Communications on Applied Nonlinear Analysis 

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Stability Analysis of Two Point Boundary Value Problem on Time 

Scales 

 

G. V. Ramana1, M. Bala Prabhakar2 and N. Veerraju3* 
1Associate Professor, Department of Mathematics, Aditya University, Surampalem-533 437, A. P., India. 

Mail: ramanaginjala9@gmail.com 
2Associate Professor, Department of Mathematics, Aditya University, Surampalem-533 437, A. P., India. 

Mail: prabhakar_mb@yahoo.co.in 
3Assistant Professor, Department of Mathematics, SRKR Engineering College, Bhimavaram-534 204, A.P., India. 

Mail:veerrajunalla@gmail.com 

*Corresponding Author. 

 

Article History: 

Received: 20-08-2024 

Revised: 01-10-2024 

Accepted: 19-10-2024 

Abstract:  

V. Lakshmi Kantham and M. Rama Mohana Rao contributed a lot for the theory of 

integro- differential equations of Volterra type and its various properties. In this 

paper, an attempt is made to explore the existence of a solution for a matrix integro-

dynamical equation of Volterra type (MIDT) with two-point boundary value problem 

(TBVP) in terms of Green’s matrix (GM) on time scales (TS). And properties of GM 

and stability of the system on TS are discussed. 

Keywords: Green’s matrix, Stability, Time scales. 

     AMS Subject Classification: 34N05. 

 

1. Introduction 

Mathematical models, that describe physical phenomenon are expressed or described in most cases 

either differential or difference equations with respect to continuous cases or discrete cases 

respectively. In previous years, so many authors have worked on dynamical systems either in 

continuous cases or discrete cases. Stefan Hilger observed some interesting patterns in nature.  

According to Stefan Hilger observation, some plants in poles show growing nature for 6 months and 

non-growing nature for remaining 6 months. Further, in some species existing in Japan, Mother dies 

soon after it gives birth to its baby. After having noticed this physical phenomenon, Stefan Hilger 

drew a conclusion that there exists some systems in real world which contains both discrete and 

continuous nature. Based on this, Stefan Hilger invented a new technique TS in 1998 as an integral 

part of his Ph.D thesis which can work with systems involving both cases[5]. In recent years, owing 

to the unified treatment of analysing both continuous and discrete systems at a time [1, 3, 6], research 

related to dynamic systems on TS experienced significant significance in diversified fields. The TS 

applications has a tremendous potential in biology, engineering, economics, physics, neural 

networks, etc. Burton studied the an integro-differential equations and stability theory for Volterra 

equations [2]. Making use of asymptotic stability, G. V. S. R. Deekshitulu and G. V. Ramana studied 

Lyapunov type linear matrix Volterra integro dynamical system on TS [4]. Shah, Zada and Sabrina 

Streipert are discussed the stability of non-linear Volterra integro dynamic equations on TS [7, 8].  

Having seen the potential exploring possibilities in this, an endeavour is made to obtain a solution of 

BVP concerned with MIDT. 



Communications on Applied Nonlinear Analysis 

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389 
 

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And the obtained solution is tabled in the form of GM. Further, properties of GM and stability are 

also discussed for the system on TS. 

In this paper, the following matrix integro-dynamical equation on TS has been considered. 

𝑥∆(𝑡) = 𝐴(𝑡)𝑥(𝑡) + ∫ 𝐾(𝑡, 𝑠) 𝑥(𝑠)∆𝑠
𝜎(𝜌(𝑡))

𝑡0
+ ℥(𝑡, 𝑥(𝑡)),                           (1.1) 

                                           𝑤𝑎 𝑥(𝑎)  + 𝑤𝑏 𝑥(𝑏) = 0                                                          (1.2)  

Here 𝐴𝑛×𝑛 is rd-continuous in 𝕋 and 𝐾(𝑡, 𝑠) ∈ 𝐶𝑟𝑑[𝕋 × 𝕋, ℝ𝑛×𝑛], 𝑤𝑎 and 𝑤𝑏are(𝑛 × 𝑛) constant 

matrices, ℥(𝑡, 𝑥(𝑡)) ∈ 𝐶𝑟𝑑[𝕋 × 𝕋, ℝ𝑛]and 𝑥(𝑡) is a column vector. 

 

2. Preliminary Results 

Definition 2.1.[Bohner et al. 2003] The mappings σ and ρ: T → R are defined as 

(i) σ(t) = inf {m ∈ T: m > t},  

(ii) (ii) ρ(t) = sup {m ∈ T: m < t}. 

Definition 2.2.[Bohner et al. 2003] The Graininess function µ: T → [0, ∞] is defined as µ(t) = σ(t) – 

t and 

(i) if T = R, then µ(t) = 0 and 

(ii) if T = Z, then µ(t) = 1. 

Theorem 2.3.[Bohner et al. 2003] Assume v: T → R and let t ∈ Tk and v△(t) exist then 

 v(σ(t))  =  v(t)  +  µ(t)v△(t). 

Lemma 2.4.[Bohner et al. 2003] Let U, V and W be matrix-valued of order (n × n) and 

differentiable. Then 

(i) (U + V)△ = U△ + V△ 

(ii) ( αU)△ =  αU△ if α is constant 

(iii) (UV)△ = U△Vσ + UV△ = UσV△ + U△V 

(iv) (UVW)△ = U△VW + UσV△Wσ + UσVW△ = U△VW + UσV△W + UσVσW△ 

 

Definition 2.5. [Bohner et al. 2003] U is a matrix of order (n × n) on a time scale T is said to be 

regressive if 

 I + µ(t)U(t) ≠ 0 for all t ∈ Tk. 

Lemma 2.6. [Bohner et al. 2001] Let u ∈  Crd(T, R), ξ ∈  R+, ξ ≥ 0, β∈ R. The 

              u(t) ≤  β + ∫ u(s)ξ(s) △ s

𝑡

0

 

implies  u(t)  ≤  β𝑒ξ(𝑡, 𝑡0),  for all t ∈ T. 

Lemma 2.7. [Bohner et al. 2003] Let ξ ∈  R+. Then eξ(ϱ, ς) ≥ 1 +  ξ(ϱ − ς), for all ϱ ≥ ς. 

 

 



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3. Main Results 

Theorem 3.1. Let 𝐴(𝑡), 𝐾(𝑡, s), 𝑤𝑎 , 𝑤𝑏 and ℥(𝑡, 𝑥(𝑡)) be previously defined. Then the solution of 

(1.1) and (1.2) is  

𝑥(𝑡) = ∫ 𝐺(𝑡, 𝜎(s)) 𝐻(s, 𝜏)∆𝑠
𝑏

𝑎
                           (3.1) 

Where  𝐻(𝑠, 𝜏) = ∫ 𝐾(s, 𝜏) 𝑦(𝜏)∆𝜏 + ℥(𝜏. 𝑥(𝜏))
𝜎(𝜌(s))

𝑡0
 and 𝐺(𝑡, 𝜎(s)) is the GM. 

 

Proof: Consider the non-homogeneous BVP (1.1) and (1.2)                 

Where 𝐴𝑛×𝑛is rd-continuous in 𝕋 and 𝐾(𝑡, s) ∈ 𝐶𝑟𝑑[𝕋 × 𝕋, ℝ𝑛×𝑛], 𝑤𝑎 and 𝑤𝑏 are (𝑛 × 𝑛) constant 

matrices, ℥(𝑡, 𝑥(𝑡)) ∈ 𝐶𝑟𝑑[𝕋 × 𝕋, ℝ𝑛]. 

Assume that the homogeneous BVP corresponding to given non-homogeneous BVP is incompatible. 

Let ∅(𝑡) be a fundamental matrix of homogeneous equation (HE) in the form 𝑥(𝑡) = ∅(𝑡)𝐶, where 

𝐶 is a constant matrix. 

Let 𝑥(𝑡)̅̅ ̅̅ ̅̅  be a particular solution of non-homogeneous equation (NHE). Then any solution of the 

given NHE takes form 

𝑥(𝑡) = ∅(𝑡)𝐶 + 𝑥(𝑡)̅̅ ̅̅ ̅̅ .                    (3.2) 

Since the particular solution of (1.1) is  

𝑥(𝑡)̅̅ ̅̅ ̅̅ = ∅(𝑡) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0
+ ℥(s, 𝑥(s))] ∆s

𝑡

𝑎
                         (3.3) 

The general solution of (1.1) is  

𝑥(𝑡) = ∅(𝑡)𝐶 + ∅(𝑡) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0
+ ℥(s, 𝑥(s))] ∆s

𝑡

𝑎
.         (3.4) 

Now calculate the general solution in the given boundary condition 

0 = 𝑤(𝑎)𝑥(𝑎) + 𝑤(𝑏)𝑥(𝑏) 

= 𝑤(𝑎) [∅(𝑎)𝐶 + ∅(𝑎) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑎

𝑎

]

+ 𝑤(𝑏) [∅(𝑏)𝐶 + ∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

] 

= 𝑤(𝑎)∅(𝑎)𝐶 + 𝑤(𝑏)∅(𝑏)𝐶 + 𝑤𝑏∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

 

= [𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)]𝐶 + 𝑤(𝑏)∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

 

The characteristic matrix 𝐷 = [𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)]. 

Hence the above identity becomes 

0 = 𝐷𝐶 + 𝑤(𝑏)∅(𝑏) ∫ ∅−1(𝜎(𝑠)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

 



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Since the homogeneous BVP is incompatible, the index of compatibility is zero and hence 𝐷 is non-

singular. 

𝐷𝐶 = −𝑤(𝑏)∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

 

𝐶 = −𝐷−1𝑤(𝑏)∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

 

From equation (3.4) 

𝑥(𝑡) = −∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑏

𝑎

+ ∅(𝑡) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑡

𝑎

 

= −∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏)[∫ + ∫ ] ∅−1(𝜎(s))
𝑏

𝑡

𝑡

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s

+ ∅(𝑡) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑡

𝑎

 

= −∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∫  ∅−1(𝜎(s))
𝑡

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s

+ ∅(𝑡) ∫ ∅−1(𝜎(s)) [∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s
𝑡

𝑎

− ∅(𝑡)𝐷−1𝑤𝑏∅(𝑏) ∫  ∅−1(𝜎(s))
𝑏

𝑡

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s 

 

= ∅(𝑡)[𝐼 − 𝐷−1𝑤(𝑏)∅(𝑏)] ∫  ∅−1(𝜎(s))
𝑡

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s

− ∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∫  ∅−1(𝜎(s))
𝑏

𝑡

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s 

= ∅(𝑡)𝐷−1[𝐷 − 𝑤(𝑏)∅(𝑏)] ∫  ∅−1(𝜎(s))
𝑡

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s

− ∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∫  ∅−1(𝜎(s))
𝑏

𝑡

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s 

= ∅(𝑡)𝐷−1𝑤(𝑎)∅(𝑎)] ∫  ∅−1(𝜎(s))
𝑡

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s

− ∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∫  ∅−1(𝜎(s))
𝑏

𝑡

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s 

𝑥(𝑡) = ∫ 𝐺(𝑡, 𝜎(s))
𝑏

𝑎

[∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

+ ℥(s, 𝑥(s))] ∆s 

= ∫ 𝐺(𝑡, 𝜎(s))
𝑏

𝑎
𝐻(s, 𝜏)∆s. 



Communications on Applied Nonlinear Analysis 

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Where 𝐻(𝑠, 𝜏) = ∫ 𝐾(s, 𝜏) 𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0
+ ℥(s, 𝑥(s)) and 𝐺(𝑡, 𝜎(s)) is the Greens matrix and is 

given by  

𝐺(𝑡, 𝜎(s)) = {
∅(𝑡)𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(𝜎(s)),   𝑎 ≤ s ≤ 𝑡 ≤ 𝑏

−∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)),   𝑎 ≤ 𝑡 ≤ s ≤ 𝑏.
 

Next, we discuss the two extreme cases of 𝕋 and obtain some interesting results. 

Corollary 3.2. If 𝕋 = ℝ+, we have 𝜎(𝑡) = 𝑡 and  ∅−1(𝜎(s)) =  ∅−1(s) in Theorem 3.1 the state 

equation is  

𝑥′(𝑡) = 𝐴(𝑡)𝑥(𝑡) + ∫ 𝐾(𝑡, s) 𝑥(s)∆s
𝑡

𝑡0
+ ℥(𝑡, 𝑥(𝑡)),

0 = 𝑤(𝑎)𝑥(𝑎) + 𝑤(𝑏)𝑥(𝑏)
}        (3.5) 

Where 𝐴𝑛×𝑛 is a continuous matrix on ℝ+, 𝐾(𝑡, s)𝑛×𝑛 is a continuous matrix for 

0 ≤ s ≤ 𝑡 ≤ ∞ and 𝐹 ∈ 𝐶[ℝ+ × ℝ+, ℝ𝑛], then the solution of (3.5) is  

𝑥(𝑡) = ∫ 𝐺(𝑡, s)
𝑏

𝑎

𝐻(s, 𝜏)∆s 

Where 𝐻(s, 𝜏) = ∫ 𝐾(s, 𝜏) 𝑥(𝜏)𝑑𝜏
s

𝑡0
+ ℥(𝜏, 𝑥(𝜏)) and 𝐺(𝑡, s) is the GM and is given by  

𝐺(𝑡, s) = {
∅(𝑡)𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(s),   𝑎 ≤ s ≤ 𝑡 ≤ 𝑏

−∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(s),   𝑎 ≤ 𝑡 ≤ s ≤ 𝑏.
 

Where ∅(𝑡) is a fundamental matrix and 𝐷 = [𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)]. 

Corollary 3.3. If 𝕋 = ℕ, we have 𝜎(𝜉) = 𝜉 + 1 and  ∅−1(𝜎(𝜉)) =  ∅−1(𝜉 + 1) in Theorem 3.1 the 

state equation is  

∆𝑥(𝜉) = 𝐴(𝜉)𝑥(𝜉) + ∑ 𝐾(𝜉, s) 𝑥(s)𝜉−1
s=𝜉0

+ ℥(𝜉, 𝑥(𝜉)),

0 = 𝑤(𝑎)𝑥(𝑎) + 𝑤(𝑏)𝑥(𝑏)
}          (3.6) 

Where 𝐴(𝜉) and 𝐾(𝜉, s) are 𝜂 × 𝜂 matrices for each𝜉, s ∈ ℕ and 

 𝐹: ℕ𝜉0

+ × ℕ𝜉0

+ → ℝ𝑑 , ℕ𝜉0

+ = {𝜉0, 𝜉0 + 1 … 𝜉0 + 𝑘, … }, 𝜉0, 𝑘 ∈ ℕ then the solution of (3.6) is 

𝑥(𝜉) = ∑ 𝐺(𝜉, s + 1)𝑏−1
s=𝑎 𝐻(s, 𝜏). 

Where 𝐻(s, 𝜏) = ∑ 𝐾(s, 𝜏) 𝑥(𝜏) +s−1
𝜏=𝜉0

℥(𝜉, 𝑥(𝜉)) and 𝐺(𝜉, s + 1) is the GM and is given by  

𝐺(𝜉, s + 1) = {
∅(𝜉)𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(s + 1),   𝑎 ≤ s ≤ 𝜉 ≤ 𝑏 − 1

−∅(𝜉)𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(s + 1),   𝑎 ≤ 𝜉 ≤ s ≤ 𝑏 − 1.
 

Where ∅(𝜉) is a fundamental matrix and 𝐷 = [𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)]. 

Theorem 3.4. GM exhibits the below mentioned characteristics: 

i. The components of 𝐺(𝑡, 𝜎(s)) are functions of 𝑡 withfixeds, possess rd-continuous delta 

differentiable on the intervals [𝑎, s)and (s, 𝑏]. At the point 𝑡 = s, 𝐺 exhibits an upward jump of 

magnitude ∅(𝑎) ∅−1(𝜎(s)). 



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i.e.,𝐺(s+, 𝜎(s)) − 𝐺(s−, 𝜎(s)) = ∅(𝑎) ∅−1(𝜎(s)). 

ii. 𝐺(𝑡, 𝜎(s))is a formal solution of the homogeneous BVP 

𝑥∆(𝑡) = 𝐴(𝑡)𝑥(𝑡)

0 = 𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)
}                (3.7) 

     

𝐺, ceases to be a true solution solely due to its jump condition at 𝑡 = s. 

iii. The Green’s matrix 𝐺(𝑡, 𝜎(s)) having the properties (i) and (ii) is unique. 

Proof:(i). The GM for the BVP 

𝑥∆(𝑡) = 𝐴(𝑡)𝑥(𝑡)

                           0 = 𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏).
 

 is 

𝐺(𝑡, 𝜎(s)) = {
∅(𝑡)𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(𝜎(s)),   𝑎 ≤ s ≤ 𝑡 ≤ 𝑏

−∅(𝑡)𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)),   𝑎 ≤ 𝑡 ≤ s ≤ 𝑏
 

Where 𝐺  is considered as a function of 𝑡 with fixed s, the GM can be written as  

𝐺(𝑡, 𝜎(s)) = {
∅(𝑡)𝐻+,   s ≤ 𝑡

∅(𝑡)𝐻−,   𝑡 ≤ s.
 

Where the matrices 

𝐻+ =   𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(𝜎(s)) 

𝐻− =  −𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) 

They are independent of 𝑡.Consequently, the components of 𝐺(𝑡, 𝜎(s)) possess rd-continuous first 

delta derivatives with respect to𝑡on [𝑎, s) and (s, 𝑏]. Further  

𝐻+ − 𝐻− =   𝐷−1𝑤(𝑎)∅(𝑎) ∅−1(𝜎(s))+𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) 

  =   𝐷−1[𝑤(𝑎)∅(𝑎) + 𝑤(𝑏)∅(𝑏)] ∅−1(𝜎(s)) 

=   𝐷−1𝐷 ∅−1(𝜎(s)) 

=  ∅−1(𝜎(s)). 

Therefore 

𝐺(s+, 𝜎(s)) − 𝐺(s−, 𝜎(s)) = ∅(s)𝐻+ − ∅(s)𝐻− 

                                               = ∅(s)[𝐻+ − 𝐻−] 

                                                 = ∅(s)∅−1(𝜎(s)). 

(ii). We want to show that 𝐺(𝑡, 𝜎(s)) is a matrix solution of homogeneous BVP. The representation 

of 𝐺(𝑡, 𝜎(s))is the solution of (3.7) on [𝑎, s) and (s, 𝑏]. Now to show that 𝐺(𝑡, 𝜎(s)) satisfies the 

given boundary condition, we obtain 

𝑤(𝑎)𝐺(𝑎, 𝜎(s)) + 𝑤(𝑏)𝐺(𝑏, 𝜎(s)) = 𝑤(𝑎)∅(𝑎)𝐻− + 𝑤(𝑏)∅(𝑏)𝐻+ 

  = 𝑤(𝑎)∅(𝑎)𝐻− + 𝑤(𝑏)∅(𝑏)𝐻+ + 𝑤(𝑏)∅(𝑏)𝐻− − 𝑤(𝑏)∅(𝑏)𝐻− 



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     = 𝑤(𝑎)∅(𝑎)𝐻− + 𝑤(𝑏)∅(𝑏)𝐻− + 𝑤(𝑏)∅(𝑏)[𝐻+ − 𝐻−] 

                = 𝑤(𝑎)[∅(𝑎) + 𝑤(𝑏)∅(𝑏)]𝐻− + 𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) 

                                             = 𝐷𝐻− + 𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) 

                                            = 𝐷[ −𝐷−1𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s))] + 𝑤(𝑏)∅(𝑏)∅ −1(𝜎(s)) 

                           = −𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) + 𝑤(𝑏)∅(𝑏) ∅−1(𝜎(s)) 

                                       = 0 

Thus 𝐺 is a solution of the homogeneous BVP. 

(iii). Now, let us prove the uniqueness of G, let 𝐺1(𝑡, 𝜎(s)) and 𝐺2(𝑡, 𝜎(s)) be rd-continuous 

matrices with (i) and (ii). Write 

𝐿(𝑡, s) = 𝐺1(𝑡, 𝜎(s)) − 𝐺2(𝑡, 𝜎(s)). Clearly 𝐿 satisfies the following at𝑡 = s 

𝐿(s+, s) − 𝐿(s−, s) = [𝐺1(s+, 𝜎(s)) − 𝐺2(s+, 𝜎(s))] −  [𝐺1(s−, 𝜎(s)) − 𝐺2(s−, 𝜎(s))] 

= [𝐺1(s+, 𝜎(s)) − 𝐺1(s−, 𝜎(s))] − [𝐺2(s+, 𝜎(s)) − 𝐺2(s−, 𝜎(s))] 

                                = ∅(s) ∅−1(𝜎(s)) − ∅(s) ∅−1(𝜎(s)) 

                                = 0. 

Thus,  𝐿 exhibits a removable jump condition at 𝑡 = s and   

𝑤(𝑎)𝐿(𝑎, s) + 𝑤(𝑏)𝐿(𝑏, s) = 𝑤(𝑎)[𝐺1(𝑎, 𝜎(s)) − 𝐺2(𝑎, 𝜎(s))] + 𝑤(𝑏)[𝐺1(𝑏, 𝜎(s)) − 𝐺2(𝑏, 𝜎(s))] 

= [𝑤(𝑎)𝐺1(𝑎, 𝜎(s)) + 𝑤(𝑏)𝐺1(𝑏, 𝜎(s))] − [𝑤(𝑎)𝐺2(𝑎, 𝜎(s)) + 𝑤(𝑏)𝐺2(𝑏, 𝜎(s))] 

                    = 0. 

Since 𝐿 is a solution of homogeneous BVP and 𝐿(𝑡, s) = 0. i.e., 𝐺1(𝑡, 𝜎(s)) − 𝐺2(𝑡, 𝜎(s)) =

0 implies 𝐺1(𝑡, 𝜎(s)) = 𝐺2(𝑡, 𝜎(s)). Thus G is unique. 

Theorem 3.5. Assume that  

(a) |∅(𝑡)| ≤ 𝑅, 

(b) |∅(𝑡) ∅−1(𝜎(𝑡))| ≤ 𝑆, 

(c) sup∫ |𝐾(s, 𝜏)||𝑥(𝜏)|∆𝜏 + ℥(s, 𝑥(s))
𝜎(𝜌(s))

𝑡0
≤ 𝑇|𝑥(𝑡)|, 

Where 𝑅, 𝑆, 𝑇 > 0. Then solution of (1.1) is uniformly stable. 

Proof:  For any 휀 > 0, let 𝛿(휀) <
𝜀

𝑅𝑒𝑆𝑇(𝑡,𝑎)
 and |𝐶| ≤ 𝛿𝜀.  Suppose ∃, 𝑡1 ≥ 𝑎 such that |𝑥(𝑡1)| = 휀 

and |𝑥(𝑡)| < 휀 on [𝑎, 𝑡1). 

From (3.4),  

𝑥(𝑡) = ∅(𝑡)𝐶 + ∅(𝑡) ∫  ∅−1(𝜎(s))[∫ 𝐾(s, 𝜏)𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

𝑡

𝑎
+ ℥(s, 𝑥(s)]∆s   on [𝑎, 𝑡1] 



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|𝑥(𝑡)| ≤ |∅(𝑡)||𝐶| + |∅(𝑡)| ∫ | ∅−1(𝜎(s))|[∫ |𝐾(s, 𝜏)||𝑥(𝜏)|∆𝜏
𝜎(𝜌(s))

𝑡0

𝑡

𝑎

+ |℥(s, 𝑥(s)|]∆s 

≤ 𝑅𝛿𝜀 + 𝑆𝑇 ∫ |𝑥(𝑡)|∆𝑡
𝑡

𝑎

 

By Lemma 2.6,   

|𝑥(𝑡)| ≤ 𝑅𝛿𝜀𝑒𝑆𝑇(𝑡, 𝑎) 

     ≤ 𝑅𝑒𝑆𝑇(𝑡, 𝑎)
𝜀

𝑅𝑒𝑆𝑇(𝑡,𝑎)
 

= 휀 

Therefore |𝑥(𝑡1)| < 휀, which contradicts. Thus, the solution of (1.1) is uniformly stable. 

Theorem 3.6. Assume  

(a) |∅(𝑡)||𝐶| ≤ 𝑅𝑒𝑃(𝑡0, 𝑡) 

(b) |∅(𝑡) ∅−1(𝜎(𝑡))| ≤ 𝑆𝑒𝑃(0, 𝑡), 

(c) sup∫ |𝐾(s, 𝜏)||𝑥(𝜏)|∆𝜏 + ℥(s, 𝑥(s))
𝜎(𝜌(s))

𝑡0
≤ 𝐿𝑒𝑃(s, 0)𝑥(s), 

where S, T, L > 0 and (P ⊖ LT) > 0. Then each solution U(t) of (3.1) tends to 0, as t → ∞. 

Proof: From (3.4), we have 

𝑥(𝑡) = ∅(𝑡)𝐶 + ∅(𝑡) ∫  ∅−1(𝜎(s))[∫ 𝐾(s, 𝜏)𝑥(𝜏)∆𝜏
𝜎(𝜌(s))

𝑡0

𝑡

𝑎
+ ℥(s, 𝑥(s)]∆s  on  [𝑎, 𝑡1] 

|𝑥(𝑡)| ≤ |∅(𝑡)||𝐶| + |∅(𝑡)| ∫ | ∅−1(𝜎(s))|[∫ |𝐾(s, 𝜏)||𝑥(𝜏)|∆𝜏
𝜎(𝜌(s))

𝑡0

𝑡

𝑎

+ |℥(s, 𝑥(s)|]∆s 

= ReP(t0, t) + SLeP(0, t) ∫ eP(s, 0)|x(s)| △ s
t

t0

 

= ReP(t0, 0)eP(0, t) + SLeP(0, t) ∫ eP(s, 0)|x(s)| △ s
t

t0

 

|U(t)|eP(t, 0) ≤ ReP(t0, 0) + SL ∫ eP(s, 0)|x(s)| △ s
t

t0

 

By Lemma 2.6,   

|x(t)|eP(t, 0) ≤ 𝑅eP(t0, 0)eSL(t, t0) 

|x(t)| ≤ ReP(t0, 0)eP(0, t)eSL(t, t0) 

 = 𝑅eP(t0, 0)eP(0, t)e⊖SL(t0, t) 

= 𝑅eP(t0, 0)eP(0, t)e⊖SL(t0, 0)e⊖SL(0, t) 

= 𝑅eP⊖SL(t0, 0)eP⊖SL(0, t) 

 



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By Lemma 2.7,  

eP⊖SL(0, t) ≤
1

1+(P⊖SL)t
 , so we obtain 

|x(t)| ≤ 𝑅eP⊖SL(t0, 0)
1

1 + (P ⊖ SL)t
 

Since (P ⊖ SL) > 0. Hence, we obtain desired result. 

4. Example 

Example 4.1. Consider non-homogeneous boundary value problem on time scales 

𝑥∆(𝑡) = [
0 1
0 0

] 𝑥(𝑡) + ∫ [
1 0
0 1

] 𝑥(s)
𝜎(𝜌(𝑡))

0
∆s + [

𝑡
𝑡
] ,

0 = [
1 −1
0 0

] 𝑥(0) + [
0 0
1 1

] 𝑥(10),
}            (4.1) 

for  0 ≤ s ≤ 𝑡 < ∞. 
 

Case(a): If 𝕋 = ℝ+ then (4.1) is  

𝑥′(𝑡) = [
0 1
0 0

] 𝑥(𝑡) + ∫ [
1 0
0 1

] 𝑥(s)
𝑡

0
𝑑s + [

𝑡
𝑡
] ,

0 = [
1 −1
0 0

] 𝑥(0) + [
0 0
1 1

] 𝑥(10).
}                                              (4.2) 

The fundamental matrix of  𝑥′(𝑡) = [
0 1
0 0

] 𝑥(𝑡) is ∅(𝑡) = [
1 𝑡
0 1

]. 

Hence the solution of (4.2) is  

𝑥(𝑡) =
1

12
∫ [

11 − 𝑡 −11 − s(11 − 𝑡) + 𝑡
−1 s + 1

] [
𝑒s − 1
𝑒s − 1

]
𝑡

0

 𝑑s

−
1

12
∫ [

1 + 𝑡 1 + 𝑡 + 10(1 + 𝑡) − s(1 + 𝑡)
1 11 − s

] [
𝑒s − 1
𝑒s − 1

]
10

𝑡

 𝑑s 

=
1

12
[
59 + 24𝑒𝑡 + 59𝑡 − 6𝑡2 − 3𝑒10(1 + 𝑡)

71 − 3𝑒10 + 12𝑒𝑡 − 12𝑡
]. 

 

 

Figure 1. Continuous solution graph 

Case(b): If 𝕋 = ℕ then (4.1) is  



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∆𝑥(𝜉) = [
0 1
0 0

] 𝑥(𝜉) + ∑ [
1 0
0 1

] 𝑥(s) +
𝜉−1
s=0 [

𝜉
𝜉

] ,

0 = [
1 −1
0 0

] 𝑥(0) + [
0 0
1 1

] 𝑥(10),
}                                 (4.3) 

The fundamental matrix of  ∆𝑥(𝜉) = [
0 1
0 0

] 𝑥(𝜉) is∅(𝜉) = [
1 𝜉
0 1

]. 

Using Laplace transformation on discrete case then the solution of (4.3) is 

𝑥(𝜉) =
1

12
∑ [

11 − 𝜉 −11 + 𝜉 + (11 − 𝜉)(−1 − s)
−1 s + 2

] [
2s − 1
2s − 1

]

𝜉−1

s=0

−
1

12
∑ [

1 + 𝜉 1 + 𝜉 + 10(1 + 𝜉) + (1 + 𝜉)(−1 − s)
1 10 − s

] [
2s − 1
2s − 1

]

9

s=𝜉

 

           =
1

12
[
6(−503 + 22+𝜉 − 502𝜉 − 𝜉2)

6(−501 + 21+𝜉 − 2𝜉)
]  

 

Figure 2. Discrete solution graph (take n = 𝜉) 

Using Z- transformation on discrete case then the solution of (4.3) is  

𝑥(𝜉) =
1

12
∑ [

11 − 𝜉 −11 + 𝜉 + (11 − 𝜉)(−1 − s)
−1 s + 2

] [
−1
−1

]

𝜉−1

s=0

−
1

12
∑ [

1 + 𝜉 1 + 𝜉 + 10(1 + 𝜉) + (1 + 𝜉)(−1 − s)
1 10 − s

] [
−1
−1

]

9

s=𝜉

 

       =
1

12
[
65 + 59𝜉 − 6𝜉2

65 − 12𝜉
] 

 



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Figure 3. Discrete solution graph (take n = 𝜉) 

5. Conclusion 

In this work, a solution is obtained for integro-dynamical system of Volterra type in the form of 

greens matrix. Further, properties of obtained green matrix and stability analysis for the said system 

are discussed. Most importantly, even the system is not continuous on some finite piecewise 

continuous intervals, still the solution is obtained with the help of green matrix. This work is 

authenticated with numerical examples. Graphs are also provided to explore the existence of solution 

and analyse stability of system. 

Future work: This work can be extended onto the systems of lyapunov type, impulsive and 

fractional order. This work may make better some processes concerned with robotics, image 

processing, system trajectories and various other fields. 

Acknowledgement: The authors thank their respective college managements for their continuous 

support and constant encouragement. 

References 

[1] Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. 

[2] Burton, T.A., Stability theory for Volterra equations, J. Diff Equs., 32 (1979), 101-118.  

[3] Burton, T.A., an integro differential equations, Proc. Amer. Math. Soc., 79 (1980), 393-399. 

[4] G.V.S.R. Deekshitulu G.V. Ramana, Asymptotic stability of Lyapunov type matrix Volterra integro dynamic 

system on time scales. IJET, (2017), 179-185. 

[5] Hilger, S., Ein Mabkettenkalkulmit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Universitat 

Wurzburg, 1988.  

[6] Lakshmikantham, V., Sivasundaram, S., Kaymakcalan, B., Dynamic Systems on Measure Chains, Kluwer, 1996. 

[7] Sabrina Streipert. (2023). Dynamic equations on time scales. Nonlinear systems. 

[8] S. O. Shah and A. Zada, Stability of non-linear Volterra integro dynamic equations on time scales, di Matematica, 

(2019), NO.2. 57-69.