DSpace at VNU: ON INITIAL AND BOUNDARY VALUE PROBLEMS FOR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES

Acta Math Vietnam DOI 10.1007/s40306-013-0026-z ON INITIAL AND BOUNDARY VALUE PROBLEMS FOR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES Nguyen Chi Liem Received: 29 December 2011 / Revised: 10 September 2012 / Accepted: 24 September 2012 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013 Abstract This paper is concerned with the index concept, the unique solvability of the initial value problem, the two point boundary-value problem and Green’s function for a class of linear implicit dynamic equations with index-1 on time scales The results are a generalization of the previous ones for differential and difference-algebraic equations Keywords Time scales · Implicit dynamic equation · Linear dynamic equation · Index · Boundary-value problem · Green’s function Mathematics Subject Classification (2000) 39A11 · 34A09 · 65L80 · 65F20 · 34D20 Introduction In the last decades, the theory of differential-algebraic equations (DAEs for short) has been an intensively discussed field in both theory and practice The general form of DAEs is f t, x (t), x(t) = 0, (1) At x (t) = Bt x(t) + qt , (2) and its linearization has the form where A and B are given matrix functions Equations (1) and (2) can be seen in many real problems, such as in electric circuits, chemical reactions, vehicle systems, If the matrices At are invertible for all t ∈ R, we can multiply both sides of (2) by A−1 t to obtain an ordinary differential equation However, in the case where there is at least one t0 ∈ R such that At0 is singular, some further assumptions need to be posed One of the ways B N.C Liem ( ) Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam e-mail: liemlkqn2005@gmail.com N.C LIEM to investigate (2) is to introduce the index concept of the equation Based on this concept, we can study (2) by decomposing it into an ordinary differential equation and algebraic relations Results on the solvability of the Cauchy problem for (2) can be found in [10]; for the boundary value problems, we can refer to [17] Together with the theory of differential-algebraic equations, there has been a great interest in singular difference equations (SDEs) (also referred to as descriptor systems, implicit difference equations) because of their appearance in many practical areas, such as in the Leontiev dynamic model of multisector economy, the Leslie population growth model, singular discrete optimal control problems and so forth (see [6, 7]) On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and partial differential-algebraic equations , which have already attracted much attention of researchers (cf [6, 10, 13]) Bondarenko and his colleagues in [5] considered a special class of implicit nonautonomous difference equation T (n)x(n + 1) + x(n) = f (n), where T (n) are degenerate matrices, and established the solvability of initial value problems (IVPs) and periodic boundary-value problems (BVPs) for this special class of SDEs The index notion for linear SDEs with time-varying coefficients An x(n + 1) = Bn x(n) + qn (3) was introduced in [8, 16] and the solvability of IVPs as well as multipoint BVPs are studied in [1, 3] Later on, the index notion has been extended to nonlinear cases f (n, x(n + 1), x(n)) = [2] There is a close relation between linear SDEs and linear DAEs, namely, the explicit Euler method applied to a linear index-1 DAE leads to a linear index-1 SDE (see [1, 3]) and the unique solutions of the discretized IVP/BVP converge to the solutions of the corresponding continuous problems Further, in recent years, to unify the continuous and the discrete analyses or to describe the process of numerical calculation with non-constant steps, a new theory was born and is more and more extensively concerned, that is, the theory of the analysis on time scales The most popular examples of time scales are T = R and T = Z Using the “language” of time scales, we rewrite (2) and (3) in the form At x Δ = Bt x + qt , (4) or in the general form f t, x Δ (t), x(t) = 0, with t in time scale T and Δ being the derivative operator on T A natural question is whether the existing results for (2) and (3) can be extended and unified for the implicit dynamic equations of the form (4) The purpose of this paper is to answer a part of that question We will study the solvability of the Cauchy problem and some matters concerning the boundary-value problem of (4) The organization of this paper is as follows In Sect we summarize some results about the analysis on time scales In Sect 3, we introduce the index-1 concept and deal with the Cauchy problem of linear implicit dynamic equations (4) (LIDEs) The technique we use in this section is somewhat similar to the one in [8, 10] However, we need some improvements because of the complicated structure of a time scale Section deals with the solution uniqueness for the two point boundary-value problem and constructs the Green function IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES Preliminaries This section surveys some notions on the theory of the analysis on time scales which was introduced by Stefan Hilger in 1988 [11] A time scale is a nonempty closed subset of the real numbers R, and we usually denote it by the symbol T We assume throughout that a time scale T is endowed with the topology inherited from the real numbers with the standard topology We define the forward jump operator and the backward jump operator σ, ρ : T → T by σ (t) = inf{s ∈ T : s > t} (supplemented by inf ∅ = sup T) and ρ(t) = sup{s ∈ T : s < t} (supplemented by sup ∅ = inf T) The graininess μ : T → R+ ∪ {0} is given by μ(t) = σ (t) − t A point t ∈ T is said to be right-dense if σ (t) = t and t < sup T, right-scattered if σ (t) > t , left-dense if ρ(t) = t and t > inf T, left-scattered if ρ(t) < t , and isolated if t is right-scattered and left-scattered For every a, b ∈ T, by [a, b] we mean the set {t ∈ T : a t b} The set Tk is defined to be T if T does not have a left-scattered maximum; otherwise it is T without this left-scattered maximum Let f be a function defined on T, valued in Rm We say that f is delta differentiable (or simply: differentiable) at t ∈ Tk provided there exists a vector f Δ (t) ∈ Rm , called the derivative of f , such that for all > there is a neighborhood V around t with f (σ (t)) − f (s) − f Δ (t)(σ (t) − s) |σ (t) − s| for all s ∈ V If f is differentiable for every t ∈ Tk , then f is said to be differentiable on T If T = R then delta derivative is f (t) from continuous calculus; if T = Z, the delta derivative is the forward difference, Δf , from discrete calculus A function f defined on T is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point The set of all rd-continuous functions from T to a Banach space X is denoted by Crd (T, X) A matrix function f from T to Rm×m is said to be regressive if det(I + μ(t)f (t)) = for every t ∈ Tk Theorem (See [4]) Let A(·) be an rd-continuous m × m-matrix function Then, for any t0 ∈ Tk , the IVP x Δ = A(t)x, x(t0 ) = x0 (5) t0 Further, if A(·) is regressive, this solution exists has a unique solution x(·) defined on t on t ∈ Tk The solution of the corresponding matrix-valued IVP X Δ (t) = A(t)X(t), X(t0 ) = I , which is called the Cauchy operator of the dynamic equation (5) and denoted by ΦA (t, t0 ), always exists for t t0 , even A(·) is not regressive (see [12, 19]) If we suppose further that A(·) is regressive, the Cauchy operator ΦA (t, t0 ) is defined for all t, t0 ∈ Tk It is seen that any solution x(·) of the dynamic equation (5) can be written as x(·) = ΦA (·, t0 )x0 and the cocycle property ΦA (t, τ ) = ΦA (t, s)ΦA (s, τ ) is valid for all τ s t t If A(t) commutes with its integral t0 A(s)Δs, (in particular, A(t) ≡ constant matrix satisfies this straightforwardly) then we denote eA (t, t0 ) instead of ΦA (t, t0 ) Theorem (Constant variation formula, see [12]) Let A : Tk → Rm×m and f : Tk × Rm → Rm be rd-continuous and there exists a solution x(t), t t0 for the dynamic equation x Δ = A(t)x + f (t, x), x(t0 ) = x0 Then t x(t) = ΦA (t, t0 )x0 + ΦA t, σ (s) f s, x(s) Δs, t0 t t0 (6) N.C LIEM We refer to [12, 18] for more information on analysis on time scales Linear implicit dynamic equations on time scales Let T be a time scale We consider the linear implicit dynamic equation of the form At x Δ = B t x + qt , t ∈ T (7) The homogeneous equation associated to (7) is At x Δ = Bt x, (8) where A , B ∈ Crd (Tk , Rm×m ), q ∈ Crd (Tk , Rm ) In the case where the matrices At are into obtain an ordinary vertible for every t ∈ T, we can multiply both sides of (7) by A−1 t dynamic equation −1 x Δ = A−1 t B t x + A t qt , t ∈ T, which has been well studied If there is at least a t such that At is singular, we cannot solve explicitly the leading term x Δ In fact, we are concerned with a so-called ill-posed problem where the solutions of the Cauchy problem may exist only on a submanifold or even they not exist One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of equation We introduce the so-called index-1 of (7) Suppose that rank At = r for all t ∈ T and let Tt ∈ GL(Rm ) be such that Tt |ker At is an isomorphism between ker At and ker Aρ(t) ; T ∈ Crd (Tk , Rm×m ) Let Qt be a projector onto ker At satisfying Q ∈ Crd (Tk , Rm×m ) We can find such operators Tt and Qt in the following way: let At possess a singular value decomposition At = Ut Σt Vt , where Ut , Vt are orthogonal matrices and Σt is a diagonal matrix with singular values σt1 σt2 · · · σtr > on its main diagonal Since A ∈ Crd (Tk , Rm×m ), in the above decomposition of At we can choose Vt ∈ Crd (Tk , Rm×m ) (see [7]) Hence, by putting Qt = Vt diag(0, Im−r )Vt and Tt = Vρ(t) Vt , we obtain Qt and Vt as the requirement Let Qt and Tt be such matrices and put Pt := I − Qt We suppose further that Qρ(t) is rd-continuously differentiable on Tk It is known that ρ(σ (t)) = t if and only if t is not right-dense and left-scattered at the same time We consider the case where t = t0 is right-dense and left-scattered at the same time (ρ(t0 ) < t0 = σ (t0 )) Then, from the continuity of Qρ(·) and Q· at t0 we get the equalities limt→t0 Qρ(t) = Qρ(t0 ) and limt→t0 Qρ(t) = Qt0 Therefore, Qρ(t0 ) = Qt0 Thus, by the above assumptions of the projector Qt we always have Qρ(σ (t)) = Qt for all t ∈ Tk From the relation Pρ(t) x(t) Δ = Pρ(σ (t)) x Δ (t) + (Pρ(t) )Δ x(t) = Pt x Δ (t) + (Pρ(t) )Δ x(t), for all t ∈ Tk , we get At x Δ (t) = At Pt x Δ (t) = At Pρ(t) x(t) Δ − (Pρ(t) )Δ x(t) IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES Therefore, the implicit dynamic equation (7) can be rewritten as At (Pρ(t) x)Δ = At (Pρ(t) )Δ + Bt x + qt , t ∈ Tk Thus, we should look for solutions of (7) from the space CN1 : CN1 Tk , Rm = x(·) ∈ Crd Tk , Rm :Pρ(t) x(t) is differentiable at every t ∈ Tk Note that CN1 does not depend on the choice of the projector function since the relations Pt Pt = Pt and Pt Pt = Pt are true for each two projectors Pt and Pt along the space ker At Let St = x ∈ Rm : Bt x ∈ imAt Under these notations, we have: Lemma [9] The following assertions are equivalent kerAρ(t) ∩ St = {0}, The matrix Gt = At − Bt Tt Qt is nonsingular, Rm = ker Aρ(t) ⊕ St for all t ∈ Tk Lemma [9] Suppose that the matrix Gt is nonsingular Then we have the following assertions: Pt = G−1 t At , Qt = (9) −G−1 t B t Tt Q t , − Tt Qt G−1 t Bt (10) is the projector onto ker Aρ(t) along St , −1 Pt G−1 t Bt = Pt Gt Bt Pρ(t) , Qt G−1 t Bt = Tt Qt G−1 t Qt G−1 t Bt Pρ(t) (11) (12) − Tt −1 Qρ(t) , does not depend on the choice of Tt and Qt (13) (14) Definition The LIDE (7) is said to be of index-1 if for all t ∈ Tk , the following conditions hold: (i) rankAt = r= constant (1 (ii) ker Aρ(t) ∩ St = {0} r m − 1), Assume that (7) is of index-1 We now describe briefly the decomposition technique for (7) By Lemma 1, since (7) has index-1, Gt is nonsingular for all t ∈ Tk Multiplying (7) by −1 Pt G−1 t and Qt Gt , respectively, yields Δ −1 −1 Pt G−1 t A t x = P t G t B t x + P t G t qt , −1 Δ −1 Qt Gt At x = Qt Gt Bt x + Qt G−1 t qt Applying (9), we have −1 Pt x Δ = Pt G−1 t B t x + P t G t qt , −1 −1 = Qt G t B t x + Qt G t qt (15) N.C LIEM Due to (12) and (13), (15) becomes ⎧ Δ Δ −1 −1 ⎪ ⎨(Pρ(t) x) = (Pρ(t) ) (I + Tt Qt Gt Bt )(Pρ(t) x) + Pt Gt Bt (Pρ(t) x) Δ −1 + (Pt + (Pρ(t) ) Tt Qt )Gt qt , ⎪ ⎩ −1 Qρ(t) x = Tt Qt G−1 t Bt Pρ(t) x + Tt Qt Gt qt (16) Therefore, with u(t) := Pρ(t) x(t), v(t) = Qρ(t) x(t), (16) becomes an ordinary dynamic equation on T −1 Δ −1 uΔ = (Pρ(t) )Δ I + Tt Qt G−1 t Bt u + Pt Gt Bt u + Pt + (Pρ(t) ) Tt Qt Gt qt , (17) and an algebraic relation −1 v = Tt Qt G−1 t B t u + Tt Q t G t q t (18) Let t0 ∈ Tk Solving u(t) from (17) with the initial condition u(t0 ) = Pρ(t0 ) x0 and using the relation (18), we get an expression of the solutions of the index-1 LIDE (7) x(t) = u(t) + v(t) for t t0 Inspired by the above decoupling procedure, we state initial conditions for the index-1 LIDE (7) as Pρ(t0 ) x(t0 ) − x0 = 0, x0 ∈ Rm given (19) It follows that u(t0 ) = Pρ(t0 ) x(t0 ) = Pρ(t0 ) x0 , but we not expect x(t0 ) = x0 as in the case of ordinary dynamic equations on time scales Denote Qtcan := −Tt Qt G−1 t Bt , Ptcan := I − Qtcan By (11), Qtcan projects onto ker Aρ(t) along St and is called the canonical projector for the index-1 case Note that Qtcan is rdcontinuous and independent from the choice of Qt and Tt The solutions of (7) with the initial condition (19) are represented by −1 x(t) = Pρ(t) x(t) + Qρ(t) x(t) = (I + Tt Qt G−1 t Bt )u + Tt Qt Gt qt = Ptcan u(t) + Tt Qt G−1 t qt , t (20) t0 , where u ∈ Crd solves from the inherent ordinary dynamic equation (17) with the initial condition u(t0 ) = Pρ(t0 ) x0 By multiplying both sides of the homogeneous equation associated to (17) with Qt and using the fact that = (Qρ(t) Pρ(t) )Δ = Qt (Pρ(t) )Δ + (Qρ(t) )Δ Pρ(t) =⇒ Qt (Pρ(t) ) = −(Qρ(t) ) Pρ(t) , Δ Δ one has Qt uΔ = Qt (Pρ(t) )Δ Ptcan u = −(Qρ(t) )Δ Pρ(t) Ptcan u = −(Qρ(t) )Δ Pρ(t) u IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES Further, since Qt uΔ = (Qρ(t) u)Δ − (Qρ(t) )Δ u we get (Qρ(t) u)Δ = (Qρ(t) )Δ (Qρ(t) u) Hence, if Qρ(t0 ) u(t0 ) = then Qρ(t) u(t) = for all t property: if x(t0 ) ∈ im Pρ(t0 ) t0 Therefore, (17) has the invariant then x(t) ∈ im Pρ(t) for all t ∈ Tk (21) Consider the homogeneous equation (8), i.e., qt ≡ Let Φ0 (t, t0 ) be the matrix solution of the dynamic equation −1 (Φ0 (t, t0 ))Δ = (Pρ(t) )Δ (I + Tt Qt G−1 t Bt )Φ0 (t, t0 ) + Pt Gt Bt Φ0 (t, t0 ), Φ0 (t0 , t0 ) = I t t0 , Then, due to (20), the solution of the matrix equation At (Φ(t, t0 ))Δ = Bt Φ(t, t0 ), Pρ(t0 ) (Φ(t0 , t0 ) − I ) = t t0 , can be expressed by the formula Φ(t, t0 ) = I + Tt Qt G−1 t Bt Φ0 (t, t0 )Pρ(t0 ) = Ptcan Φ0 (t, t0 )Pρ(t0 ) , t t0 (22) It is easily verified that ker Φ(t, t0 ) = ker Aρ(t0 ) and im Φ(t, t0 ) = im Ptcan = St hold The matrix solution Φ(t, t0 )t t0 is called the Cauchy operator of (8) Further, due to the invariant property (21) of the solutions of (17), we have Pρ(t) Φ(t, s) = Pρ(t) Ptcan Φ0 (t, s)Pρ(s) = Φ0 (t, s)Pρ(s) , t s (23) Therefore Φ(t, s)Φ(s, τ ) = Φ(t, τ ) for all τ s t, and the unique solution of (7) with the initial condition (19) can be given by the constant variation formula x(t) = Ptcan Φ0 (t, t0 )Pρ(t0 ) x0 t + Ptcan Φ0 t, σ (s) Pρ(σ (s)) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t0 + Tt Qt G−1 t qt , t t0 , or equivalently t x(t) = Φ(t, t0 )Pρ(t0 ) x0 + Φ t, σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t0 + Tt Qt G−1 t qt , t (24) t0 Now suppose that u = Pρ(t) x satisfies the homogeneous equation associated (17) corresponding to some operators Tt and Qt , i.e., (Pρ(t) x)Δ = (Pρ(t) )Δ Ptcan Pρ(t) x + Pt G−1 t Bt Pρ(t) x N.C LIEM Let Tt be another linear transformation from Rm onto Rm such that Tt |ker At is an isomorphism between ker At and ker Aρ(t) and Qt be a projector onto ker At such that Qt is rd-continuous and Qρ(t) is rd-continuously differentiable By denoting Pt = I − Qt , Gt = At − Bt Tt Qt , we have (Pρ(t) x)Δ = (Pρ(t) Pρ(t) x)Δ = Pt (Pρ(t) x)Δ + (Pρ(t) )Δ Pρ(t) x Δ = Pt (Pρ(t) )Δ Ptcan Pρ(t) x + Pt G−1 t Bt Pρ(t) x + (Pρ(t) ) Pρ(t) x = (Pρ(t) Pρ(t) )Δ Ptcan Pρ(t) Pρ(t) x − (Pρ(t) )Δ Pρ(t) Ptcan Pρ(t) x Δ + Pt G−1 t Bt Pρ(t) x + (Pρ(t) ) Pρ(t) x = (Pρ(t) )Δ Ptcan Pρ(t) x − (Pρ(t) )Δ Pρ(t) x Δ + Pt G−1 t Bt Pρ(t) x + (Pρ(t) ) Pρ(t) x = (Pρ(t) )Δ Ptcan Pρ(t) x + Pt G−1 t Bt Pρ(t) x −1 Moreover, it is easy to prove that Pt G−1 t = Pt Gt which implies −1 −1 −1 Pt G−1 t Bt Pρ(t) x = Pt Gt Bt Pρ(t) x = Pt Gt Bt Pρ(t) Pρ(t) x = Pt Gt Bt Pρ(t) x Therefore we obtain (Pρ(t) x)Δ = (Pρ(t) )Δ Ptcan Pρ(t) x + Pt G−1 t Bt Pρ(t) x Furthermore, we also have the relation Qρ(t) x = −Qtcan Pρ(t) x ⇒ Qρ(t) x = −Qtcan Pρ(t) x This shows that the Cauchy operator of (8) does not depend on the choice of Tt and Qt , and hence neither the expression of x by (24) Consider the case where the right-hand side of the homogeneous equation associated to (17), i.e., the matrix At = (Pρ(t) )Δ Ptcan + Pt G−1 t Bt , is regressive (obviously, At is rd-continuous) With this assumption, the inherent dynamic equation (17) has a unique solution defined on Tk Thus, Theorem Given an index-1 LIDE (7), then for each t0 ∈ Tk , x0 ∈ Rm , q ∈ Crd (Tk , Rm ), the LIDE (7) with the initial condition (19) is uniquely solvable for t t0 , further, with the assumption At to be regressive, exactly one solution of the homogeneous equation x(t) of (8) passes through each x0 ∈ St0 at t0 Remark 1 When T = R (ρ(t) = t for all t ∈ R) we choose Tt = −I to see the result mentioned in [10] For the case T = Z, the result can be seen in [3] IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES On the solvability of IVP for the quasi-linear implicit dynamic equations At x Δ = Bt x + f (t, x), with small perturbation f (t, x) and for the equations At x Δ = f (t, x), with the assumption of differentiability for f (t, x), we can refer to [9] To finish this section, we give an example ∞ k=0 [2k, 2k Example Consider the time scale T = qt , with At = t , t Bt = t −1 + 1] and the equation At x Δ = Bt x + t −1 , −1 qt = We have ker At = span{(1, −1)T }, rank At = for all t ∈ T It is easy to verify that Qt = 1 −1 is a projector onto ker At Let us choose Tt = I and observe that −1 1 t −1 − t t G t = A t − B t Tt Q t = t −1 1 −1 −1 −1 = t− t+ Since det Gt = = 0, (7) has index-1 We get t + 12 −t + 12 G−1 t = Tt Qt G−1 t Bt = t −2 , −t + Tt Qt G−1 t qt = Note that At = −1 , 1 t−1 t−1 t−1 t−1 Ptcan = t2 − t −t + t t2 − t + −t + t t2 − t + , −t + t − t2 − t + −t + t is regressive (and rd-continuous) Indeed, det I + μ(t)At = 2t if t ∈ if t ∈ ∞ k=0 [2k, 2k + 1) ∞ k=0 {2k + 1} =0 for all t ∈ T Therefore the equation At x Δ = Bt x + qt with the initial condition Pρ(t0 ) (x(t0 ) − x0 ) = is uniquely solvable on Tk and its solution is represented by x = Ptcan u + Tt Qt G−1 t qt = t2 − t + −t + t t2 − t + t −2 u+ , −t + t −t + t−1 t−1 → uΔ = where u satisfies uΔ = 12 t−1 t−1+ Pρ(t0 ) x0 Put h(t) := 12 (t − 1), we can find t−1 t−1 t−1 t−1 , also the initial condition u(t0 ) = t Φ0 (t, t0 ) = e2h (t, t0 ) − t0 e2h (t, σ (s))h(s)Δs t e2h (t, t0 ) − − t0 e2h (t, σ (s))h(s)Δs t t0 e2h (t, σ (s))h(s)Δs t + t0 e2h (t, σ (s))h(s)Δs ∀t, t0 ∈ Tk Therefore the Cauchy operator of the above equation is Φ(t, t0 ) = Ptcan Φ0 (t, t0 )Pρ(t0 ) = 2e2h (t, t0 )Ptcan N.C LIEM Boundary value problem and Green’s function By using the results mentioned in Sect 3, we study the existence and uniqueness of solutions of two point BVPs of LIDEs Let t0 , t1 ∈ Tk , we consider the dynamic equation A t x Δ = B t x + qt (25) with t ∈ [t0 , t1 ], and the boundary condition for this equation is D1 x(t0 ) + D2 x(t1 ) = b, (26) where D1 , D2 are constant square matrices of order m, and b ∈ M := im((D1 , D2 )) Throughout this section we suppose that (25) is of index-1 By (16), if x(t) is a solution of (25), it must be defined on t ∈ [t0 , σ (t1 )], instead of on [t0 , t1 ], because we have to define the derivative of u(t) = Pρ(t) x(t) at t = t1 We recall the notion of Moore–Penrose pseudoinverse of a matrix Let X be an m × m-matrix Then there exists a unique matrix Y satisfying the conditions (i) Y XY = Y , (ii) XY X = X, (iii) XY = R⊥ , (iv) Y X = P⊥ , where Q⊥ = I − P⊥ (resp R⊥ ) is the orthogonal projector onto ker X (resp onto im X) We call the matrix Y - the Moore–Penrose pseudoinverse of X and denote it by X + If X is nonsingular then X + = X −1 We keep all notations and hypotheses as in Sect on the matrices At , Bt and on the operators Tt , Qt Note that here the assumption that these matrices and operators are defined only on [t0 , σ (t1 )] instead of on T is sufficient By these assumptions, (25) with the initial condition Pρ(t0 ) (x(t0 ) − x0 ) = has a unique solution x(t; t0 , x0 ) given by (24), where Φ(t, t0 ) is given by (22) For the sake of simplicity, we denote Φ(t, t0 ) = Φ(t), t xq (t) = −1 Φ t, σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs + Tt Qt Gt qt t0 Then x(t; t0 , x0 ) = Φ(t)Pρ(t0 ) x0 + xq (t) (27) It is easy to see that xq (t) is a partial solution of (25) satisfying Pρ(t0 ) xq (t0 ) = We are now in a position to find the initial condition x0 such that x(·; t0 , x0 ) is the solution of the dynamic equation (25) satisfying the boundary condition (26) Substituting (27) into (26) we obtain D1 Φ(t0 )x0 + xq (t0 ) + D2 Φ(t1 )x0 + xq (t1 ) = b, or D1 Φ(t0 ) + D2 Φ(t1 ) x0 = b − D1 xq (t0 ) − D2 xq (t1 ) (28) IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES The matrix D = D1 Φ(t0 ) + D2 Φ(t1 ) (29) is called the characterization matrix of the BVP (25), (26) It is easy, in general, to see that im D ⊂ M, ker Aρ(t0 ) ⊂ ker D (30) Theorem The homogeneous BVP At x Δ = Bt x, t ∈ [t0 , t1 ], D1 x(t0 ) + D2 x(t1 ) = (31) has only the trivial solution if and only if ker Aρ(t0 ) = ker D The BVP (25), (26) is uniquely solvable for each arbitrary q ∈ Crd ([t0 , t1 ], Rm ), b ∈ M if and only if im D = M, ker Aρ(t0 ) = ker D (32) Proof Assume that the homogeneous BVP (31) has only the trivial solution x(t) ≡ Let x0 ∈ ker D The initial value problem At x Δ = Bt x, Pρ(t0 ) (x(t0 ) − x0 ) = has a unique solution x(·; t0 , x0 ) = Φ(·)x0 Since Dx0 = 0, we see that Φ(·)x0 is a solution of the dynamic equation (31) which implies that Φ(t)x0 = for all t ∈ [t0 , t1 ] Since ker Φ(t) = ker Aρ(t0 ) we get x0 ∈ ker Aρ(t0 ) Hence ker D ⊂ ker Aρ(t0 ) Taking into account (30), we obtain ker Aρ(t0 ) = ker D Conversely, suppose that ker Aρ(t0 ) = ker D The condition Dx0 = says that x0 ∈ ker D = ker Aρ(t0 ) Thus, x(t; t0 , x0 ) = Φ(t)x0 = Φ(t)Pρ(t0 ) x0 = Assume the BVP (25), (26) is uniquely solvable for each arbitrary q ∈ Crd ([t0 , t1 ], Rm ), b ∈ M Due to the item 1, we immediately get ker Aρ(t0 ) = ker D Further, for any b ∈ im M, denote by x(·) the solution corresponding to q = of (25) with the boundary condition (26) Using (28) we have Dx0 = b This means that b ∈ im D Thus, M ⊂ imD Hence, by (30) we get M = im D We now suppose that im D = M and ker D = ker Aρ(t0 ) and let q ∈ Crd ([t0 , t1 ], Rm ), b ∈ M Since b = b − D1 xq (t0 ) − D2 xq (t1 ) ∈ M, there is x0 ∈ Rm such that Dx0 = b The initial value problem A t x Δ = B t x + qt , Pρ(t0 ) (x(t0 ) − x0 ) = has a unique solution, namely x(·; t0 , x0 ) = Φ(·)x0 + xq (·) Substituting this relation into (26) we obtain D1 x(t0 ; t0 , x0 ) + D2 x(t1 ; t0 , x0 ) = D1 Φ(t0 )x0 + xq (t0 ) + D2 Φ(t1 )x0 + xq (t1 ) = Dx0 + D1 xq (t0 ) + D2 xq (t1 ) = b + D1 xq (t0 ) + D2 xq (t1 ) = b This means that x(·; t0 , x0 ) is the solution of the BVP (25), (26) The proof of the theorem is complete N.C LIEM We now go to the construction of the Green’s function for the BVP (25) and (26) Assume that the condition (32) holds Denote by Q⊥ (resp R⊥ ) the orthogonal projector onto ker D = ker Aρ(t0 ) (resp onto im D) Let D + be the Moore–Penrose pseudoinverse of D Substituting the expression (24) of the solution x(t) of (25) into the boundary condition (26) we get t1 DPρ(t0 ) x0 = b − D2 Φ t1 , σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs, (33) t0 −1 + where b = b − D1 Tt0 Qt0 G−1 t0 qt0 − D2 Tt1 Qt1 Gt1 qt1 Multiplying both sides of (33) by D , we obtain P ⊥ x0 = D + b − D + D t1 Φ t1 , σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t0 Substituting again this equality into (24) and paying attention to Φ(t)P⊥ = Φ(t)Pρ(t0 ) P⊥ = Φ(t)Pρ(t0 ) we get x(t) = Φ(t)D + b − Φ(t)D + D2 t1 Φ t1 , σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t0 t + −1 Φ t, σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs + Tt Qt Gt qt t0 = Φ(t)D + b + t Ptcan Φ0 t, σ (s) Pρ(σ (s)) − Φ0 (t, t0 )Pρ(t0 ) D + D2 Φ t1 , σ (s) t0 × Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t1 − Ptcan Φ0 (t, t0 )Pρ(t0 ) D + D2 Φ t1 , σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs t + Tt Qt G−1 t qt := Φ(t)D + b + t1 −1 Ptcan G t, σ (s) Ps + (Pρ(s) )Δ Ts Qs G−1 s qs Δs + Tt Qt Gt qt , t0 where Φ0 (t, σ (s))Pρ(σ (s)) − Φ0 (t, t0 )Pρ(t0 ) D + D2 Φ(t1 , σ (s)) −Φ0 (t, t0 )Pρ(t0 ) D + D2 Φ(t1 , σ (s)) if t0 if t0 t1 , t1 (34) The expression (34) is called Green’s function (matrix) for the two point BVP (25), (26) We see that the formula (34) is not symmetric In order to improve it, we need to suppose that the matrix G t, σ (s) := At = (Pρ(t) )Δ Ptcan + Pt G−1 t Bt s