Abstract Linear Dynamical Equations On Time Scales

Throughout this paper, R, C, Z and N denote the real numbers, the complex numbers, the integers and the natural numbers respectively. We denote by X a real or complex Banach space, H a complex separable infinite dimensinal Hilbert space. Let B(X)(or B(H)) denote the algebra of all bounded.linear operators on X(or H). Let T always denote a times scales which is defined as an arbitrary nonempty closed subset of R.The theory of time scales was introduced by Hilger in his 1988 PhD disserta-tion [51]. The calculus of time scales unifies and extends the fields of discrete and continuous calculus. The delta derivative defined by Hilger [51,52] is equal to f' (the usual derivative) if T=R, and it is equal toΔf(the usual forward difference) if T=Z. So the study of dynamical equations on time scales allows a simultaneous treatment of differential and difference equations. And this field has attracted many researchers'attention [23,26,18,79,20]. In 2002, Atici and Guseinov [9] defined the nabla derivative by replacing the forward jump operatorσby backward jump operator p and researched the V dynamical equations on time scales.It is well known that linear systems of differential or difference equations are applied to many fields, such as mechanic systems, electric circuit and biological systems [23,79,10]. So it is meaningful to research the linear dynamical systems on time scales both in theory and applications.For linear dynamical equations on time scales, we mainly consider the explicit formula of solutions in matrix algebra and Banach algebra, and explor the existence and uniqueness of solutions in Banach space. Let us consider a linear systemwhere A(·), X(·) are functions from a time scale T to Mm.In the case that T=R or T=Z and A(·)= A, the study of the explicit constructions of solutions to equation (0.1) has attracted many researchers'attention [88,65,78,49,5,83,92]. With the development of dynamical equations on time scales, some results have been obtained in calculating the solutions of equation (0.1) on general time scales. In 1990, Hilger [52] defined a generalized exponential function on time scale and verified that it is the solution to the scalar form of equation (0.1) in the field of real numbers. For matrix equation (0.1) in Mm, Bohner and Peterson [23] defined the solution to equation (0.1) as the generalized matrix exponential function directily. They solved the exponential of a constant matrix on time scales by Putzer algorithm [88]. Then Harris algorithm and Leonard algorithm which were introduced firstly to calculate the matrix exponential eAt were extended to calculate the exponential of a constant matrix on time scales [15,95,96].'In 2007, Verde-Star [92] introduced an elementary method to solve linear matrix differential and difference equations and obtained explicit formulas for the classic exponential of a matrix eAt and An, n∈N.In chapter two, we extend the method in paper [92] to solve system (0.1). Let us introduce the calculating algorithm and the explicit constructions of solutions to equation (0.1). Letωbe any monic polynomial that is divisible by the minimal polynomial of A,i.e., where b0=1. The solution f of initial values problemis called the dynamic solution associated withω, where D denotes theΔderivative operator.Defineωk(x)=b0xk+b1xk-1+…+bk(0< k< n+1) be the Horner polynomials ofω. Then the solution of equation (0.1) can be represented by the polynomial of A with coefficients determined by the dynamic solution associated withω. Theorem 0.1 Let A be any square matrix,ωbe a monic polynomial as (0.2) andω0,ω1,…,ωn be the Horner polynomials ofω. If f(t) be the dynamic solution associated withω. Then we have the solution of equation (0.1), i.e., the generalized matrix exponential function as followsNote that the dependence on t of eA(t,0) is completely determined by the dynamic solution f(t), which is in turn determined by the polynomialω.Next, let us define the basic generalized exponential polynomials on the complex planewhere ea(t, s) is solution of dynamical equationWe define a commutative multiplication on{ga,k: a∈C,k∈N}, called the convolution product, as follows. whereWe obtain a direct construction of the dynamic solution associated withω. Theorem 0.2 Letω(t) be as in (0.2), andThen fωis the dynamic solution associated withω.Note that we can use the recurrence relation (0.8) to compute the summands in (0.4) and (0.5). The computation of fωusing (0.8) is a straightforward repeated application of the convolution formula (0.6) and (0.7).For V matrix dynamical equation, we can obtain the similar results as equation (0.1).It is more difficult to obtain the constructions of solutions of time varying linear dynamic system (0.1). In chapter three we consider the explicit formula of eA(·,s) with time varying function A(t). We extend the definition of generalized exponential function in the set of real numbers to the unital Banach algebra under some conditions.Let (?) denote a complex Banach algebra with unityⅠ[39] and A be an element in (?). We define the spectrum of A to be the setLet A:T→(?) be rd-continuous. We define the cylinder transformation of operator value function A(t) under some spectral condition.Let G be the single-valued and analytic branch of Logz. If the spectrum of A(·) satisfies the following condition Then the following cylinder transformation of A(·) is reasonable.Therefore, we can define the delta exponential function on time scales in Banach algebra as follows.Definition 0.3 Let A= A(t) be a rd-continuous function from a time scale T to a unital Banach algebra B (or a subalgebra of B) and satisfy the spectral conditions (S). Then we can define a delta exponential function of A(·) for T in Banach algebra as followsWe can verify that the cylinder transformation of A(·), i.e., (0.9), is well defined by Riesz Functional Calculus and Spectral Mapping Theorem.For B being a commutative unital Banach algebra, we obtain that, the delta exponential function defined above is just the explicit formula of the solution to system (0.1).Theorem 0.4 Let B be a commutative (or a commutative subalgebra of) Banach algebra with unity I. If A(·) is rd-continuous and satisfies the spectral conditions (S), then y(t)= eA(t, s) is the uniqueness solution of equation (0.1).Theorem 0.4 extends the result in scalar case to commutative Banach algebra case.We also consider the nabla exponential function in unital Banach algebra and obtain the similar results as definition 0.3 and theorem 0.4.In chapter four, we consider the existence and uniqueness of global solutions to linear dynamical equations in Hilbert space on time scales from a new point of view.where A:T-> B(H), y:T→H and y0∈H. we first define a class of operators u as u:={A∈B(H):σ(A)∩R-=(?)).Then u is the largest class of operators which ensuring that equation(0.10)has exactly one global solution for any time scale T under some conditions.Theorem 0.5 u is the largest one of those classes in B(H) which satisfy the fol-lowing condition(G).(G):"for any time scale T,if operator-valued function A(.):T→u is rd-continuous in stong operator opology,then equation(0.10)has exactly one global solution on T."Then we also describe the size of the class u on Hilbert space by characterizing its closure and interior.For A∈B(H),denote by N(A)and R(A)the kernel of A and the range of A,respectively.A is called a semi-Fredholm operator,if R(A) is closed and either nulA or nulA* is finite,where nulA:=dimN(A)and nulA* dim N(A*)；in this case,ind A:=nulA-nulA* is called the index of A.In particu-lar,if-∞0}.We denote by (?) the closure of u and u°the interior of u.Theorem 0.6 Let T∈B(H).Then(i)T∈(?) if and only if for eachλ≤0 eitherλ∈σlre(T)orλ∈ρs-F(0)(T)；(ii)T∈u°if and only ifσ(T)n[R-u{0)]=(?).It follows from Theorem 0.6 that the class u has interior points.Especially in matrix algebra Mm,one can verify that (?)=Mm.Hence we can deduce that u is a very large class of operators.Symmetrically,we discuss the existence and uniqueness of global solutions to nabla linear dynamical equations and obtain the similar results as theorem(0.5)and theorem(0.6) In 2006,Bohner and Guseinov[19]unified and extended the concepts of classic and discrete analytic functions to a concept of analytic functions on an arbitrary time scale complex plane T1+iT2 which will be called the△analytic functions, where T1 and T2 are arbitrary time scales.In 2007,Sinan[63]extended the results of paper[19]to the case of the V analytic functions.In chapter five,We consider the relationship between the△(or (?))analytic functions and the continuous analytic functions and derive the sufficient and nec-essary conditins for local△(or (?))analytic continuation in some cases. We also research the△(or V)analyticity for monomial pn(z)=zn and obtain some results.Let a function f:T1+iT2→C be△(or V)analytic on T1+iT2. For z0=x0+iy0∈T1k+iT2k,if there exists a classic analytic function g(z)on a neighborhood Uδ(z0)in the complex plan such that g(z)=f(z) and g'(z)=f△(z) (or g'(z)=f(?)(z))for any z∈Uδ(z0)n(T1k+iT2k).Then the function g(z)is said to be the local analytic continuation of the function f(z) at point z0.If there exists locally analytic continuation of f(z) at any point z0∈T1k+iT2k,then the function f(z) is said to be locally analytic continuable on T1+iT2.we derive the sufficient and necessary conditins for local△analytic continuation in some cases.Theorem 0.7 Let T1=[a,b] be a closed interval in R(T1 is permitted to be unbounded interval such as[a,+∞),(-∞,b]or(-∞,+∞)),and T2={yk}kN=1 be at most countable and without cluster point,here 2≤N≤+∞.DenoteΓk=[a,b)+i{yk},where k=1,2,3,…,N-1 for N<∞and k=1,2,3, for N=∞.Supppose f:T1+iT2→C be a△analytic function on T1+iT2 and f|Γk=u(x,yk)+ix(x,yx).Then there exists a unique local analytic continuation of f(z) onΓk if and only if the functions u(x,yk)and v(x,yk)are both real analytic function on (a,b),where k=1,2,3,…,N一1 for N<∞and k=1,2,3,…for N=∞.We get some results of the△(or V) analyticity for monomial pn(z)=zn, especially for n=3,n=4.Theorem 0.8 Suppose n∈N and n≥3.Let T1,T2 be two time scales with at least one right-scattered point in T1 and at least n right-dense points in T2.Then there exists a point z0∈T1+iT2 at which pn(z)=zn is not△analytic.Theorem 0.9 Let T1,T1 be two time scales,X1 and X2 be sets of all right-scattered points of T1 and T2 separately.Then there exists at most one point z0= x0+iy0 in T1,T2 such that p3(z)=z3 is△analytic at it.Furthermore,if p3(z)=x3 is△analytic at z0 ,thenσ1(x0)=一2x0；σ2(y0)=一2y0 must be satisfied.Theorem 0.10 Let T1,T2 be two time scales.If there exists right-scattered point in T1 and T2={yk}k=0∞is a monotonic sequence converging to y0.Then there exists a point z0 in T1+iT2 such that p4(z)=z4 is not△analytic at z0.We also obtain the parallel results of△analytic functions for V analytic functions on time scales.