Impulsive Boundary Value Problems for Planar Hamiltonian Systems

and Applied Analysis 3 from which we have

The corresponding homogeneous BVP takes the form Note that if we take then we obtain as a special case of (6a), (6b), and (6c) the impulsive BVP for second-order differential equations of 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 linear 2 × 2 Hamiltonian systems under impulse effect.
The existence and uniqueness of linear impulsive boundary value problem for the first-order equations are considered in [1][2][3][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][8][9][10].In [11], 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 [12].Basic theory of impulsive differential equations is contained in the seminal book [13].
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 [14] and for systems [15][16][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.

Preliminaries
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 [20] in the sense that 2|()| is replaced by |()|.
Using ( 13) and ( 14) we obtain Integrating ( 16) from  1 to  2 and using ( 15) and ( 17) lead to from which we have On the other hand, from the first equation in (13), we have [() exp (∫ If we integrate (20) from  1 to  0 , we see that and so Using the obvious estimate and then applying Cauchy-Schwarz inequality, we have Similarly, by integrating (21) from  0 to  2 and following the above procedure, we get Now we recall the elementary inequality: for  ≥ 0 and  ≥ 0. In view of ( 26) and ( 27) setting we have Combining ( 19) and (30) results in Finally, since ( 0 ) ≥ () for  ∈ [ 1 ,  2 ], from (31) we obtain the desired inequality: (32)

Green's Function.
Here we derive Green's function to be used for the representation of the solutions of the inhomogeneous BVP.Let be a fundamental matrix for (8a), (8b) and set Define the rectangles and the triangles (36) 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 matrix Φ( 1 )+Φ( 2 ) exists in view of the assumption (see also the proof of Theorem 4).
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 that (, ) is left continuous function at  =   .Let us consider the interval [ 1 , ).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.
Remark 3. One can easily rewrite Green's function (pair) in terms of the solutions of system (8a), (8b).Indeed, Abstract and Applied Analysis we may write ] . (46)

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.
For the existence, we start with the variation of parameters formula and write the general solution of system (6a), (6b) as where  = (, )  .Since we have the uniqueness of solutions, the matrix Φ( 1 ) + Φ( 2 ) must have an inverse.Setting we may solve  from (52) uniquely: (56)