On the perturbation of Volterra integro-differential equations

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Applied Mathematics Letters

journal homepage:www.elsevier.com/locate/aml

On the perturbation of Volterra integro-differential equations

Soon-Mo Jung

^a

, Sebaheddin Şevgin

^b,^∗

, Hamdullah Şevli

^c

aMathematics Section, College of Science and Technology, Hongik University, 339-701 Jochiwon, Republic of Korea

bYuzuncu Yil University, Faculty of Art and Science, Department of Mathematics, 65080 Van, Turkey

cDepartment of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University, 34672 Uskudar, Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 23 August 2012

Received in revised form 23 October 2012 Accepted 23 October 2012

Keywords:

Volterra integro-differential equation Volterra integral equation of the second

kind Perturbation Hyers–Ulam stability

Generalized Hyers–Ulam stability

a b s t r a c t

In this work, we will prove that every solution of a perturbed Volterra integro-differential equation can be approximated by a solution of the Volterra integro-differential equation.

1. Introduction

Let I be either

(−∞,

^b

] ,

R

, [

^a

, ∞)

, or a closed interval

[

a

,

^b

]

with

−∞ <

^a

<

^b

< ∞

, let c be a fixed point of I, and let

ϕ :

^I

→ [

0

, ∞)

be a continuous function.

Recently, S.-M. Jung [1] proved that if a continuous function u

:

I

→

C satisfies the perturbed Volterra integral equation of the second kind



u

(

^t

) − 

t c

F

(τ,

^u

(τ))

^d

τ



≤ ϕ(

^t

)

for all t

∈

I, then under some additional conditions, there exist a unique continuous function u₀

:

I

→

C and a constant C

>

0 such that

u₀

(

^t

) =



t c

F

(τ,

^u0

(τ))

^d

τ

^and

|

u

(

^t

) −

^u0

(

^t

)| ≤

^C

ϕ(

^t

)

for all t

∈

I.

In this work, we will investigate the solutions u

:

I

→

R of the integro-differential inequality



u^′

(

^t

) +

^p

(

^t

)

^u

(

^t

) +

^q

(

^t

) + 

t c

K

(

^t

, τ)

^u

(τ)

^d

τ



≤ ϕ(

^t

)

^(1.1)

∗Corresponding author. Tel.: +90 5055938215; fax: +90 4322251806.

E-mail addresses:smjung@hongik.ac.kr(S.-M. Jung),ssevgin@yahoo.com,ssevgin@yyu.edu.tr(S. Şevgin),hsevli@yahoo.com(H. Şevli).

doi:10.1016/j.aml.2012.10.018

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which is the perturbed form of the Volterra integro-differential equation

u^′

(

^t

) +

^g

(

^t

)

^u

(

^t

) +

^h

(

^t

) +



t c

K

(

^t

, τ)

^u

(τ)

^d

τ =

⁰

.

^(1.2)

More precisely, we will prove that if p

:

I

→

_R

,

^q

:

I

→

_R

,

^K

:

I

×

I

→

_{R and}

ϕ :

^I

→ [

0

, ∞)

are sufficiently smooth functions and if a continuously differentiable function u

:

I

→

R satisfies the perturbed Volterra integro-differential equation(1.1)for all t

∈

I, then there exists a unique solution u₀

:

I

→

R of the Volterra integro-differential equation(1.2) such that

|

u

(

^t

) −

^u0

(

^t

)| ≤

^exp



−



t c

p

(τ)

^d

τ  

b t

ϕ(ξ)

^exp



ξ c

p

(τ)

^d

τ



d

ξ

for all t

∈

I (provided that I

= (−∞,

^b

]

or

[

a

,

^b

]

).

2. The main results

In the following theorem, we will investigate the perturbation problem of the Volterra integro-differential equation(1.2) by using ideas from [1–3]. More precisely, we provide a characterization of the integro-differential inequality(1.1)for the class of real-valued functions.

Theorem 2.1. Let I

= [

a

,

^b

]

for given real numbers a

,

^{b with a}

<

b and let c

∈

I. Assume that p

:

I

→

_R

,

^q

:

I

→

_{R, and} K

:

I

×

I

→

R are continuous functions such that p

(

^t

)

^{and exp}



t

cp

(τ)

^d

τ

^q

(

^t

)

are integrable on

(

^c

,

^d

)

^{for each d}

∈

I with d

>

c. Moreover, suppose

ϕ :

^I

→ [

0

, ∞)

is a function such that

ϕ(

^t

)

^exp



t

cp

(τ)

^d

τ

is integrable on I. If a continuously differentiable function u

:

I

→

R satisfies the integro-differential inequality(1.1)for all t

∈

I, then there exists a unique

α ∈

R such that



u

(

^t

) − α

^exp



−



t c

p

(τ)

^d

τ

 +

_exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ +

exp



−



t c

p

(τ)

^d

τ  

t c



_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



_ξ

c

p

(τ)

^d

τ



d

ξ



≤

exp



−



t c

p

(τ)

^d

τ  

b t

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

^(2.1)

for every t

∈

I.

Proof. We will now introduce an auxiliary function z

:

I

→

R defined by z

(

^t

) :=

^exp



t

c

p

(τ)

^d

τ



u

(

^t

) + 

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ +



t c



ξ c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



ξ c

p

(τ)

^d

τ



d

ξ

for every t

∈

I. By the fundamental theorem of calculus and in view of(1.1), we get

|

z

(

^t

) −

^z

(

^s

)| =



exp



t c

p

(τ)

^d

τ



u

(

^t

) −

^exp



s c

p

(τ)

^d

τ



u

(

^s

) +



t s



q

(ξ) +



_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



ξ c

p

(τ)

^d

τ



d

ξ



=





t s

d d

ξ



exp



_ξ

c

p

(τ)

^d

τ



u

(ξ)



d

ξ +



t s



q

(ξ) + 

_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



_ξ

c

p

(τ)

^d

τ



d

ξ



=





t s

exp



_ξ

c

p

(τ)

^d

τ

 

u^′

(ξ) +

^p

(ξ)

^u

(ξ) +

^q

(ξ) + 

_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



d

ξ



≤





t s

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ



(2.2)

for any s

,

^t

∈

I.

(3)

Since

ϕ(

^t

)

^exp



t

c p

(τ)

^d

τ

is assumed to be integrable on I, the inequality(2.2)implies that for any given

ε >

^{0 there} exists a t₀

∈

I such that s

,

^t

≥

t₀implies

|

z

(

^t

) −

^z

(

^s

)| ≤ ε

^{. Indeed,}

{

z

(

^s

)}

s∈Iis a Cauchy net and hence there exists a real number

α

such that z

(

^s

) → α

^{as s}

→

b, since R is complete.

Finally, it follows from(2.2)and the above argument that for any t

∈

I,



u

(

^t

) − α

^exp



−



t c

p

(τ)

^d

τ

 +

exp



−



t c

p

(τ)

^d

τ

 

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ +

exp



−



t c

p

(τ)

^d

τ  

t c



_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



_ξ

c

p

(τ)

^d

τ



d

ξ



=



exp



−



t c

p

(τ)

^d

τ



(

^z

(

^t

) − α)



≤

exp



−



t c

p

(τ)

^d

τ



|

z

(

^t

) −

^z

(

^s

)| +

^exp



−



t c

p

(τ)

^d

τ



|

z

(

^s

) − α|

≤

exp



−



t c

p

(τ)

^d

τ

 





t s

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ



 +

exp



−



t c

p

(τ)

^d

τ



|

z

(

^s

) − α|

→

exp



−



t c

p

(τ)

^d

τ

 





t b

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ



as s

→

b

,

since we assumed that z

(

^s

) → α

^{as s}

→

b, which proves the validity of(2.1).

It now remains to prove the uniqueness of

α

. Assume that

β ∈

R also satisfies the inequality(2.1)in place of

α

^{. Then, we} have



exp



−



t c

p

(τ)

^d

τ

 (β − α)



≤

2 exp



−



t c

p

(τ)

^d

τ  

b t

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

for any t

∈

I. It follows from the integrability hypotheses that

| β − α| ≤

²



b t

ϕ(ξ)

^exp



ξ c

p

(τ)

^d

τ



d

ξ →

⁰ ^{as t}

→

b

.

This implies the uniqueness of

α

^.

Remark 2.1. It is not difficult to show that for any real number

α

^, u

(

^t

) = α

^exp



−



t c

p

(τ)

^d

τ



−

exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

−

exp



−



t c

p

(τ)

^d

τ  

t c



_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



_ξ

c

p

(τ)

^d

τ



d

ξ

is the general solution of the Volterra integro-differential equation(1.2), where

α

is an arbitrary real number.

In the preceding theorem, we have investigated the perturbed Volterra integro-differential equation(1.2)defined on a bounded and closed interval. We will now prove the theorem for the case of unbounded intervals. More precisely, Theorem 2.1is also true if I is replaced by an unbounded interval such as

[

a

, ∞)

. We can prove the following theorem by replacing b with

∞

in the proof ofTheorem 2.1. Hence, we omit the proof.

Theorem 2.2. For any given real number a, let I

= [

a

, ∞)

and c be a fixed point of I. Assume that p

:

I

→

_R

,

^q

:

I

→

_R, and K

:

I

×

I

→

(

^t

)

^{and exp}



t

c p

(τ)

^d

τ

^q

(

^t

)

are integrable on

(

^c

,

^d

)

^{for each d}

∈

I with d

>

ϕ :

^I

→ [

0

, ∞)

ϕ(

^t

)

^exp



t

c p

(τ)

^d

τ

is integrable on I. If a continuously differentiable function u

:

I

→

R satisfies the integro-differential inequality(1.1)for all t

∈

I, then there exists a unique

α ∈

R such that



u

(

^t

) − α

^exp



−



t c

p

(τ)

^d

τ

 +

_exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ +

exp



−



t c

p

(τ)

^d

τ  

t c



_ξ

c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



_ξ

c

p

(τ)

^d

τ



d

ξ



≤

exp



−



t c

p

(τ)

^d

τ  

∞ t

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

for any t

∈

I.

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Applying an idea from [3] and usingTheorem 2.2, we can proveTheorem 2.1for the case of I

=

_R.

Corollary 2.3. Let c be a fixed real number. Assume that p

:

_R

→

_R

,

^q

:

_R

→

_{R, and K}

:

_R

×

_R

→

R are continuous functions such that p

(

^t

)

^{and exp}



t

c p

(τ)

^d

τ

^q

(

^t

)

are integrable on

(

^c

,

^d

)

^{for each d}

∈

_{R with d}

>

ϕ :

R

→ [

0

, ∞)

ϕ(

^t

)

^exp



t

c p

(τ)

^d

τ

is integrable on R. If a continuously differentiable function u

:

_R

→

R satisfies the integro-differential inequality(1.1)for all t

∈

R, then there exists a unique

α ∈

R such that



u

(

^t

) − α

^exp



−



t c

p

(τ)

^d

τ

 +

exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



ξ c

p

(τ)

^d

τ



d

ξ +

exp



−



t c

p

(τ)

^d

τ  

t c



ξ c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



ξ c

p

(τ)

^d

τ



d

ξ



≤

exp



−



t c

p

(τ)

^d

τ  

∞ t

ϕ(ξ)

^exp



ξ c

p

(τ)

^d

τ



d

ξ

^(2.3)

for all t

∈

_R.

Proof. For any n

∈

_N₀, we define I_n

= [

c

−

n

, ∞)

. According toTheorem 2.2, there exists a unique

α

n

∈

R such that



u

(

^t

) − α

nexp



−



t c

p

(τ)

^d

τ

 +

exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



ξ c

p

(τ)

^d

τ



d

ξ +

exp



−



t c

p

(τ)

^d

τ  

t c



ξ c

K

(ξ, τ)

^u

(τ)

^d

τ



exp



ξ c

p

(τ)

^d

τ



d

ξ



≤

exp



−



t c

p

(τ)

^d

τ

 

∞

t

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

^(2.4)

for any t

∈

I_n. The uniqueness of

α

nimplies that if t

∈

I_n, then

α

n

= α

n+1

= α

n+2

= · · · .

In general, we conclude that

α

0

= α

1

= α

2

= · · · = α

n

= · · · =: α.

^(2.5)

Consequently, the validity of(2.3)follows from(2.4)and(2.5)because for each t

∈

R, there exists an n

∈

_N₀with t

∈

I_n. The uniqueness of

α

can be proved by applying the same argument as was given in the proof ofTheorem 2.1.

If we set K

(

^t

,

^s

) ≡

^{0 in}Corollary 2.3, then we obtain the generalized Hyers–Ulam stability of the first-order linear differential equations, u^′

(

^t

) +

^p

(

^t

)

^u

(

^t

) +

^q

(

^t

) =

0 (cf. [4,5] and refer to [6–11] for the definition of Hyers–Ulam stability).

In the following corollary, we will deal with the case of I

=

_{R only.}

Corollary 2.4. Let c be a fixed real number. Assume that p

:

_R

→

_{R and q}

:

_R

→

(

^t

)

and exp



t

cp

(τ)

^d

τ

^q

(

^t

)

are integrable on

(

^c

,

^d

)

^{for each d}

∈

_{R with d}

>

ϕ :

R

→ [

0

, ∞)

is a function such that

ϕ(

^t

)

^exp



t

c p

(τ)

^d

τ

is integrable on R. If a continuously differentiable function u

:

_R

→

R satisfies the differential inequality

|

u^′

(

^t

) +

^p

(

^t

)

^u

(

^t

) +

^q

(

^t

)| ≤ ϕ(

^t

)

for all t

∈

R, then there exists a unique

α ∈

R such that



u

(

^t

) − α

^exp



−



t c

p

(τ)

^d

τ

 +

exp



−



t c

p

(τ)

^d

τ  

t c

q

(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ



≤

exp



−



t c

p

(τ)

^d

τ  

∞ t

ϕ(ξ)

^exp



_ξ

c

p

(τ)

^d

τ



d

ξ

for all t

∈

_R.

Acknowledgments

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

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References

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