Skip to main content

Stability of the second order partial differential equations

Abstract

We say that a functional equation (ξ) is stable if any function g satisfying the functional equation (ξ) approximately is near to a true solution of (ξ).

In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms:

y x ( x , t ) = f ( x , t , y ( x , t ) ) , a y x ( x , t ) + b y t ( x , t ) = f ( x , t , y ( x , t ) ) , p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) - p x ( x , t ) y t ( x , t ) = f ( x , t , y ( x , t ) ) , p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) = f ( x , t , y ( x , t ) ) .

2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99.

1. Introduction

Let X be a normed space over a scalar field K, and let I be an open interval. Assume that, for any function f : IX (y = f (x)) satisfying the differential inequality

| | a n ( t ) y ( n ) ( t ) + a n - 1 ( t ) y ( n - 1 ) ( t ) + + a 1 ( t ) y ( t ) + a 0 ( t ) y ( t ) + h ( t ) | | ε

for all t I, where ε ≥ 0, there exists a function f0 : IX satisfying

y 0 = f 0 ( x ) , a n ( t ) y 0 ( n ) ( t ) + a n - 1 ( t ) y 0 ( n - 1 ) ( t ) + + a 1 ( t ) y 0 ( t ) + a 0 ( t ) y 0 ( t ) + h ( t ) = 0

and ||f (t) - f0 (t)|| ≤ K (ε) for any t I.

Then we say that the above differential equation has the Hyers-Ulam stability. If the above statement is also true, then we replace ε and K(ε) by φ(t) and ϕ(t), where φ, ϕ : I → [0, ∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability.

In 1998, the Hyers-Ulam stability of differential equation y' = y was first investigated by Alsina and Ger [1]. In 2002, this result has been generalized by Takahasi et al. [2] for the Banach space-valued differential equation y' = λy. In 2005, Jung [3] proved the generalized Hyers-Ulam stability of a linear differential equation of the first order. For more results on stability of differential equations, see also [47] and [8] and, for more details on the Hyers-Ulam stability and related topics, the readers refer to [917] and [1820].

In this paper, we prove the Hyers-Ulam-Rassias stability of the following partial differential equations:

  1. (1)

    The first order nonlinear partial differential equation:

    y x ( x , t ) = f ( x , t , y ( x , t ) ) ;
  2. (2)

    The first order nonlinear partial differential equation:

    a y x ( x , t ) + b y t ( x , t ) = f ( x , t , y ( x , t ) )

for all a, b ;

  1. (3)

    The second order nonlinear partial differential equation:

    p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) = f ( x , t , y ( x , t ) )
    (1.1)

under the following condition:

p x x ( x , t ) = q x ( x , t ) .
(1.2)

The differential equation (1.1) is the second order nonlinear partial differential equation, and we call it exact if the condition (1.2) holds.

  1. (4)

    The mixed type second order nonlinear partial differential equation:

    p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) - p x ( x , t ) y t ( x , t ) = f ( x , t , y ( x , t ) )

under the following condition:

p x t ( x , t ) = q t ( x , t ) .

Theorem 1.1. (Banach's Contraction Principle) Let (X, d) be a complete matric space and T : XX be a contraction, that is, there exists α [0,1) such that

d ( T x , T y ) α d ( x , y )

for all x, y X. Then, there exists a unique a X such that Ta = a. Moreover, a = lim n →∞ Tn x and

d ( a , x ) 1 1 - α d ( x , T x )

for all x X.

2. Main results

In this section, let I = [a, b] be a closed interval with a < b and C (I × I) = {f : I × I: f is continuous}. For the sake of convenience, assume that all the integrals and all the derivatives exist.

Theorem 2.1. Let c I φ : I × I → (0, ∞) be a continuous function, L : I × I → [1,∞) be an integrable function and K : I × I × be a continuous function. Assume that there exists 0 < β < 1 such that

c x L ( τ , t ) φ ( τ , t ) d τ < β φ ( x , t ) ;
(2.1)
| K ( x , t , u ( x , t ) ) - K ( x , t , v ( x , t ) ) | L ( x , t ) | u ( x , t ) - v ( x , t ) |
(2.2)

for all x, t I and u, v C (I × I). Let y : I × I be such that

| y x ( x , t ) - K ( x , t , y ( x , t ) ) | φ ( x , t )
(2.3)

for all x, t I. Then, there exists a unique continuously differentiable function y0 : I × I such that

y 0 ( x , t ) = y ( c , t ) + c x K ( τ , t , y 0 ( τ , t ) ) d τ

(consequently, y0 is a solution to y x (x, t) = K(x, t, y(x,t))) and

| y ( x , t ) - y 0 ( x , t ) | β 1 - β φ ( x , t )

for all x, t I.

Proof. Let X be the set of all continuously differentiable functions u : I × I. We define a metric d and an operator T on X as follows, respectively:

d ( u , v ) = su p x , t I | u ( x , t ) - v ( x , t ) | φ ( x , t )

and the operator

( T u ) ( x , t ) = y ( c , t ) + c x K ( τ , t , u ( τ , t ) ) d τ

for all u X. Using (2.1) and (2.2), we have

d ( T u , T v ) = su p x , t I c x K ( τ , t , u ( τ , t ) ) - K ( τ , t , v ( τ , t ) ) d τ φ ( x , t ) su p x , t I c x L ( τ , t ) | u ( τ , t ) - v ( τ , t ) | d τ φ ( x , t ) = su p x , t I c x L ( τ , t ) φ ( τ , t ) | u ( τ , t ) - v ( τ , t ) | φ ( τ , t ) d τ φ ( x , t ) su p x , t I c x L ( τ , t ) φ ( τ , t ) su p τ , t I | u ( τ , t ) - v ( τ , t ) | φ ( τ , t ) d τ φ ( x , t ) = d ( u , v ) su p x , t I c x L ( τ , t ) φ ( τ , t ) d τ φ ( τ , t ) β d ( u , v ) .

Now, by Theorem 1.1, there exists a unique y0 X such that T y 0 = y 0 , that is,

y 0 ( x , t ) = y ( c , t ) + c x K ( τ , t , y 0 ( τ , t ) ) d τ .

Moreover, by Theorem 1.1, we have

d ( y 0 , y ) 1 1 - β d ( y , T y )
(2.4)

for all y X. It follows from (2.3) that

- φ ( x , t ) y x ( x , t ) - K ( x , t , y ( x , t ) ) φ ( x , t )

for all x, t I. If we integrate each term in the above inequality from c to x, then we get

| y ( x , t ) ( y ( c , t ) c x K ( τ , t , y ( τ , t ) ) d τ | c x φ ( τ , t ) d τ c x L ( τ , t ) φ ( τ , t ) d τ β φ ( x , t ) .

Now, we have

| y ( x , t ) - ( T y ) ( x , t ) | φ ( x , t ) β su p x , t I | y ( x , t ) - ( T y ) ( x , t ) | φ ( x , t ) β .

Thus, we get

d ( y , T y ) β .
(2.5)

Therefore, by (2.4) and (2.5), we see that

| y ( x , t ) - y 0 ( x , t ) | β 1 - β φ ( x , t )

for all x, t I. This completes the proof. □

Theorem 2.2. Let c I, p, q : I × I be continuous functions with p(x, t) 0 for all x, t I, φ : I × I → (0, ∞) be a continuous function, L : I × I → [1, ∞) be an integrable function, and f : I × I × be a continuous function. Assume that there exists 0 < β < 1 such that

c x L ( τ , t ) φ ( τ , t ) d τ β φ ( x , t ) ; h ( c , t ) = - [ p ( c , t ) y x ( c , t ) - p x ( c , t ) y ( c , t ) + q ( c , t ) y ( c , t ) ] ; K ( x , t , y ( x , t ) ) = - ( p ( x , t ) ) - 1 ( p x ( x , t ) - q ( x , t ) ) y ( x , t ) + h ( c , t ) - c x f ( τ , t , y ( τ , t ) d τ

and

| K ( x , t , u ( x , t ) ) - K ( x , t , v ( x , t ) ) | L ( x , t ) | u ( x , t ) - v ( x , t ) |

for all c, x, t I and h, u, v, y C (I × I). Let y : I × I be a function such that:

| p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) - f ( x , t , y ( x , t ) ) | φ ( x , t )
(2.7)

for all x, t I and (1.2) holds. Then, there exists a unique solution y0 : I × I of (1.1) such that

| y ( x , t ) - y 0 ( x , t ) | β 1 - β φ ( x , t ) .

Proof. It follows from (1.2) and (2.7) that

| p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) f ( x , t , y ( x , t ) ) | = | ( p ( x , t ) y x ( x , t ) p x ( x , t ) y ( x , t ) ) x + ( q ( x , t ) y ( x , t ) ) x + [ p x x ( x , t ) q x ( x , t ) ] y ( x , t ) f ( x , t , y ( x , t ) ) | = | ( p ( x , t ) y x x ( x , t ) p x ( x , t ) y ( x , t ) ) x + ( q ( x , t ) y ( x , t ) ) x f ( x , t , y ( x , t ) ) | φ ( x , t ) .

Thus, we have

- φ ( x , t ) p ( x , t ) y x x ( x , t ) - p x ( x , t ) y ( x , t ) x + ( q ( x , t ) y ( x , t ) ) x - f ( x , t , y ( x , t ) ) φ ( x , t ) .
(2.8)

By using (2.8), we get

| p ( x , t ) y x ( x , t ) - p x ( x , t ) y ( x , t ) + q ( x , t ) y ( x , t ) + h ( c , t ) - c x f ( τ , t , y ( τ , t ) ) d τ | = | p ( x , t ) | y x ( x , t ) + ( p ( x , t ) ) - 1 ( q ( x , t ) - p x ( x , t ) ) y ( x , t ) + h ( c , t ) - c x f ( τ , t , y ( τ , t ) ) d τ c x φ ( τ , t ) d τ ,
(2.9)

where

h ( c , t ) = - [ p ( c , t ) y x ( c , t ) - p x ( c , t ) y ( c , t ) + q ( c , t ) y ( c , t ) ] .

From (2.9), it follows that

y x ( x , t ) + ( p ( x , t ) ) - 1 ( q ( x , t ) - p x ( x , t ) ) y ( x , t ) + h ( c , t ) - c x f ( τ , t , y ( τ , t ) ) d τ | p ( x , t ) | - 1 c x φ ( τ , t ) d τ .

From p ( x , t ) = p ( x , t ) [ 1 + ( p ( x , t ) ) 2 ] 1 + ( p ( x , t ) ) 2 , without less of generality, we can assume that |p(x, t)| ≥ 1.

Now, By putting

K ( x , t , y ( x , t ) ) = - ( p ( x , t ) ) - 1 ( p x ( x , t ) - q ( x , t ) ) y ( x , t ) + h ( c , t ) - c x f ( τ , t , y ( τ , t ) d τ

in the above inequality, we get

| y x ( x , t ) - K ( x , t , y ( x , t ) ) | | p ( x , t ) | - 1 c x φ ( τ , t ) d τ c x φ ( τ , t ) d τ c x L ( τ , t ) φ ( τ , t ) d τ β φ ( x , t ) φ ( x , t ) .

Thus, the conclusions of the Theorem follows from Theorem 2.1. This completes the proof. □

If (1.1) is multiplied by a function μ(x, t) such that the resulting equation is exact, that is,

μ ( x , t ) [ p ( x , t ) y x x ( x , t ) + q ( x , t ) y x - f ( x , t , y ( x , t ) ) ] = 0
(2.10)

and

( μ ( x , t ) p ( x , t ) ) x x - ( q ( x , t ) μ ( x , t ) ) x = 0 ,
(2.11)

then we say that μ(x, t) is an integrating factor of the partial differential equation (1.1).

Corollary 2.3. Let p, q, μ : I × I be continuous functions such that p(x, t) 0 and μ(x, t) 0 for all x, t I, and (2.10) holds. Assume that c I, L : I × I → [1, ∞) is an integrable function and f : I × I × is a continuous function. Suppose that there exists 0 < β < 1 such that

c x L ( x , t ) φ ( τ , t ) d τ β φ ( x , t ) ; h ( c , t ) = - [ μ ( c , t ) p ( x , t ) y x ( c , t ) - ( μ p ) x ( c , t ) y ( c , t ) + μ ( c , t ) q ( c , t ) y ( c , t ) ] ; K ( x , t , y ( x , t ) ) = - ( μ ( x , t ) p ( x , t ) ) - 1 ( μ ( x , t ) q ( x , t ) - ( μ q ) x ( x , t ) ) y ( x , t ) + h ( c , t ) - c x μ ( τ , t ) f ( τ , t , y ( τ , t ) ) d τ

and

| K ( x , t , u ( x , t ) ) - K ( x , t , v ( x , t ) ) | L ( x , t ) | u ( x , t ) - V ( x , t ) | .

for all c, x, t I and h, u, v C (I × I). Let y : I × I be a function such that

| μ ( x , t ) | | p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) - f ( x , t , y ( x , t ) ) | φ ( x , t )

for all x, t I and the condition (2.11) holds. Then, There exists a unique solution y0 : I × I of (2.10) such that

| y ( x , t ) - y 0 ( x , t ) | β 1 - β φ ( x , t ) .

Proof. It follows from Theorem 2.2 that

y 0 ( x , t ) = y ( c , t ) + c x K ( τ , t , y ( τ , t ) ) d τ

with

K ( x , t , y ( x , t ) ) = - ( μ ( x , t ) p ( x , t ) ) - 1 ( μ ( x , t ) q ( x , t ) - ( μ q ) x ( x , t ) ) y ( x , t ) + h ( c , t ) - c x μ ( τ , t ) f ( τ , t , y ( τ , t ) ) d τ

and

h ( c , t ) = - [ μ ( c , t ) p ( x , t ) y x ( c , t ) - ( μ p ) x ( c , t ) y ( c , t ) + μ ( c , t ) q ( c , t ) y ( c , t ) ]

has the required properties. This completes the proof. □

Remark 2.4. In 2009, Jung [7] proved the Hyers-Ulam stability of linear partial differential equation of the first order of the following form:

a y x ( x , t ) + b y t ( x , t ) + g ( x ) y ( x , t ) + h ( x ) = 0

for all a ≥ 0 and b > 0.

Now, we consider the generalization of this equation as follows:

a y x ( x , t ) + b y t ( x , t ) = f ( x , t , y ( x , t ) )
(2.12)

for all a, b with a ≠ 0 and b ≠ 0. Let ζ and η be defined by

ζ = x - a b t , η = 1 b t .
(2.13)

If we define y ̃ ( ζ , η ) =y ( ζ + a η , b η ) =y ( x , t ) , then, by (2.13), we have

y x ( x , t ) = y ̃ ζ ( ζ , η ) ζ x + y ̃ ( ζ , η ) η x , y t ( x , t ) = y ̃ ζ ( ζ , η ) ζ t + y ̃ η ( ζ , η ) η t = - a b y ̃ ζ ( ζ , η ) + 1 b y ̃ η ( ζ , η ) .

Thus, we see that a y x ( x , t ) +b y t ( x , t ) = y ̃ η ( ζ , η ) , and so we can rewrite the equation (2.12) as follows:

y ̃ η ( ζ , η ) = f ̃ ( ζ , η , y ̃ η ( ζ , η ) ) .
(2.14)

Now, we can use Theorem 2.1 for the generalized Hyers-Ulam stability of (2.14).

We consider the mixed type second order nonlinear partial differential equation:

p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) - p x ( x , t ) y t ( x , t ) = f ( x , t , y ( x , t ) ) .
(2.15)

Now, we prove the Hyers-Ulam-Rassias stability of (2.15) under the condition:

p x t ( x , t ) = q t ( x , t )
(2.16)

Theorem 2.5. Let c Ip, q : I × I be continuous functions with p(x, t) 0 for all x, t I, φ : I × I → (0, ∞) be a continuous function, L : I × I → [1, ∞) be an integrable function, and f : I × I × be a continuous function. Assume that there exists 0 < β < 1 such that

c x L ( τ , t ) φ ( τ , t ) d τ β φ ( x , t ) ; h ( x , c ) = [ p ( x , c ) y x ( x , c ) p x ( x , c ) y ( x , c ) + q ( x , c ) y ( x , c ) ] ; K ( x , t , y ( x , t ) ) = ( p ( x , t ) ) 1 [ ( p x ( x , t ) q ( x , t ) ) y ( x , t ) + h ( x , c ) c t f ( x , τ , y ( x , τ ) d τ ]

and

| K ( x , t , u ( x , t ) ) - K ( x , t , v ( x , t ) ) | L ( x , t ) | u ( x , t ) - v ( x , t ) |

for all c, x, t I and h, y, u, v C (I × I). Let y : I × I be a function such that

| p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) - p x ( x , t ) y t ( x , t ) - f ( x , t , y ( x , t ) ) | φ ( x , t )
(2.17)

for all x, t I and the condition (2.16) holds. Then, there exists a unique solution y 0 : I × I of (2.15) such that

| y ( x , t ) - y 0 ( x , t ) | β 1 - β φ ( x , t ) .

Proof. By (2.17) and (2.16), we see that

| p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) p x ( x , t ) y t ( x , t ) f ( x , t , y ( x , t ) ) | = | ( p ( x , t ) y x ( x , t ) p x ( x , t ) y ( x , t ) + q ( x , t ) y ( x , t ) ) t + [ p x t ( x , t ) q t ( x , t ) ] y ( x , t ) f ( x , t , y ( x , t ) ) | = | ( p ( x , t ) y x ( x , t ) p x ( x , t ) y ( x , t ) + q ( x , t ) y ( x , t ) ) t f ( x , t , y ( x , t ) ) |

Thus, we have

- φ ( x , t ) ( p ( x , t ) y x ( x , t ) - p x ( x , t ) y ( x , t ) + q ( x , t ) y ( x , t ) ) t - f ( x , t , y ( x , t ) ) φ ( x , t ) .
(2.18)

It follows from (2.18) that

p ( x , t ) y x ( x , t ) - p x ( x , t ) y ( x , t ) + q ( x , t ) y ( x , t ) + h ( c , t ) - c t f ( x , τ , y ( x , τ ) ) d τ = | p ( x , t ) | - 1 y x ( x , t ) + ( p ( x , t ) ) - 1 ( q ( x , t ) - p x ( x , t ) ) y ( x , t ) + h ( c , t ) - c t f ( x , τ , y ( x , τ ) ) d τ c t φ ( x , τ ) d τ ,
(2.19)

where

h ( x , c ) = - [ p ( x , c ) y x ( x , c ) - p x ( x , c ) y ( x , c ) + q ( x , c ) y ( x , c ) ] .

From (2.19), we obtain

y x ( x , t ) + ( p ( x , t ) ) - 1 ( q ( x , t ) - p x ( x , t ) ) y ( x , t ) + h ( c , t ) - c t f ( x , τ , y ( x , τ ) ) d τ | p ( x , t ) | - 1 c t φ ( x , τ ) d τ .

The rest of the proof is similar to that of Theorem 2.2. This completes the proof. □

Remark 2.6. We can define the integrating factor for the equation (2.15) and prove a corollary similar to Corollary 2.3 for Theorem 2.6.

References

  1. 1.

    Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J Inequal Appl 1998, 2: 373–380. 10.1155/S102558349800023X

    MathSciNet  Google Scholar 

  2. 2.

    Takahasi SE, Miura T, Miyajima S: On the Hyers-Ulam stability of Banach space-valued differential equation y ' = λy , Bull. Korean Math Soc 2002, 39: 309–315. 10.4134/BKMS.2002.39.2.309

    MathSciNet  Article  Google Scholar 

  3. 3.

    Jung SM: Hyers-Ulam stability of linear differential equations of firsty order III. J Math Anal Appl 2005, 311: 139–146. 10.1016/j.jmaa.2005.02.025

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jung SM: Hyers-Ulam stability of linear differential equations of firsty order II. Appl Math Lett 2006, 19: 854–858. 10.1016/j.aml.2005.11.004

    MathSciNet  Article  Google Scholar 

  5. 5.

    Jung SM: A fixed point approach to the stability of differential equations y ' = F ( x , y ). Bull Malays Math Sci Soc 2010, 33: 47–56.

    MathSciNet  Google Scholar 

  6. 6.

    Jung SM, Min S: On approximate Eular differential equations. Abstr Appl Anal 2009., 2009: (Article ID 537963), 8 pages

    Google Scholar 

  7. 7.

    Jung SM: Hyers-Ulam stability of linear partial differential equations of first order. Appl Math Lett 2009, 22: 70–74. 10.1016/j.aml.2008.02.006

    MathSciNet  Article  Google Scholar 

  8. 8.

    Li Y, Shen Y: Hyers-Ulam stability of linear differential equations of scecond order. Appl Math Lett 2010, 23: 306–309. 10.1016/j.aml.2009.09.020

    MathSciNet  Article  Google Scholar 

  9. 9.

    Baker J: The stability of the cosin equation. Proc Amer Math Soc 1980, 80: 411–416. 10.1090/S0002-9939-1980-0580995-3

    MathSciNet  Article  Google Scholar 

  10. 10.

    Cholewa PW: Remarks on the stability of functional equations. Aequat Math 1984, 27: 76–86. 10.1007/BF02192660

    MathSciNet  Article  Google Scholar 

  11. 11.

    Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59–64. 10.1007/BF02941618

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gordji ME, Karimi T, Gharetapeh SK: Approximately n -Jordan homomorphisms on Banach algebras. J Ineq Appl 2009., 2009: (Article ID 870843), 8 pages

    Google Scholar 

  13. 13.

    Eshaghi Hordji M, Ghaemi MB, Kaboli Gharetapeh S, Shams S, Ebadian A: On the stability of J *-derivations. J Geom Phy 2010, 60: 454–459. 10.1016/j.geomphys.2009.11.004

    Article  Google Scholar 

  14. 14.

    Găvruta P, Găvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. Intern J Nonlinear Anal Appl 2010, 1: 11–18.

    Google Scholar 

  15. 15.

    Gordji ME, Zolfaghari S, Rassias JM, Savadkouhi MB: Solution and stability of a mMixed type cubic and qqartic functional equation in quasi-Banach spaces. Abstr Appl Anal 2009., 2009: (Article ID 417473), 14 pages

    Google Scholar 

  16. 16.

    Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhausar, Basel; 1998.

    Google Scholar 

  18. 18.

    Khodaei H, Rassias TM: Approximately generalized additive functions in several variable. Intern J Nonlinear Anal Appl 2019, 1: 22–41.

    Google Scholar 

  19. 19.

    Rassias TM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ulam SM: Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York; 1940.

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

Author information

Affiliations

Authors

Corresponding author

Correspondence to YJ Cho.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

AB carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. JY carried out the immunoassays. MT participated in the sequence alignment. ES participated in the design of the study and performed the statistical analysis. FG conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gordji, M.E., Cho, Y., Ghaemi, M. et al. Stability of the second order partial differential equations. J Inequal Appl 2011, 81 (2011). https://doi.org/10.1186/1029-242X-2011-81

Download citation

Keywords

  • generalized Hyers-Ulam stability
  • linear differential equation
  • Banach's contraction principle