Functional Differential and Difference Equations with Applications 2013View this Special Issue
Impulsive Boundary Value Problems for Planar Hamiltonian Systems
We give an existence and uniqueness theorem for solutions of inhomogeneous impulsive boundary value problem (BVP) for planar Hamiltonian systems. Green's function that is needed for representing the solutions is obtained and its properties are listed. The uniqueness of solutions is connected to a Lyapunov type inequality for the corresponding homogeneous BVP.
The planar Hamiltonian system of 2-linear first-order equations has the form where is a symmetric matrix with piecewise continuous real-valued entries, and is the so called symplectic identity. Setting the Hamiltonian system can be rewritten in a more convenient way as
Our aim in this work is to prove an existence and uniqueness theorem for solutions of the related BVP for inhomogeneous Hamiltonian system under impulse effect of the formwhere(i),,,, andare real sequences forwith (ii) , whereand is continuous on each interval, the limitsexist andfor(iii)forandfor;andare given real numbers.
We also setandfor convenience.
The corresponding homogeneous BVP takes the form
To the best of our knowledge although many results have been obtained for linear impulsive boundary value problems by using different techniques, there is little known for the linearHamiltonian systems under impulse effect.
The existence and uniqueness of linear impulsive boundary value problem for the first-order equations are considered in [1–4]. For the second-order case we refer to [5, 6] in which the integral representation of the solution of second order linear impulsive boundary value problems is given by using Green’s function and the existence and uniqueness of the solutions are obtained. Variational technique approach for the existence of the solutions of linear and nonlinear impulsive boundary value problems can be found in [7–10]. In , the method of upper and lower solutions is employed for the existence of solutions of nonlinear impulsive boundary value problems. For a detailed discussion on boundary value problems for higher-order linear impulsive equations we refer to . Basic theory of impulsive differential equations is contained in the seminal book .
Our method of proof is based on Green’s function formulation and Lyapunov type inequalities for linear Hamiltonian system under impulse effect. There are many studies on Lyapunov type inequalities and their applications for linear ordinary differential equations  and for systems [15–17] as well as for linear impulsive differential equations and systems [18, 19]. However, the use of a Lyapunov type inequality in connection with BVPs seems to be limited.
2.1. Lyapunov Type Inequality for Homogeneous Problem
In this section we provide a Lyapunov type inequality to be used for the uniqueness of the inhomogeneous BVP. The obtained inequality is sharper than the one given by the present authors in  in the sense thatis replaced by.
forand, where we put again,and make a convention thatif.
It is not difficult to see from (8a), (8b), (8c), and (12) that
Since we assumed that,is continuous on. Moreover,,, andfor all. We may assume without loss of generality thaton.
Using (13) and (14) we obtain Integrating (16) fromtoand using (15) and (17) lead to from which we have
On the other hand, from the first equation in (13), we have Let If we integrate (20) fromto, we see that and so Using the obvious estimate and then applying Cauchy-Schwarz inequality, we have Similarly, by integrating (21) fromtoand following the above procedure, we get
Now we recall the elementary inequality: forand. In view of (26) and (27) setting we have Combining (19) and (30) results in Finally, sincefor, from (31) we obtain the desired inequality:
2.2. Green’s Function
Here we derive Green’s function to be used for the representation of the solutions of the inhomogeneous BVP.
Define the rectangles and the triangles
Green’s function (pair) and its properties are given in the next theorem.
Theorem 2. Suppose that the homogeneous BVP (8a)–(8c) has only the trivial solution. Let
Note that the inverse of matrixexists in view of the assumption (see also the proof of Theorem 4).
Then the pair of functions constitutes Green’s function for (6a), (6b), and (6c). Moreover, we have the following properties: (G1)is continuous and bounded on,(G2)is continuous and bounded on the rectangleswithand on the trianglesand,(G3)satisfies the following jump conditions:(a)where(b),(c),(G4), considered as a function of, is left continuous and satisfies whereis any of the intervalsor(G5)(G6), considered as a function of, is left continuous and satisfies (39).
Proof. (G1) and (G2) are trivial. Let us consider (G3)(a) follows from
To see (b), we write for,
For (c), let; then
Next, we consider (G4). By definition, it is easy to see thatis left continuous function at. Let us consider the interval. The later is similar. The first equation in (39) is direct consequences of (c) and the definition of. Clearly,
The proofs of (G5) and (G6) are similar to (a) and (G4), respectively.
3. The Main Result
Our main result is the following theorem.
Proof. We first prove the uniqueness. It suffices to show that the homogeneous BVP (8a)–(8c) has only the trivial solution. Leton. By Theorem 1, we see that Lyapunov type inequality (11) holds contradicting the inequality (47). Thusfor all. Moreover, by (6a), (6b), and (6c) we have
which results infor. Taking limit we see that. As a result we obtainfor all. This completes the uniqueness of the solutions.
For the existence, we start with the variation of parameters formula and write the general solution of system (6a), (6b) as Clearly, the boundary condition is satisfied if where.
Since we have the uniqueness of solutions, the matrixmust have an inverse. Setting we may solvefrom (52) uniquely: Hence, Therefore the unique solution of the BVP (6a)–(6c) can be expressed as
Let us now consider the BVP (10a), (10b), (10c), and (10d). In this case it is not difficult to see that the corresponding Green’s function (pair) becomes whereis the first row of the (Wronskian) matrix:
Corollary 5. Suppose thatandare piece-wise continuous on,, andfor.If then the BVP (10a), (10b), (10c), and (10d) has a unique solutionwhich is expressible as where and Green’s function pairis given by (57).
Remark 6. The results in this work are new even if the impulses are absent. The statements of the corresponding theorems are left to the reader.
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