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

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 form

We also set

By a solution of the impulsive BVP (

The corresponding homogeneous BVP takes the form

Note that if we take

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

The existence and uniqueness of linear impulsive boundary value problem for the first-order equations are considered in [

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 [

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 [

If the homogeneous BVP (

Define

It is not difficult to see from (

Since we assumed that

Using (

On the other hand, from the first equation in (

Now we recall the elementary inequality:

Here we derive Green’s function to be used for the representation of the solutions of the inhomogeneous BVP.

Let

Define the rectangles

Green’s function (pair) and its properties are given in the next theorem.

Suppose that the homogeneous BVP (

Then the pair of functions

where

(G1) and (G2) are trivial. Let us consider (G3)(a) follows from

Next, we consider (G4). By definition, it is easy to see that

The proofs of (G5) and (G6) are similar to (a) and (G4), respectively.

One can easily rewrite Green’s function (pair) in terms of the solutions of system (

Our main result is the following theorem.

Let (i)–(iii) hold. If

We first prove the uniqueness. It suffices to show that the homogeneous BVP (

For the existence, we start with the variation of parameters formula and write the general solution of system (

Since we have the uniqueness of solutions, the matrix

Let us now consider the BVP (

Suppose that

The results in this work are new even if the impulses are absent. The statements of the corresponding theorems are left to the reader.