Pseduo Almost Automorphic Problems in Differential EquationsAuthor: Wang YanMajor: Probability and StatisticsThesis advisor: Professor Song Lixing and Li YongIn 1924-1926, Danish mathematician H. Bohr studied the theory of almost periodic functions first, which has played an important roll in the study of harmonic analysis on group, topological and smooth dynamical systems. Since 1930's, mathematicians, such as Favard, Levitan, Simon, Fink ([16, 17, 35, 36, 33, 34, 52, 53]) and so on, have given more results in the work of almost periodic differential equations.Almost automorphic fuctions is a generalized almost periodic functions, which was first introduced by S. Bochner in 1955 ([20]). Mathematicians, such as Veech, Terras ([78, 79, 80, 76]) and so on, have given a series of foundation results in theire works. In symbolic dynamical systems, almost automorphic phenomena indicates somewhat complexity and chaos ([39]).Since it was not clear to interpret the importance of the almost automorphyin the study of differential dynamical systems, we have ignored the study of almost automorphic differential dynamical systems. In 1980's, Johnson ([49, 50]) found that the almost automorphic differential dynamical systems can be generated by some linea scalar ordinary differential equationswith almost periodic coefficients. During recetent 20 years, Shen, Yi, N'Guererkata, Diagana, Minh ([29, 41, 42, 73]) have given notable results.As the study of almost periodic and almost automorphic functions, Zhang introduced the notion of pseudo almost periodicity, as a generalizationto almost periodicity ([90, 91]). In Zhang's work, he has given the existenceof pseudo almost periodic problems. Recently, Aitdads, Arino, Ezzinbi, Diagana ([11, 12, 29, 30]) have made contribution in studying that. In this paper we present a conception of a more general class of functions called pseduo almost automorphic functions, that is, functions of the typewhere f is almost automorphic andφis continuous, bounded and M(‖φ‖) = 0. M(·) is the "asymptotic" mean value, defined byWe study the existence of pseudo almost automorphic solutions for nonautonomousdifferential equations. Now let us introduc the main results of the paper.Almost automorphic and pseudo almost automorphic functionsTheorem 1 Suppose that f∈AA(R×X, X), and f(t,·) is uniformly continuousin bounded subsets of Banach space X uniformly in t∈R. Letφ: R→X is almost automorphic. Then the map F : R→X defined as F(t) = f(t,φ(t)) is almost automorphic.Theorem 2 Let X is a Banach space. Suppose thatΦ: R→Gl(m,m), f: R→R~m are almost automorphic. ThenΦf : R→R~m, t(?)Φ(t)f(t) is almost automphic. Definition 1 Setf∈BC(R, X) is called pseudo almost automorphic ifwith g∈AA{X) andφ∈AA_0(X).Theorem 3 Assume f, f_1, and f_2 art pseudo almost automorphic andλisany scalar, then the following holds true. i)λf,f_1 + f_2 are pseudo almost automorphic;ii) f_τ(t) = f(t +τ),τ∈R., is pseudo almost automorphic;iii) f(t) = f(-t), t∈R is pseudo almost automorphic;iv) sup_(t∈R)‖f(t)‖<∞.Theorem 4 Let f∈PAA(X) be decomposed as f = g +φwith g∈AA(X) andφ∈AA_0(X). Then we haveTheorem 5 If f∈PAA(X), then we havethat is, the function f∈PAA(X) has a unique decomposition.Theorem 6 PAA(X) is a Banach space with the sup normDefinition 2 f:R×X→X is called pseudo almost automorphic, if there exist functions g,φ:R×X→X with g(·, x)∈AA(X) for each x∈X andφ∈AA_0(R×X, X) such that f = g +φ, whereThoerem 7 Let X, Y be Banach spaces andφ: R→X be a pseudo almost automorphic function. If f : X→Y satisfies the following conditions:(f1) f(K) is bounded for every bounded subset K (?) X; (f2) f is uniformly continuous in each bounded subset of X uniformly in t∈R. More explicitly, givenε> 0 and K (?) X bounded, there exists aδ> 0 such that for x, y∈K and‖x - y‖<δThen f(φ(·)): K→Y is pseudo almost automorphic.Theorem 8 Let f : R×X→X andφ: R→X be pseudo almostautomorphic. Suppose that the following conditions hold:(f1') f(R,K) is bounded for every bounded subset K (?) X.(f2') f is uniformly continuous in each bounded subset of X uniformly in t∈R.Then F : R→X, F(t) = f(t,φ(t)) is pseudo almost automorphic.Pseudo almost automorphic problems in ordinary differential equationsConsider the linear ordinary differential equation:and nonlinear ordinary differential equations:Theorem 1 Let matrix A(t) : R→Gl(m, m) is almost automorphic, that is, for any real sequence {t_n~'}_(n=1)~∞there exists a subsequence {t_n}_(n=1)~∞, such that hold pointwisely. Suppose that the ordinary differential equation (1) has an exponential trichotomy with projections P, Q and constants K,α. Then the ordinary differential equationposses an exponential trichotomy. Furthermore, X(t)PX~(-1)(t), X(t)QX~(-1)(t) are almost automorphic, where X(t) is the fundamental matrix for the linear differential equation (1) with X(0) = 1.Theorem 2 Let the ordinary differential equation (1) has an exponential trichotomywith projections P, Q and constants K, a. Suppose that the matrix A(t) :R→Gl(m,m) is almost automorphic, and f∈PA4(R~m). Then the ordinary differential equation (2) admits a pseudo almost automorphic solution.Pseudo almost automorphic problems in evolution equationsConsider the nonlinear evolution equationsandSuppose that A(t), U(t, s) satisfy the following conditions:(H1) (a) There exists constantsω≥0,φ∈(2/π,π), L, M≥0, andμ, v∈(0,1] withμ+ v > 1 such thatand (b) The evolution family (U(t, s))_(t≥s) is generated by (A(t))_(t∈R).(H2) The evolution family (U(t, s))_(t≥s)has an exponential stable:(H3) R(ω,A(·))∈AP(R,(?)(X)).Definition 1 A function u∈D(A(t)) is called s-mild solution of equation (5), ifTheorem 1 LetX be a Banach space, and f∈PAA(X). Suppose that (H1), (H2) and (H3) hold. Then Eq. (5) has a unique pseduo almost automorphic s-mild solution.Theorem 2 Let (H1), (H2) and (H3) hold. Assume that f∈PAA(R×X, X) satisfies conditions (f1'), (f2'), and it is Lipschitz continuous in x uniformly for all t∈R, i. e.where L <α/N. Then Eq. (6) has a unique pseudo almost automorphic s-mildsolution.Theorem 3 Consider following equations:andwhere f,g∈PAA(R×X, X) satify the conditions (f1') and (f2'). Assume the following conditions hold:(i) There exists an open bounded subsetΩ(?) PAA(X) such that every possiblesolution u of Eq. (8) satisfies u (?) (?)Ω. (ii)whereΦ(u)(t) =∫_(-∞)~t U(t, s)f(s,u(s))ds is s-mild solution of Eq. (7).(iii) Conditions (H1), (H2) and (H3) hold, and for any t∈R, h > 0, (U(t, s))_(t-s≥h) is compact.Then Eq. (8) has a pseudo almot automrophic s- mild solution u∈Ω.Theorem 4 Consider following equations:andwhere f∈PAA(X), g∈PAA(R×X, X) satify the conditions (f1') and (f2'). Assume the following conditions hold:(i) There exists an open bounded subsetΩ(?) PAA(X) such that every possiblesolution u of Eq. (10) satisfies u (?)(?)Ω.(ii) Eq. (9) has a unique pseudo almost automorphic s-mild solution u∈Ω.(iii) Conditions (H1), (H2) and (H3) hold, and for any t∈R, h > 0, (U(t,s))_(t-s≥h) is compact.Then Eq. (10) has a pseudo almot automrophic s- mild solution u∈Ω.