Stability of the second order partial differential equations

Gordji, Cho Yj, Ghaemi Mb, Alizadeh, Alizadeh B

Description

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 , .

Sujets

• Linear differential equation
• Banach fixed point theorem

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 16
• Langue: English

Extrait

Gordjiet al.Journal of Inequalities and Applications2011,2011:81 http://www.journalofinequalitiesandapplications.com/content/2011/1/81

R E S E A R C HOpen Access Stability of the second order partial differential equations 1,2,3 4*5 6,7 M Eshaghi Gordji, YJ Cho, MB Ghaemiand B Alizadeh

* Correspondence: [e-mail] 4 Department of Mathematics Education and the Rins, Gyeongsang National University, Chinju 660701, Korea Full list of author information is available at the end of the article

Abstract We say that a functional equation (ξ) is stable if any functiongsatisfying 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)),   ay x (x,t) +by t (x,t) =f(x,t,y(x,t)), p(x,t)y xt (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 xx (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 LetXbe a normed space over a scalar field K , and letIbe an open interval. Assume that, for any functionf:I®X(y=f(x)) satisfying the differential inequality

(n) (n−1) ||a n (t)y(t) +a n−1 (t)y(t) +∙ ∙ ∙+a 1 (t)y(t) +a 0 (t)y(t) +h(t)|| ≤ε

for alltÎI, whereε≥0, there exists a functionf 0 :I®Xsatisfying  y 0 =f 0 (x), (n) (n−1)  a n (t)y(t) +a n−1 (t)y(t) +a 0 (t)y 0 (t) +h(t) = 0 0 0 ∙ ∙ ∙+a 1 (t)y 0 (t) +

and ||f(t) f 0 (t)||≤K(ε) for anytÎI. Then we say that the above differential equation has theHyersUlam stability. If the above statement is also true, then we replaceεandK(ε) by(t) andj(t), where,j:I ®[0,∞) are functions not depending onfandf 0 explicitly, then we say that the corre sponding differential equation has theHyersUlamRassias stabilityor thegeneralized HyersUlam stability. In 1998, the HyersUlam stability of differential equationy’=ywas 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 equationy’=ly. In 2005, Jung [3] proved the generalized HyersUlam stability of a linear differential equation of the first order. For

© 2011 Gordji et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.