g1(t,x)∈CT and g2(t,x)∈C0. Moreover, g(t,x) is uniformly continuous in any bounded subset of Rn uniformly for t∈R1 and satisfies that |g(t,x)|≤a|x|+b,(?)t, x, where a, b> 0 are constants. If the corresponding homogeneous linear differential equation x’=A(t)x has an exponential tri-chotomy with projections Pi, P2 and constants K,α satisfying the condition that 2Ka/α<1, then equation x’= A(t)x+g(t,x(t)) admits a unique pseudo (Q,T)-affine-periodic solution.These complete the thesis.