Impulsive Boundary Value Problems for Planar Hamiltonian Systems

Abstract

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.

1. Introduction

The planar Hamiltonian system of 2-linear first-order equations has the form 𝑦^′=𝐽𝐻(𝑡)𝑦,𝑡∈ℝ, (1) where 𝐻(𝑡)=𝑐(𝑡)𝑎(𝑡)𝑎(𝑡)𝑏(𝑡) (2) is a symmetric matrix with piecewise continuous real-valued entries, and 𝐽=01−10 (3) is the so called symplectic identity. Setting 𝑦₁(𝑡)=𝑥(𝑡),𝑦₂(𝑡)=𝑢(𝑡), (4) the Hamiltonian system can be rewritten in a more convenient way as 𝑥^′(𝑡)(𝑡)𝑢=𝑎𝑥+𝑏𝑢,_′=−𝑐(𝑡)𝑥−𝑎(𝑡)𝑢. (5)

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 form𝑥^′=𝑎(𝑡)𝑥+𝑏(𝑡)𝑢+𝑓₁(𝑡),𝑢^′=−𝑐(𝑡)𝑥−𝑎(𝑡)𝑢+𝑓₂(𝑡),𝑡∈𝑡₁,𝑡₂⧵𝜏_𝑖,𝑥𝜏⁺_𝑖=𝛼_𝑖𝑥𝜏⁻_𝑖+𝑎_𝑖1,𝑢𝜏⁺_𝑖=−𝛽_𝑖𝑥𝜏⁻_𝑖+𝛼_𝑖𝑢𝜏⁻_𝑖+𝑎_𝑖2,𝑥𝑡𝑖=1,2,…,𝑝,₁=𝐴,𝑥𝑡₂=𝐵, (6a)(6b)(6c)where(i){𝜏_𝑖},{𝛼_𝑖},{𝛽_𝑖},{𝑎_𝑖1}, and{𝑎_𝑖2}are real sequences for𝑖=1,2,…,𝑝with 𝑡₁<𝜏₁<𝜏₂<⋯<𝜏_𝑝<𝑡₂; (7)(ii) 𝑎,𝑏,𝑐,𝑓₁,𝑓₂∈PLC(𝐽₀), where𝐽₀=[𝑡₁,𝑡₂]and PLC(𝐽₀)={𝜔∶𝐽₀→ℝis continuous on each interval(𝜏_𝑖,𝜏_𝑖+1), the limits𝑤(𝜏^±_𝑖)exist and𝑤(𝜏⁻_𝑖)=𝑤(𝜏_𝑖)for𝑖=1,2,…,𝑝};(iii)𝑏(𝑡)>0for𝑡∈(𝑡₁,𝑡₂)and𝛼_𝑖≠0for𝑖=1,2,…,𝑝;𝐴and𝐵are given real numbers.

We also set𝜏₀=𝑡₁and𝜏_𝑝+1=𝑡₂for convenience.

By a solution of the impulsive BVP (6a)–(6c), we mean nontrivial functions𝑥,𝑢∈PLC(𝐽₀)such that(𝑥,𝑢)satisfies system (6a)–(6c) for all𝑡∈𝐽₀.

The corresponding homogeneous BVP takes the form 𝑥^′=𝑎(𝑡)𝑥+𝑏(𝑡)𝑢,𝑢^′(𝑡)(𝑡)𝑡=−𝑐𝑥−𝑎𝑢,𝑡∈₁,𝑡₂⧵𝜏_𝑖,𝑥𝜏⁺_𝑖=𝛼_𝑖𝑥𝜏⁻_𝑖,𝑢𝜏⁺_𝑖=−𝛽_𝑖𝑥𝜏⁻_𝑖+𝛼_𝑖𝑢𝜏⁻_𝑖,𝑥𝑡𝑖=1,2,…,𝑝,₁=0,𝑥𝑡₂=0. (8a)(8b)(8c)

Note that if we take 𝑎(𝑡)(𝑡)=1≡0,𝑏𝑝(𝑡)(𝑡)(𝑡),𝑓,𝑐=𝑞₁(𝑡)≡0,𝑓₂(𝑡)=𝑓(𝑡), (9) then we obtain as a special case of (6a), (6b), and (6c) the impulsive BVP for second-order differential equations of the form 𝑝(𝑡)𝑥^′′+𝑞(𝑡)𝑥=𝑓(𝑡),𝑡∈𝑡₁,𝑡₂⧵𝜏_𝑖,𝑥𝜏⁺_𝑖=𝛼_𝑖𝑥𝜏⁻_𝑖+𝑎_𝑖1,𝑝𝑥^′𝜏⁺_𝑖=−𝛽_𝑖𝑥𝜏⁻_𝑖+𝛼_𝑖𝑝𝑥^′𝜏⁻_𝑖+𝑎_𝑖2𝑥𝑡,𝑖=1,2,…,𝑝,₁=𝐴,𝑥𝑡₂=𝐵. (10a)(10b)(10c)(10d)

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 linear2×2Hamiltonian 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 [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 [15–17] 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. 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 that2|𝑎(𝑡)|is replaced by|𝑎(𝑡)|.

Theorem 1. If the homogeneous BVP (8a), (8b), and (8c) has a real solution(𝑥(𝑡),𝑢(𝑡))such that𝑥(𝑡)≢0on(𝑡₁,𝑡₂), then one has the Lyapunov type inequality: exp^𝑡2_𝑡1|𝑎(𝑡)|𝑑𝑡^𝑡2_𝑡1𝑏(𝑡)×𝑑𝑡_𝑡2𝑡1𝑐⁺(𝑡)𝑑𝑡+_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖⁺≥4, (11) where𝑐⁺(𝑡)=max{𝑐(𝑡),0}and(𝛽_𝑖/𝛼_𝑖)⁺=max{𝛽_𝑖/𝛼_𝑖,0}.

Proof. Define 𝑧(𝑡)=1𝛼₁𝛼₂⋯𝛼_𝑖𝑥(𝑡)(𝑡)=1,𝑣𝛼₁𝛼₂⋯𝛼_𝑖𝑢(𝑡) (12) for𝑡∈(𝜏_𝑖,𝜏_𝑖+1)and𝑖=0,1,…,𝑝, where we put again𝜏₀=𝑡₁,𝜏_𝑝+1=𝑡₂and make a convention that𝛼₁𝛼₂⋯𝛼_𝑖=1if𝑖=0.
It is not difficult to see from (8a), (8b), (8c), and (12) that 𝑧^′=𝑎(𝑡)𝑧+𝑏(𝑡)𝑣,𝑣^′(𝑡)(𝑡)𝑡=−𝑐𝑧−𝑎𝑣,𝑡∈₁,𝑡₂⧵𝜏_𝑖,𝑧𝜏⁺_𝑖=𝑧𝜏⁻_𝑖,𝑣𝜏⁺_𝑖𝛽=−^𝑖𝛼_𝑖𝑧𝜏⁻_𝑖+𝑣𝜏⁻_𝑖,𝑧𝑡𝑖=1,2,…,𝑝,₁=0,𝑧𝑡₂=0. (13)(14)(15)
Since we assumed that𝑧(𝜏_𝑖)=𝑧(𝜏⁻_𝑖),𝑧(𝑡)is continuous on[𝑡₁,𝑡₂]. Moreover,𝑧^′∈PLC(𝐽₀),𝑧(𝑡₁)=𝑧(𝑡₂)=0, and𝑧(𝑡)≢0for all𝑡∈(𝑡₁,𝑡₂). We may assume without loss of generality that𝑧(𝑡)≥0on(𝑡₁,𝑡₂).
Using (13) and (14) we obtain (𝑣𝑧)^′=−𝑐(𝑡)𝑧²+𝑏(𝑡)𝑣²,𝑡≠𝜏_𝑖,(𝑣𝑧)𝜏⁺_𝑖−(𝑣𝑧)𝜏⁻_𝑖𝛽=−^𝑖𝛼_𝑖𝑧²𝜏_𝑖. (16)(17) Integrating (16) from𝑡₁to𝑡₂and using (15) and (17) lead to _{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖𝑧²𝜏_𝑖=^𝑡2_𝑡1𝑏(𝑡)𝑣²(𝑡)−𝑐(𝑡)𝑧²(𝑡)d𝑡, (18) from which we have ^𝑡2_𝑡1𝑏(𝑡)𝑣²(𝑡)d𝑡≤^𝑡2_𝑡1𝑐⁺(𝑡)𝑧²(𝑡)+d𝑡_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖⁺𝑧²𝜏_𝑖. (19)
On the other hand, from the first equation in (13), we have 𝑧(𝑡)exp−^𝑡_𝑡1𝑎(𝑢)d𝑢^′=𝑏(𝑡)𝑣(𝑡)exp−^𝑡_𝑡1𝑎(𝑢)d𝑢,𝑧(𝑡)exp^𝑡2_𝑡𝑎(𝑢)d𝑢^′=𝑏(𝑡)𝑣(𝑡)exp^𝑡2_𝑡𝑎(𝑢)d𝑢. (20)(21) Let max|𝑧(𝑡)|∶𝑡∈𝑡₁,𝑡₂=𝑧𝑡₀>0. (22) If we integrate (20) from𝑡₁to𝑡₀, we see that 𝑧𝑡₀=^𝑡0_𝑡1𝑏(𝑡)𝑣(𝑡)exp^𝑡0_𝑡𝑎(𝑢)d𝑢d𝑡 (23) and so 𝑧𝑡₀≤^𝑡0_𝑡1𝑏(𝑡)|𝑣(𝑡)|exp^𝑡0_𝑡|𝑎(𝑢)|d𝑢d𝑡. (24) Using the obvious estimate ^𝑡0_𝑡|𝑎(𝑢)|d𝑢≤^𝑡0_𝑡1|𝑎(𝑢)|d𝑢 (25) and then applying Cauchy-Schwarz inequality, we have 𝑧²𝑡₀≤exp2^𝑡0_𝑡1|𝑎(𝑢)|d𝑢^𝑡0_𝑡1𝑏(𝑡)×d𝑡_𝑡0𝑡1𝑏(𝑡)𝑣²(𝑡)d𝑡. (26) Similarly, by integrating (21) from𝑡₀to𝑡₂and following the above procedure, we get 𝑧²𝑡₀≤exp2^𝑡2_𝑡0|𝑎(𝑢)|d𝑢^𝑡2_𝑡0𝑏(𝑡)×d𝑡_𝑡2𝑡0𝑏(𝑡)𝑣²(𝑡)d𝑡. (27)
Now we recall the elementary inequality: 𝑥²𝛼+𝑦²𝛽≥4𝑥𝑦,𝛼,𝛽>0,𝛼+𝛽=1 (28) for𝑥≥0and𝑦≥0. In view of (26) and (27) setting ∫𝛼=^𝑡0𝑡1𝑏(𝑡)d𝑡∫^𝑡2_𝑡1𝑏(𝑡)∫d𝑡,𝛽=^𝑡2𝑡0𝑏(𝑡)d𝑡∫^𝑡2_𝑡1𝑏(𝑡),𝑡d𝑡𝑥=𝑧₀exp−^𝑡0_𝑡1|𝑎(𝑢,𝑡)|d𝑢𝑦=𝑧₀exp−^𝑡2_𝑡0|𝑎(𝑢)|d𝑢 (29) we have 4𝑧²𝑡₀exp∫^𝑡2_𝑡1|𝑎(𝑡≤)|d𝑡^𝑡2𝑡1𝑏(𝑡)d𝑡^𝑡2_𝑡1𝑏(𝑡)𝑣²(𝑡)d𝑡. (30) Combining (19) and (30) results in 4𝑧²𝑡₀exp∫^𝑡2_𝑡1|𝑎(𝑡≤)|d𝑡_t2𝑡1𝑏(𝑡)×d𝑡_𝑡2𝑡1𝑐(𝑡)𝑧²(𝑡)d𝑡+_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖𝑧²𝜏_𝑖. (31) Finally, since𝑧(𝑡₀)≥𝑧(𝑡)for𝑡∈[𝑡₁,𝑡₂], from (31) we obtain the desired inequality: exp^𝑡2_𝑡1|𝑎(𝑡)|d𝑡^𝑡2_𝑡1𝑏(𝑡)×d𝑡_𝑡2𝑡1𝑐⁺(𝑡)d𝑡+_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖⁺≥4. (32)

2.2. Green’s Function

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

Let Φ(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)𝑢₁(𝑡)𝑢₂(𝑡),Φ(0)=𝐼 (33) be a fundamental matrix for (8a), (8b) and set 𝑀=1000,𝑁=0010. (34)

Define the rectangles 𝑅₁₁=𝑡₁,𝜏₁×𝑡₁,𝜏₁,𝑅_𝑖1=𝜏_𝑖−1,𝜏_𝑖×𝑡₁,𝜏₁𝑅,𝑖=2,3,…,𝑝+1,_1𝑗=𝑡₁,𝜏₁×𝜏_𝑗−1,𝜏_𝑗𝑅,𝑗=2,3,…,𝑝+1,_𝑖𝑗=𝜏_𝑖−1,𝜏_𝑖×𝜏_𝑗−1,𝜏_𝑗,𝑖,𝑗=2,3,…,𝑝+1, (35) and the triangles 𝑇^𝑢=(𝑡,𝑠)∈𝑡₁,𝑡₂×𝑡₁,𝑡₂,𝑇∶𝑠>𝑡_𝑙=(𝑡,𝑠)∈𝑡₁,𝑡₂×𝑡₁,𝑡₂,𝑇∶𝑠<𝑡_𝑢𝑖𝑖=(𝑡,𝑠)∈𝑅_𝑖𝑖𝑇∶𝑠>𝑡,𝑖=1,2,3,…,𝑝+1,_𝑙𝑖𝑖=(𝑡,𝑠)∈𝑅_𝑖𝑖∶𝑠<𝑡,𝑖=1,2,3,…,𝑝+1. (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 𝐾=−𝑀Φ(𝑡₁)+𝑁Φ(𝑡₂)⁻¹𝑁Φ𝑡₂. (37) Note that the inverse of matrix𝑀Φ(𝑡₁)+𝑁Φ(𝑡₂)exists in view of the assumption (see also the proof of Theorem 4).
Then the pair of functions 𝐺()=Φ(𝑡)Φ𝑡,𝑠)(𝐼+𝐾⁻¹(𝑠)Φ(𝑡),𝑠<𝑡,𝐾Φ⁻¹(𝑠)𝐺,𝑠≥𝑡,𝑡,𝜏⁺_𝑖=Φ(𝑡)(𝐼+𝐾)Φ⁻¹𝜏⁺_𝑖,𝜏_𝑖Φ(𝑡)<𝑡,𝐾Φ⁻¹𝜏⁺_𝑖,𝜏_𝑖≥𝑡, (38) constitutes Green’s function for (6a), (6b), and (6c). Moreover, we have the following properties: (G1)𝐺(𝑡,𝑠)is continuous and bounded on𝑅_𝑖𝑗,(G2)(𝜕𝐺(𝑡,𝑠))/𝜕𝑡is continuous and bounded on the rectangles𝑅_𝑖𝑗with𝑖≠𝑗and on the triangles𝑇^𝑢_𝑖𝑖and𝑇^𝑙_𝑖𝑖,(G3)𝐺(𝑡,𝑠)satisfies the following jump conditions:(a)𝐺(𝜏⁺_𝑖,𝜏_𝑖)−𝐺(𝜏⁻_𝑖,𝜏_𝑖)=𝐵_𝑖+(𝐵_𝑖−𝐼)𝐺(𝜏⁻_𝑖,𝜏_𝑖)where𝐵_𝑖=^𝛼𝑖0_{−𝛽𝑖𝛼𝑖},(b)𝐺(𝑠⁺,𝑠)−𝐺(𝑠⁻,𝑠)=𝐼,𝑠≠𝜏_𝑖,(c)(𝜕𝐺(𝑠⁺,𝑠)/𝜕𝑡)−(𝜕𝐺(𝑠⁻,𝑠)/𝜕𝑡)=𝐽𝐻(𝑠),𝑠≠𝜏_𝑖,(G4)𝐺(𝑡,𝑠), considered as a function of𝑡, is left continuous and satisfies 𝑦^′=𝐽𝐻(𝑡)𝑦,𝑡∈𝐽_𝑠⧵𝜏_𝑖,𝑦𝜏⁺_𝑖=𝐵_𝑖𝑦𝜏⁻_𝑖,𝑖∈𝑖∶𝜏_𝑖∈𝐽_𝑠,𝑀𝑦𝑡₁+𝑁𝑦𝑡₂=0, (39) where𝐽_𝑠is any of the intervals[𝑡₁,𝑠)or(𝑠,𝑡₂],(G5)Δ|_{𝑡=𝜏𝑖}𝐺(𝑡,𝜏⁺_𝑖)=𝐺(𝜏⁺_𝑖,𝜏⁺_𝑖)−𝐺(𝜏⁻_𝑖,𝜏⁺_𝑖)=(𝐵_𝑖−𝐼)𝐺(𝜏⁻_𝑖,𝜏⁺_𝑖),(G6)𝐺(𝑡,𝑠), considered as a function of𝑡, is left continuous and satisfies (39).

Proof. (G1) and (G2) are trivial. Let us consider (G3)(a) follows from 𝐺𝜏⁺_𝑖,𝜏_𝑖−𝐺𝜏⁻_𝑖,𝜏_𝑖=Φ𝜏⁺_𝑖(𝐼+𝐾)Φ⁻¹𝜏_𝑖−Φ𝜏⁻_𝑖𝐾Φ⁻¹𝜏_𝑖=𝐵_𝑖+𝐵_𝑖−𝐼𝐺𝜏⁻_𝑖,𝜏_𝑖. (40) To see (b), we write for𝑠≠𝜏_𝑖, 𝐺𝑠⁺,𝑠−𝐺(𝑠⁻,𝑠)=Φ𝑠⁺(𝐼+𝐾)Φ⁻¹(𝑠)−Φ(𝑠⁻)𝐾Φ⁻¹(𝑠)=𝐼. (41) For (c), let𝑡≠𝜏_𝑖; then 𝜕𝐺(𝑡,𝑠)=Φ𝜕𝑡^′(𝑡)(𝐼+𝐾)Φ⁻¹(𝑠)=𝐽𝐻(𝑡)Φ(𝑡)(𝐼+𝐾)Φ⁻¹(𝑠)Φ,𝑠<𝑡,_′(𝑡)𝐾Φ⁻¹(𝑠)=𝐽𝐻(𝑡)Φ(𝑡)𝐾Φ⁻¹(𝑠)𝑠,𝑠≥𝑡,𝜕𝐺⁺,𝑠−(𝑠𝜕𝑡𝜕𝐺⁻,𝑠)(𝑠)Φ(𝑠)Φ𝜕𝑡=𝐽𝐻)(𝐼+𝐾⁻¹(𝑠)−𝐽𝐻(𝑠)Φ(𝑠)𝐾Φ⁻¹(𝑠)=𝐽𝐻(𝑠). (42)
Next, we consider (G4). By definition, it is easy to see that𝐺(𝑡,𝑠)is left continuous function at𝑡=𝜏_𝑖. Let us consider the interval[𝑡₁,𝑠). The later is similar. The first equation in (39) is direct consequences of (c) and the definition of𝐺(𝑡,𝑠). Clearly, 𝐺𝜏⁺_𝑖,𝑠=Φ𝜏⁺_𝑖𝐾Φ⁻¹(𝑠)=𝐵_𝑖Φ𝜏⁻_𝑖𝐾Φ⁻¹(𝑠)=𝐵_𝑖𝐺𝜏⁻_𝑖,𝑡,𝑠𝑀𝐺₁,𝑠+𝑁𝐺𝑡₂𝑡,𝑠=𝑀Φ₁𝐾Φ⁻¹(𝑠)+𝑁Φ𝑡₂(𝐼+𝐾)Φ⁻¹(𝑠)=𝑀Φ𝑡₁+𝑁Φ𝑡₂𝐾Φ⁻¹(𝑠)+𝑁Φ𝑡₂Φ⁻¹(𝑠)=0. (43)
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, 𝐾=−𝑀Φ𝑡₁+𝑁Φ𝑡₂⁻¹𝑁Φ𝑡₂=1𝑥₁𝑡₁𝑥₂𝑡₂−𝑥₁𝑡₂𝑥₂𝑡₁×𝑥₁𝑡₂𝑥₂𝑡₁𝑥₂𝑡₁𝑥₂𝑡₂−𝑥₁𝑡₁𝑥₁𝑡₂−𝑥₁𝑡₁𝑥₂𝑡₂, (44) and since detΦ(𝑡)=detΦ(0)exp^𝑡₀((𝑠×trace𝐽𝐻))d𝑠_𝑝𝑖=1det𝐵_𝑖=^𝑝_𝑖=1𝛼²_𝑖, (45) we may write Φ⁻¹(𝑡)=1(𝑡)𝑢detΦ²(𝑡)−𝑥₂(𝑡)−𝑢₁(𝑡)𝑥₁(𝑡)=^𝑝_𝑖=1𝛼_𝑖⁻²𝑢₂(𝑡)−𝑥₂(𝑡)−𝑢₁(𝑡)𝑥₁(𝑡). (46)

3. The Main Result

Our main result is the following theorem.

Theorem 4. Let (i)–(iii) hold. If exp^𝑡2_𝑡1|𝑎(𝑡)|𝑑𝑡^𝑡2_𝑡1𝑏(𝑡)×𝑑𝑡_𝑡2𝑡1𝑐⁺(𝑡)𝑑𝑡+_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖⁺<4, (47) then BVP (6a), (6b), and (6c) has a unique solution(𝑥(𝑡),𝑢(𝑡)). Moreover,𝑦=(𝑥(𝑡),𝑢(𝑡))is expressible as 𝑦(𝑡)=𝑤(𝑡)+^𝑡2_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠)d𝑠+_{𝑡1≤𝜏𝑖<𝑡2}𝐺𝑡,𝜏⁺_𝑖𝑎_𝑖, (48) where 𝑤(𝑡)=Φ(𝑡)𝑀Φ(𝑡₁)+𝑁Φ(𝑡₂)⁻¹𝜂,𝜂=(𝐴,𝐵)^𝑇, (49) and Green’s function pair(𝐺,𝐺)is given by (38).

Proof. We first prove the uniqueness. It suffices to show that the homogeneous BVP (8a)–(8c) has only the trivial solution. Let𝑥(𝑡)≢0on(𝑡₁,𝑡₂). By Theorem 1, we see that Lyapunov type inequality (11) holds contradicting the inequality (47). Thus𝑥(𝑡)=0for all𝑡∈[𝑡₁,𝑡₂]. Moreover, by (6a), (6b), and (6c) we have 𝑏(𝑡)𝑢=0,𝑡≠𝜏_𝑖, (50) which results in𝑢(𝑡)=0for𝑡≠𝜏_𝑖. Taking limit we see that𝑢(𝜏^±_𝑖)=0. As a result we obtain(𝑥(𝑡),𝑢(𝑡))=(0,0)for all𝑡∈[𝑡₁,𝑡₂]. 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 𝑦(𝑡)=Φ(𝑡)𝑐+^𝑡_𝑡1Φ(𝑡)Φ⁻¹(𝑠)𝑓(𝑠)+d𝑠_{𝑡1≤𝜏𝑖<𝑡}Φ(𝑡)Φ⁻¹𝜏⁺_𝑖𝑎_𝑖. (51) Clearly, the boundary condition is satisfied if 𝑀Φ𝑡₁+𝑁Φ𝑡₂𝑐=𝜂−𝑁Φ𝑡₂×^𝑡2_𝑡1Φ⁻¹(𝑠)𝑓(𝑠)d𝑠+_{𝑡1≤𝜏𝑖<𝑡2}Φ⁻¹𝜏⁺_𝑖𝑎_𝑖, (52) where𝜂=(𝐴,𝐵)^𝑇.
Since we have the uniqueness of solutions, the matrix𝑀Φ(𝑡₁)+𝑁Φ(𝑡₂)must have an inverse. Setting 𝐾=−𝑀Φ(𝑡₁)+𝑁Φ(𝑡₂)⁻¹𝑁Φ𝑡₂, (53) we may solve𝑐from (52) uniquely: 𝑐=𝑀Φ𝑡₁+𝑁Φ𝑡₂⁻¹𝜂+𝐾^𝑡2_𝑡1Φ⁻¹(𝑠)𝑓(𝑠)d𝑠+_{𝑡1≤𝜏𝑖<𝑡2}Φ⁻¹𝜏⁺_𝑖𝑎_𝑖. (54) Hence, 𝑦(𝑡)=Φ(𝑡)𝑀Φ𝑡₁+𝑁Φ𝑡₂⁻¹(𝑡)×𝜂+Φ)(𝐼+𝐾_𝑡𝑡1Φ⁻¹(𝑠)𝑓(𝑠)d𝑠+_{𝑡1≤𝜏𝑖<𝑡}Φ⁻¹𝜏⁺_𝑖𝑎_𝑖+Φ(𝑡)𝐾^𝑡2_𝑡Φ⁻¹(𝑠)𝑓(𝑠)d𝑠+_{𝑡≤𝜏𝑖<𝑡2}Φ⁻¹𝜏⁺_𝑖𝑎_𝑖. (55) Therefore the unique solution of the BVP (6a)–(6c) can be expressed as 𝑦(𝑡)=𝑤(𝑡)+^𝑡2_𝑡1𝐺()𝑓(𝑠)+𝑡,𝑠d𝑠_{𝑡1≤𝜏𝑖<𝑡2}𝐺𝑡,𝜏⁺_𝑖𝑎_𝑖. (56)

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 𝐺()=𝜓(𝑡)Ψ𝑡,𝑠)(𝐼+𝐾⁻¹(𝑠)1𝑝(𝑠)𝑒₂𝜓(𝑡),𝑠<𝑡,𝐾Ψ⁻¹(𝑠)1𝑝(𝑠)𝑒₂𝐺,𝑠≥𝑡,𝑡,𝜏⁺_𝑖=𝜓(𝑡)(𝐼+𝐾)Ψ⁻¹𝜏⁺_𝑖,𝜏_𝑖𝜓(𝑡)<𝑡,𝐾Ψ⁻¹𝜏⁺_𝑖,𝜏_𝑖≥𝑡, (57) where𝜓(𝑡)=[𝜓₁,𝜓₂]is the first row of the (Wronskian) matrix: Ψ(𝑡)=𝜓₁(𝑡)𝜓₂(𝑡)𝜓^′₁(𝑡)𝜓^′₂(𝑡),𝐾=−𝑀Ψ𝑡₁+𝑁Ψ𝑡₂⁻¹𝑁Ψ𝑡₂,𝑒₂=[0,1]^𝑇. (58)

Corollary 5. Suppose that𝑝and𝑐are piece-wise continuous on[𝑡₁,𝑡₂],𝑝(𝑡)>0, and𝛼_𝑖≠0for𝑖=1,2,…,𝑝.If ^𝑡2_𝑡11𝑝(𝑡)𝑑𝑡^𝑡2_𝑡1𝑐⁺(𝑡)𝑑𝑡+_{𝑡1≤𝜏𝑖<𝑡2}𝛽_𝑖𝛼_𝑖⁺<4, (59) then the BVP (10a), (10b), (10c), and (10d) has a unique solution𝑥(𝑡)which is expressible as 𝑥(𝑡)=𝑤(𝑡)+^𝑡2_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠+_{𝑡1≤𝜏𝑖<𝑡2}𝐺𝑡,𝜏⁺_𝑖𝑎_𝑖, (60) where 𝑤(𝑡)=𝜓(𝑡)𝑀Ψ(𝑡₁)+𝑁Ψ(𝑡₂)⁻¹𝜂, (61) and Green’s function pair(𝐺,𝐺)is 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.