- Open Access
Rofe-Beketov formula for symplectic systems
© Zemánek; licensee Springer. 2012
- Received: 17 March 2012
- Accepted: 18 June 2012
- Published: 10 July 2012
We establish the Rofe-Beketov formula for symplectic systems on time scales. This result generalizes the well-known d’Alembert formula (or the Reduction of Order Theorem) and the Rofe-Beketov formula published for the second order Sturm-Liouville equations on time scales. Moreover, this result is new even in the discrete time case.
MSC:34N05, 26E70, 39A12.
- Rofe-Beketov formula
- symplectic system
- time scale
- Reduction of Order Theorem
on a time scale . We unify the Rofe-Beketov formulas published recently in the literature for the second order Sturm-Liouville differential, difference, and dynamic equations and also for the linear Hamiltonian differential systems and we generalize them by establishing its dynamic counterpart for system (). We point out that this result is new even in the discrete time case (see Remark 4(v)) and, moreover, it can be viewed as an improvement of the corresponding Reduction of Order Theorem, see , Remark 6] and , Theorem 3.32], respectively.
represents the second linearly independent solution of equation (1) on J and it holds . This statement was established in , Lemma 2] and it is a generalization of the original Rofe-Beketov formula presented for equation (1) with in , Lemma 2]. An application of identity (3) can be found in the study of the relative oscillation theory and spectral properties of differential operators associated with (1), see [5, 7].
where A, B, C are matrix-valued locally integrable complex functions and , , see , Theorem 6.5]. This result is recalled in the following proposition, where we denote the Hermitian component of the matrix A by ReA, i.e., , see e.g., pp.268-269] or , Facts 3.7.27-3.7.29].
Proposition 1 (Rofe-Beketov formula for (H))
forms the second linearly independentmatrix-valued solution of (H) satisfying the condition, i.e., the solutionsandare normalized. Moreover, the conditionsandhold also true.
was presented in , Remark 1(ii)]. In this article we complete this treatment by proving the Rofe-Beketov formula for symplectic dynamic (and consequently for difference) systems.
The time scale calculus was originally published by Hilger in his dissertation , see also . It is well known that this theory provides suitable tools for a study of differential and difference (among others) equations under the unified framework. The time scale theory has been developed in the last 20 years intensively and we refer to [13, 14] for the fundamental results achieved in this field.
respectively, and simultaneously we put and . The graininess function is defined as .
where we used the notation . We note that is not well defined if exists and is left-scattered.
where I denotes an appropriate identity matrix. A function f is called regulated if its right-hand limit exists (finite) at all right-dense points and the left-hand limit exists (finite) at all left-dense points . Provided a function f is regulated and it is continuous at each right-dense point , it is called rd-continuous (we write ) on . A function f is said to be piecewise rd-continuous () on if it is regulated and if f is rd-continuous at all but possibly finitely many right-dense points . A function f is said to be rd-continuously Δ-differentiable () on if exists for all and on . Finally, a function f is said to be piecewise rd-continuously Δ-differentiable () on if f is continuous on and exists at all except of possibly finitely many points , and on .
for any antiderivative F of f.
Moreover, we note that identity (8) implies the symplecticity of the matrix on , i.e., . Since every symplectic matrix is invertible, it follows that the matrix-valued function is regressive on . Hence the existence of a unique solution for any (vector or matrix) initial value problem is a consequence of , Theorem 5.7] or , Theorem 5.8], see also , Remark 1].
The theory of symplectic difference systems was initiated in  as the discrete counterpart of system (H), while the study of system () originates in . It is a well-known fact that system () includes many special cases, e.g., equations (1), (4) and (5), system (H), discrete symplectic systems, and any even order Sturm-Liouville equation, see .
where and are any , , solutions of (). It is a direct consequence of that the Wronskian matrix takes a constant value on .
A solution of () is said to be a conjoined solution if , i.e., is a Hermitian matrix at one (and hence at any) . Two solutions are called normalized if . A solution is said to be a basis if on . It was shown in , Proposition 2.2] that the value of is also constant on .
The Reduction of Order Theorem for system () was published in , Remark 6] and it is recalled in the following proposition.
Proposition 2 (Reduction of Order Theorem for ())
is also a conjoined basis of (). Moreover, Z andare linearly independent (even normalized), i.e., they form a basis of the solution space for ().
Now, we improve this proposition in the main result by dropping the invertibility of X, i.e., we state and prove the Rofe-Beketov formula for system ().
Theorem 3 (Rofe-Beketov formula for ())
for a fixedchosen without any restriction, solves also system (). Moreover, is a basis of () and it holds (i.e., they are linearly independent and normalized). In addition, constitutes also a conjoined basis if.
i.e., G is determined by (11). Consequently, identity (16) yields that the matrix-valued function F is given by (12) for a fixed chosen without any restriction at the outset of the proof.
i.e., is also a conjoined solution if on , and the proof is complete. □
- (i)The fact that represents a basis of (), follows also from the calculation
- (ii)With using the block identities in (9) and identity (7), the function F can be also given in the form
It follows from identities (17) and (8) that it is satisfied for a point and, consequently, the pair forms a conjoined basis if it holds . Especially if , we have , , and . In this setting, system () has the form of (H) and the statement of Theorem 3 corresponds to Proposition 1, i.e., , Theorem 6.5].
- (iv)Equation (4), where , can be written as the symplectic dynamic system
with , represents a solution of () such that , compare with , Theorem 3.32].
The research was supported by the Czech Science Foundation under grant P201/10/1032 and also by the Operational Programme “Education for Competitiveness” under the project CZ.1.07/2.3.00/30.0009. The author wishes to thank the anonymous referees for the detailed reading of the manuscript and for their comments which helped to improve the presentation of the results.
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