Open Access

Stability of the second order partial differential equations

Journal of Inequalities and Applications20112011:81

https://doi.org/10.1186/1029-242X-2011-81

Received: 8 December 2010

Accepted: 7 October 2011

Published: 7 October 2011

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.

Keywords

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

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 →∞ T n 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Semnan University
(2)
Research Group of Nonlinear Analysis and Applications (RGNAA)
(3)
Centre of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University
(4)
Department of Mathematics Education and the Rins, Gyeongsang National University
(5)
Department of Mathematics, Iran University of Science and Technology
(6)
Graduate Center, Payame Noor University, Shahnaz Alley
(7)
Tabriz College of Technology

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